Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 98.0%

Time: 19.6s

Alternatives: 18

Speedup: 23.4×

• 39.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 35.8% 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: 98.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r/ N/A

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

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

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

3. Simplified 28.8%

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

5. Taylor expanded in k around inf

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

   1. associate-/l* N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 91.2%

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

7. Simplified 91.2%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-/l/ N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. clear-num N/A

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

   12. un-div-inv N/A

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

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

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

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

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

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

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

   16. sin-lowering-sin.f64 91.8%

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

9. Applied egg-rr 91.8%

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

    1. associate-*l/ N/A

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

    2. div-inv N/A

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

    3. clear-num N/A

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

    4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

    10. *-commutative N/A

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

    11. clear-num N/A

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

    12. un-div-inv N/A

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

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

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

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

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

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

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

    16. tan-lowering-tan.f64 94.5%

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

11. Applied egg-rr 94.5%

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

    1. associate-/r* N/A

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

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

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

    3. associate-/l/ N/A

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

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

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

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

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

    6. sin-lowering-sin.f64 98.9%

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

13. Applied egg-rr 98.9%

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

Alternative 2: 93.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := \frac{2}{k \cdot \tan k}\\ \mathbf{if}\;t \leq 3.5 \cdot 10^{-147}:\\ \;\;\;\;\ell \cdot \frac{t\_2}{\frac{k \cdot t}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \frac{\frac{\ell}{k}}{\frac{t}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ l (sin k))) (t_2 (/ 2.0 (* k (tan k)))))
   (if (<= t 3.5e-147)
     (* l (/ t_2 (/ (* k t) t_1)))
     (* t_2 (/ (/ l k) (/ t t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = l / sin(k);
	double t_2 = 2.0 / (k * tan(k));
	double tmp;
	if (t <= 3.5e-147) {
		tmp = l * (t_2 / ((k * t) / t_1));
	} else {
		tmp = t_2 * ((l / k) / (t / t_1));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = l / sin(k)
    t_2 = 2.0d0 / (k * tan(k))
    if (t <= 3.5d-147) then
        tmp = l * (t_2 / ((k * t) / t_1))
    else
        tmp = t_2 * ((l / k) / (t / t_1))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = l / Math.sin(k);
	double t_2 = 2.0 / (k * Math.tan(k));
	double tmp;
	if (t <= 3.5e-147) {
		tmp = l * (t_2 / ((k * t) / t_1));
	} else {
		tmp = t_2 * ((l / k) / (t / t_1));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = l / math.sin(k)
	t_2 = 2.0 / (k * math.tan(k))
	tmp = 0
	if t <= 3.5e-147:
		tmp = l * (t_2 / ((k * t) / t_1))
	else:
		tmp = t_2 * ((l / k) / (t / t_1))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(l / sin(k))
	t_2 = Float64(2.0 / Float64(k * tan(k)))
	tmp = 0.0
	if (t <= 3.5e-147)
		tmp = Float64(l * Float64(t_2 / Float64(Float64(k * t) / t_1)));
	else
		tmp = Float64(t_2 * Float64(Float64(l / k) / Float64(t / t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = l / sin(k);
	t_2 = 2.0 / (k * tan(k));
	tmp = 0.0;
	if (t <= 3.5e-147)
		tmp = l * (t_2 / ((k * t) / t_1));
	else
		tmp = t_2 * ((l / k) / (t / t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, 3.5e-147], N[(l * N[(t$95$2 / N[(N[(k * t), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[(N[(l / k), $MachinePrecision] / N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.50000000000000004e-147

   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-/l/ N/A

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

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

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

   3. Simplified 22.1%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 87.4%

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

   7. Simplified 87.4%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. associate-/l/ N/A

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 94.4%

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

   9. Applied egg-rr 94.4%

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

   if 3.50000000000000004e-147 < t

   1. Initial program 32.5%

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

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

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

   3. Simplified 37.2%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 96.0%

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

   7. Simplified 96.0%

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

      1. clear-num N/A

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

      2. inv-pow N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. unpow-prod-down N/A

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

      6. inv-pow N/A

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

      7. clear-num N/A

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

      8. inv-pow N/A

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

      9. clear-num N/A

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

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

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

   9. Applied egg-rr 96.3%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3.5 \cdot 10^{-147}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{k \cdot \tan k}}{\frac{k \cdot t}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \tan k} \cdot \frac{\frac{\ell}{k}}{\frac{t}{\frac{\ell}{\sin k}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{k \cdot t} \cdot \frac{\frac{\ell \cdot 2}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{\sin k \cdot t}}} \cdot \frac{2}{\tan k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.95e-82)
   (* (/ (/ l (sin k)) (* k t)) (/ (/ (* l 2.0) k) k))
   (* (/ l (/ (* k k) (/ l (* (sin k) t)))) (/ 2.0 (tan k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e-82) {
		tmp = ((l / sin(k)) / (k * t)) * (((l * 2.0) / k) / k);
	} else {
		tmp = (l / ((k * k) / (l / (sin(k) * t)))) * (2.0 / 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 <= 1.95d-82) then
        tmp = ((l / sin(k)) / (k * t)) * (((l * 2.0d0) / k) / k)
    else
        tmp = (l / ((k * k) / (l / (sin(k) * t)))) * (2.0d0 / tan(k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.95e-82) {
		tmp = ((l / Math.sin(k)) / (k * t)) * (((l * 2.0) / k) / k);
	} else {
		tmp = (l / ((k * k) / (l / (Math.sin(k) * t)))) * (2.0 / Math.tan(k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.95e-82:
		tmp = ((l / math.sin(k)) / (k * t)) * (((l * 2.0) / k) / k)
	else:
		tmp = (l / ((k * k) / (l / (math.sin(k) * t)))) * (2.0 / math.tan(k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.95e-82)
		tmp = Float64(Float64(Float64(l / sin(k)) / Float64(k * t)) * Float64(Float64(Float64(l * 2.0) / k) / k));
	else
		tmp = Float64(Float64(l / Float64(Float64(k * k) / Float64(l / Float64(sin(k) * t)))) * Float64(2.0 / tan(k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.95e-82)
		tmp = ((l / sin(k)) / (k * t)) * (((l * 2.0) / k) / k);
	else
		tmp = (l / ((k * k) / (l / (sin(k) * t)))) * (2.0 / tan(k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.95e-82], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * 2.0), $MachinePrecision] / k), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(N[(k * k), $MachinePrecision] / N[(l / N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.94999999999999987e-82

   1. Initial program 40.4%

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

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

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

   3. Simplified 31.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 93.0%

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

   7. Simplified 93.0%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. associate-/l/ N/A

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 93.0%

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

   9. Applied egg-rr 93.0%

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

       1. associate-*l/ N/A

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

       2. div-inv N/A

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

       3. clear-num N/A

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

       4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

       10. *-commutative N/A

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

       11. clear-num N/A

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

       12. un-div-inv N/A

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

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

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

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

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

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

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

       16. tan-lowering-tan.f64 95.9%

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

   11. Applied egg-rr 95.9%

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

   12. Taylor expanded in k around 0

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

       1. associate-*r/ N/A

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

       2. unpow2 N/A

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

       3. associate-/r* N/A

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

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

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

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

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

       6. *-lowering-*.f64 81.7%

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

   14. Simplified 81.7%

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

   if 1.94999999999999987e-82 < k

   1. Initial program 24.3%

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

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

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

   3. Simplified 23.4%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 87.5%

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

   7. Simplified 87.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

      4. associate-*r/ N/A

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

      5. clear-num N/A

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

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

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

      7. associate-*r* N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      16. tan-lowering-tan.f64 77.2%

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

   9. Applied egg-rr 77.2%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.95 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{\ell}{\sin k}}{k \cdot t} \cdot \frac{\frac{\ell \cdot 2}{k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{\sin k \cdot t}}} \cdot \frac{2}{\tan k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return (l / ((k * tan(k)) / 2.0)) * ((l / sin(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 = (l / ((k * tan(k)) / 2.0d0)) * ((l / sin(k)) / (k * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r/ N/A

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

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

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

3. Simplified 28.8%

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

5. Taylor expanded in k around inf

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

   1. associate-/l* N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 91.2%

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

7. Simplified 91.2%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-/l/ N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. clear-num N/A

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

   12. un-div-inv N/A

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

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

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

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

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

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

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

   16. sin-lowering-sin.f64 91.8%

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

9. Applied egg-rr 91.8%

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

    1. associate-*l/ N/A

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

    2. div-inv N/A

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

    3. clear-num N/A

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

    4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

    10. *-commutative N/A

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

    11. clear-num N/A

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

    12. un-div-inv N/A

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

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

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

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

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

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

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

    16. tan-lowering-tan.f64 94.5%

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

11. Applied egg-rr 94.5%

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

12. Final simplification 94.5%

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

Alternative 5: 92.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return l * ((2.0 / (k * tan(k))) / ((k * t) / (l / sin(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 = l * ((2.0d0 / (k * tan(k))) / ((k * t) / (l / sin(k))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r/ N/A

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

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

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

3. Simplified 28.8%

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

5. Taylor expanded in k around inf

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

   1. associate-/l* N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 91.2%

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

7. Simplified 91.2%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-/l/ N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. clear-num N/A

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

   12. un-div-inv N/A

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

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

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

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

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

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

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

   16. sin-lowering-sin.f64 91.8%

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

9. Applied egg-rr 91.8%

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

10. Final simplification 91.8%

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

Alternative 6: 74.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{k \cdot k}}{\frac{k \cdot t}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 450.0)
   (* l (/ (/ 2.0 (* k k)) (/ (* k t) (/ l (sin k)))))
   (/ (/ l (/ k (/ l t))) (/ k -0.3333333333333333))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * ((2.0 / (k * k)) / ((k * t) / (l / sin(k))));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	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 <= 450.0d0) then
        tmp = l * ((2.0d0 / (k * k)) / ((k * t) / (l / sin(k))))
    else
        tmp = (l / (k / (l / t))) / (k / (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * ((2.0 / (k * k)) / ((k * t) / (l / Math.sin(k))));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 450.0:
		tmp = l * ((2.0 / (k * k)) / ((k * t) / (l / math.sin(k))))
	else:
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 450.0)
		tmp = Float64(l * Float64(Float64(2.0 / Float64(k * k)) / Float64(Float64(k * t) / Float64(l / sin(k)))));
	else
		tmp = Float64(Float64(l / Float64(k / Float64(l / t))) / Float64(k / -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 450.0)
		tmp = l * ((2.0 / (k * k)) / ((k * t) / (l / sin(k))));
	else
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 450.0], N[(l * N[(N[(2.0 / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(N[(k * t), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 450

   1. Initial program 39.5%

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

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

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

   3. Simplified 31.1%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 93.3%

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

   7. Simplified 93.3%

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

      1. associate-/r* N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. associate-/l/ N/A

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

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

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

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

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

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

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. clear-num N/A

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

      12. un-div-inv N/A

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

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

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

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

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

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

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

      16. sin-lowering-sin.f64 93.3%

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

   9. Applied egg-rr 93.3%

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

   10. Taylor expanded in k around 0

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

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

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

       2. unpow2 N/A

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

       3. *-lowering-*.f64 79.9%

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

   12. Simplified 79.9%

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

   if 450 < k

   1. Initial program 24.5%

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

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

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

   3. Simplified 23.1%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 86.0%

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

   7. Simplified 86.0%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 52.3%

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

   13. Simplified 52.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

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

       2. times-frac N/A

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

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       5. clear-num N/A

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

       6. un-div-inv N/A

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

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

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

       8. associate-/l/ N/A

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

       9. clear-num N/A

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

       10. div-inv N/A

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

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

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

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

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

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

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

       14. /-lowering-/.f64 55.1%

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

   15. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{k \cdot k}}{\frac{k \cdot t}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 3.7× speedup

Math

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

FPCore

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

C

double code(double t, double l, double k) {
	return ((l / sin(k)) / (k * t)) * (((l * 2.0) / 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 = ((l / sin(k)) / (k * t)) * (((l * 2.0d0) / k) / k)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.2%

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

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

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

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

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

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

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

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. associate-*r/ N/A

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

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

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

3. Simplified 28.8%

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

5. Taylor expanded in k around inf

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

   1. associate-/l* N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

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

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

   5. unpow2 N/A

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

   6. associate-/r* N/A

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

   7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 91.2%

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

7. Simplified 91.2%

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

   1. associate-/r* N/A

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

   2. associate-/r/ N/A

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

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

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

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

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

   5. associate-/l/ N/A

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

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

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

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

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

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

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

   9. associate-/l* N/A

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

   10. associate-*r* N/A

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

   11. clear-num N/A

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

   12. un-div-inv N/A

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

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

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

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

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

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

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

   16. sin-lowering-sin.f64 91.8%

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

9. Applied egg-rr 91.8%

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

    1. associate-*l/ N/A

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

    2. div-inv N/A

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

    3. clear-num N/A

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

    4. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

    10. *-commutative N/A

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

    11. clear-num N/A

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

    12. un-div-inv N/A

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

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

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

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

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

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

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

    16. tan-lowering-tan.f64 94.5%

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

11. Applied egg-rr 94.5%

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

12. Taylor expanded in k around 0

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

    1. associate-*r/ N/A

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

    2. unpow2 N/A

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

    3. associate-/r* N/A

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

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

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

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

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

    6. *-lowering-*.f64 73.8%

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

14. Simplified 73.8%

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

15. Final simplification 73.8%

    \[\leadsto \frac{\frac{\ell}{\sin k}}{k \cdot t} \cdot \frac{\frac{\ell \cdot 2}{k}}{k} \]
16. Add Preprocessing

Alternative 8: 48.7% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}}{0 - k \cdot k}\\ \mathbf{elif}\;k \leq 450:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{0 - \frac{t}{\ell \cdot \ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.6e-162)
   (/ (/ (* (* l l) 0.3333333333333333) t) (- 0.0 (* k k)))
   (if (<= k 450.0)
     (/ (/ 0.3333333333333333 (- 0.0 (/ t (* l l)))) (* k k))
     (/ (/ l (/ k (/ l t))) (/ k -0.3333333333333333)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.6e-162) {
		tmp = (((l * l) * 0.3333333333333333) / t) / (0.0 - (k * k));
	} else if (k <= 450.0) {
		tmp = (0.3333333333333333 / (0.0 - (t / (l * l)))) / (k * k);
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	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 <= 1.6d-162) then
        tmp = (((l * l) * 0.3333333333333333d0) / t) / (0.0d0 - (k * k))
    else if (k <= 450.0d0) then
        tmp = (0.3333333333333333d0 / (0.0d0 - (t / (l * l)))) / (k * k)
    else
        tmp = (l / (k / (l / t))) / (k / (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.6e-162) {
		tmp = (((l * l) * 0.3333333333333333) / t) / (0.0 - (k * k));
	} else if (k <= 450.0) {
		tmp = (0.3333333333333333 / (0.0 - (t / (l * l)))) / (k * k);
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.6e-162:
		tmp = (((l * l) * 0.3333333333333333) / t) / (0.0 - (k * k))
	elif k <= 450.0:
		tmp = (0.3333333333333333 / (0.0 - (t / (l * l)))) / (k * k)
	else:
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.6e-162)
		tmp = Float64(Float64(Float64(Float64(l * l) * 0.3333333333333333) / t) / Float64(0.0 - Float64(k * k)));
	elseif (k <= 450.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(0.0 - Float64(t / Float64(l * l)))) / Float64(k * k));
	else
		tmp = Float64(Float64(l / Float64(k / Float64(l / t))) / Float64(k / -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.6e-162)
		tmp = (((l * l) * 0.3333333333333333) / t) / (0.0 - (k * k));
	elseif (k <= 450.0)
		tmp = (0.3333333333333333 / (0.0 - (t / (l * l)))) / (k * k);
	else
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.6e-162], N[(N[(N[(N[(l * l), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] / t), $MachinePrecision] / N[(0.0 - N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 450.0], N[(N[(0.3333333333333333 / N[(0.0 - N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.59999999999999988e-162

   1. Initial program 40.9%

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

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

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

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

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

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

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

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. associate-*r/ N/A

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

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

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

   3. Simplified 31.2%

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

   5. Taylor expanded in k around inf

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

      1. associate-/l* N/A

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

      2. unpow2 N/A

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

      3. associate-*l* N/A

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

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

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

      5. unpow2 N/A

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

      6. associate-/r* N/A

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

      7. associate-*r/ N/A

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

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

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

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

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

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

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

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

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

      12. sin-lowering-sin.f64 94.8%

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

   7. Simplified 94.8%

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

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

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

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 42.2%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

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

       1. associate-*r/ N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

       4. associate-*r/ N/A

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

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

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

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

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

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

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

       8. unpow2 N/A

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

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

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

       10. unpow2 N/A

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

       11. *-lowering-*.f64 22.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 22.6%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. frac-2neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}\right)}{\color{blue}{\mathsf{neg}\left(k \cdot k\right)}} \]

       2. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{3} \cdot \frac{\ell \cdot \ell}{t}\right)\right), \color{blue}{\left(\mathsf{neg}\left(k \cdot k\right)\right)}\right) \]

       3. associate-*r/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)}{t}\right)\right), \left(\mathsf{neg}\left(\color{blue}{k} \cdot k\right)\right)\right) \]

       4. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)\right)}{t}\right), \left(\mathsf{neg}\left(\color{blue}{k \cdot k}\right)\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{3} \cdot \left(\ell \cdot \ell\right)\right)\right), t\right), \left(\mathsf{neg}\left(\color{blue}{k \cdot k}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\ell \cdot \ell\right) \cdot \frac{-1}{3}\right)\right), t\right), \left(\mathsf{neg}\left(k \cdot k\right)\right)\right) \]

       7. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\ell \cdot \ell\right) \cdot \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right), t\right), \left(\mathsf{neg}\left(\color{blue}{k} \cdot k\right)\right)\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\ell \cdot \ell\right), \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right), t\right), \left(\mathsf{neg}\left(\color{blue}{k} \cdot k\right)\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\mathsf{neg}\left(\frac{-1}{3}\right)\right)\right), t\right), \left(\mathsf{neg}\left(k \cdot k\right)\right)\right) \]

       10. metadata-eval N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{3}\right), t\right), \left(\mathsf{neg}\left(k \cdot k\right)\right)\right) \]

       11. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{3}\right), t\right), \left(0 - \color{blue}{k \cdot k}\right)\right) \]

       12. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{3}\right), t\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(k \cdot k\right)}\right)\right) \]

       13. *-lowering-*.f64 43.6%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \frac{1}{3}\right), t\right), \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right)\right) \]

   15. Applied egg-rr 43.6%

       \[\leadsto \color{blue}{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 0.3333333333333333}{t}}{0 - k \cdot k}} \]

   if 1.59999999999999988e-162 < k < 450

   1. Initial program 28.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. 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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 30.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 82.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 82.1%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 52.9%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

       4. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       11. *-lowering-*.f64 7.0%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 7.0%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{1}{\frac{t}{\ell \cdot \ell}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       2. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{3}}{\frac{t}{\ell \cdot \ell}}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{3}}{\frac{\frac{t}{\ell}}{\ell}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{-1}{3}\right)}{\mathsf{neg}\left(\frac{\frac{t}{\ell}}{\ell}\right)}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{3}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{t}{\ell}}{\ell}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{\frac{t}{\ell}}{\ell}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \left(0 - \frac{\frac{t}{\ell}}{\ell}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{\_.f64}\left(0, \left(\frac{\frac{t}{\ell}}{\ell}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{\_.f64}\left(0, \left(\frac{t}{\ell \cdot \ell}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \left(\ell \cdot \ell\right)\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. *-lowering-*.f64 11.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   15. Applied egg-rr 11.8%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{0 - \frac{t}{\ell \cdot \ell}}}}{k \cdot k} \]

   if 450 < k

   1. Initial program 24.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 86.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 86.0%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

       4. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       11. *-lowering-*.f64 52.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 52.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{\color{blue}{k} \cdot k} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{\frac{-1}{3}}{k}} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       5. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       6. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{\ell}{\frac{t}{\ell}}}{k}}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k}\right), \color{blue}{\left(\frac{k}{\frac{-1}{3}}\right)}\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{t}{\ell}}\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{1}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{\frac{\ell}{t}}\right)\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       14. /-lowering-/.f64 55.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \mathsf{/.f64}\left(k, \color{blue}{\frac{-1}{3}}\right)\right) \]

   15. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.8% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.9e+154)
   (/ (/ 2.0 k) (* k (/ (* k (/ (* k t) l)) l)))
   (* l (* l (/ 2.0 (* k (* k (* k (* k t)))))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.9e+154) {
		tmp = (2.0 / k) / (k * ((k * ((k * t) / l)) / l));
	} else {
		tmp = l * (l * (2.0 / (k * (k * (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 <= 1.9d+154) then
        tmp = (2.0d0 / k) / (k * ((k * ((k * t) / l)) / l))
    else
        tmp = l * (l * (2.0d0 / (k * (k * (k * (k * t))))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.9e+154) {
		tmp = (2.0 / k) / (k * ((k * ((k * t) / l)) / l));
	} else {
		tmp = l * (l * (2.0 / (k * (k * (k * (k * t))))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.9e+154:
		tmp = (2.0 / k) / (k * ((k * ((k * t) / l)) / l))
	else:
		tmp = l * (l * (2.0 / (k * (k * (k * (k * t))))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.9e+154)
		tmp = Float64(Float64(2.0 / k) / Float64(k * Float64(Float64(k * Float64(Float64(k * t) / l)) / l)));
	else
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(k * Float64(k * Float64(k * Float64(k * t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.9e+154)
		tmp = (2.0 / k) / (k * ((k * ((k * t) / l)) / l));
	else
		tmp = l * (l * (2.0 / (k * (k * (k * (k * t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.9e+154], N[(N[(2.0 / k), $MachinePrecision] / N[(k * N[(N[(k * N[(N[(k * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(l * N[(2.0 / N[(k * N[(k * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.8999999999999999e154

   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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 33.8%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 91.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 91.0%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\frac{2}{k}\right)}, \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 70.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(\color{blue}{k}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   10. Simplified 70.3%

       \[\leadsto \frac{\color{blue}{\frac{2}{k}}}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}} \]

   11. Taylor expanded in k around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \color{blue}{k}\right), \ell\right)\right), \ell\right)\right)\right) \]
   12. Step-by-step derivation

   13. Simplified 70.1%

       \[\leadsto \frac{\frac{2}{k}}{k \cdot \frac{k \cdot \frac{t \cdot \color{blue}{k}}{\ell}}{\ell}} \]

   if 1.8999999999999999e154 < t

   1. Initial program 5.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      6. *-lowering-*.f64 68.4%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 68.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{4}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot {k}^{4}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot {k}^{4}\right)\right), \ell\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4}\right)\right)\right), \ell\right), \ell\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{\left(2 + 2\right)}\right)\right)\right), \ell\right), \ell\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 79.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 79.6%

      \[\leadsto \color{blue}{\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)\right), \ell\right), \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)\right)\right), \ell\right), \ell\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot k\right) \cdot k\right)\right), \ell\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot k\right), k\right)\right), \ell\right), \ell\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right), k\right)\right), \ell\right), \ell\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot t\right)\right)\right), k\right)\right), \ell\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right)\right), \ell\right), \ell\right) \]

      10. *-lowering-*.f64 82.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right), \ell\right), \ell\right) \]

   9. Applied egg-rr 82.2%

      \[\leadsto \left(\frac{2}{\color{blue}{\left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right) \cdot k}} \cdot \ell\right) \cdot \ell \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot \frac{k \cdot \frac{k \cdot t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 450.0)
   (* l (/ (/ (* l 2.0) (* k (* k t))) (* k k)))
   (/ (/ l (/ k (/ l t))) (/ k -0.3333333333333333))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * (((l * 2.0) / (k * (k * t))) / (k * k));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	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 <= 450.0d0) then
        tmp = l * (((l * 2.0d0) / (k * (k * t))) / (k * k))
    else
        tmp = (l / (k / (l / t))) / (k / (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * (((l * 2.0) / (k * (k * t))) / (k * k));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 450.0:
		tmp = l * (((l * 2.0) / (k * (k * t))) / (k * k))
	else:
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 450.0)
		tmp = Float64(l * Float64(Float64(Float64(l * 2.0) / Float64(k * Float64(k * t))) / Float64(k * k)));
	else
		tmp = Float64(Float64(l / Float64(k / Float64(l / t))) / Float64(k / -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 450.0)
		tmp = l * (((l * 2.0) / (k * (k * t))) / (k * k));
	else
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 450.0], N[(l * N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 450

   1. Initial program 39.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      6. *-lowering-*.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{4}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot {k}^{4}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot {k}^{4}\right)\right), \ell\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4}\right)\right)\right), \ell\right), \ell\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{\left(2 + 2\right)}\right)\right)\right), \ell\right), \ell\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 74.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right), \ell\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t}\right), \ell\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)}\right), \ell\right) \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right), \ell\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)}\right), \ell\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k \cdot k}\right), \ell\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}\right), \left(k \cdot k\right)\right), \ell\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), \left(k \cdot k\right)\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \left(k \cdot \left(k \cdot t\right)\right)\right), \left(k \cdot k\right)\right), \ell\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), \left(k \cdot k\right)\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), \left(k \cdot k\right)\right), \ell\right) \]

      12. *-lowering-*.f64 79.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \ell\right), \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right), \ell\right) \]

   9. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{k \cdot k}} \cdot \ell \]

   if 450 < k

   1. Initial program 24.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 86.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 86.0%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

       4. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       11. *-lowering-*.f64 52.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 52.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{\color{blue}{k} \cdot k} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{\frac{-1}{3}}{k}} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       5. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       6. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{\ell}{\frac{t}{\ell}}}{k}}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k}\right), \color{blue}{\left(\frac{k}{\frac{-1}{3}}\right)}\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{t}{\ell}}\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{1}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{\frac{\ell}{t}}\right)\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       14. /-lowering-/.f64 55.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \mathsf{/.f64}\left(k, \color{blue}{\frac{-1}{3}}\right)\right) \]

   15. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell \cdot 2}{k \cdot \left(k \cdot t\right)}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.4% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 450.0)
   (* l (* (/ l (* k k)) (/ 2.0 (* k (* k t)))))
   (/ (/ l (/ k (/ l t))) (/ k -0.3333333333333333))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * ((l / (k * k)) * (2.0 / (k * (k * t))));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	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 <= 450.0d0) then
        tmp = l * ((l / (k * k)) * (2.0d0 / (k * (k * t))))
    else
        tmp = (l / (k / (l / t))) / (k / (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * ((l / (k * k)) * (2.0 / (k * (k * t))));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 450.0:
		tmp = l * ((l / (k * k)) * (2.0 / (k * (k * t))))
	else:
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 450.0)
		tmp = Float64(l * Float64(Float64(l / Float64(k * k)) * Float64(2.0 / Float64(k * Float64(k * t)))));
	else
		tmp = Float64(Float64(l / Float64(k / Float64(l / t))) / Float64(k / -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 450.0)
		tmp = l * ((l / (k * k)) * (2.0 / (k * (k * t))));
	else
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 450.0], N[(l * N[(N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 450

   1. Initial program 39.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      6. *-lowering-*.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{4}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot {k}^{4}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot {k}^{4}\right)\right), \ell\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4}\right)\right)\right), \ell\right), \ell\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{\left(2 + 2\right)}\right)\right)\right), \ell\right), \ell\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 74.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell} \]
   8. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right), \ell\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2 \cdot \ell}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(k \cdot k\right)}\right), \ell\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{k \cdot k}\right), \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{k \cdot k}\right), \ell\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k \cdot k}\right), \ell\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right), \ell\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{\ell}{k \cdot k}\right), \left(\frac{2}{k \cdot \left(k \cdot t\right)}\right)\right), \ell\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \left(k \cdot k\right)\right), \left(\frac{2}{k \cdot \left(k \cdot t\right)}\right)\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \left(\frac{2}{k \cdot \left(k \cdot t\right)}\right)\right), \ell\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(2, \left(k \cdot \left(k \cdot t\right)\right)\right)\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right)\right), \ell\right) \]

      12. *-lowering-*.f64 79.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{*.f64}\left(k, k\right)\right), \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right)\right), \ell\right) \]

   9. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\left(\frac{\ell}{k \cdot k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)} \cdot \ell \]

   if 450 < k

   1. Initial program 24.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 86.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 86.0%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

       4. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       11. *-lowering-*.f64 52.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 52.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{\color{blue}{k} \cdot k} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{\frac{-1}{3}}{k}} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       5. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       6. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{\ell}{\frac{t}{\ell}}}{k}}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k}\right), \color{blue}{\left(\frac{k}{\frac{-1}{3}}\right)}\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{t}{\ell}}\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{1}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{\frac{\ell}{t}}\right)\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       14. /-lowering-/.f64 55.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \mathsf{/.f64}\left(k, \color{blue}{\frac{-1}{3}}\right)\right) \]

   15. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 71.5% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 450.0)
   (* l (* l (/ 2.0 (* k (* k (* k (* k t)))))))
   (/ (/ l (/ k (/ l t))) (/ k -0.3333333333333333))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * (l * (2.0 / (k * (k * (k * (k * t))))));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	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 <= 450.0d0) then
        tmp = l * (l * (2.0d0 / (k * (k * (k * (k * t))))))
    else
        tmp = (l / (k / (l / t))) / (k / (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * (l * (2.0 / (k * (k * (k * (k * t))))));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 450.0:
		tmp = l * (l * (2.0 / (k * (k * (k * (k * t))))))
	else:
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 450.0)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(k * Float64(k * Float64(k * Float64(k * t)))))));
	else
		tmp = Float64(Float64(l / Float64(k / Float64(l / t))) / Float64(k / -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 450.0)
		tmp = l * (l * (2.0 / (k * (k * (k * (k * t))))));
	else
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 450.0], N[(l * N[(l * N[(2.0 / N[(k * N[(k * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 450

   1. Initial program 39.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      6. *-lowering-*.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{4}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot {k}^{4}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot {k}^{4}\right)\right), \ell\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4}\right)\right)\right), \ell\right), \ell\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{\left(2 + 2\right)}\right)\right)\right), \ell\right), \ell\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 74.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right) \cdot t\right)\right), \ell\right), \ell\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(\left(k \cdot k\right) \cdot t\right)\right)\right), \ell\right), \ell\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot k\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)\right), \ell\right), \ell\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(k \cdot k\right)\right)\right), \ell\right), \ell\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot k\right) \cdot k\right)\right), \ell\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(\left(k \cdot \left(k \cdot t\right)\right) \cdot k\right), k\right)\right), \ell\right), \ell\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right), k\right)\right), \ell\right), \ell\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \left(k \cdot \left(k \cdot t\right)\right)\right), k\right)\right), \ell\right), \ell\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \left(k \cdot t\right)\right)\right), k\right)\right), \ell\right), \ell\right) \]

      10. *-lowering-*.f64 78.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, \mathsf{*.f64}\left(k, t\right)\right)\right), k\right)\right), \ell\right), \ell\right) \]

   9. Applied egg-rr 78.2%

      \[\leadsto \left(\frac{2}{\color{blue}{\left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right) \cdot k}} \cdot \ell\right) \cdot \ell \]

   if 450 < k

   1. Initial program 24.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 86.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 86.0%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

       4. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       11. *-lowering-*.f64 52.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 52.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{\color{blue}{k} \cdot k} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{\frac{-1}{3}}{k}} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       5. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       6. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{\ell}{\frac{t}{\ell}}}{k}}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k}\right), \color{blue}{\left(\frac{k}{\frac{-1}{3}}\right)}\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{t}{\ell}}\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{1}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{\frac{\ell}{t}}\right)\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       14. /-lowering-/.f64 55.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \mathsf{/.f64}\left(k, \color{blue}{\frac{-1}{3}}\right)\right) \]

   15. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.7% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 450.0)
   (* l (* l (/ 2.0 (* t (* (* k k) (* k k))))))
   (/ (/ l (/ k (/ l t))) (/ k -0.3333333333333333))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * (l * (2.0 / (t * ((k * k) * (k * k)))));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	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 <= 450.0d0) then
        tmp = l * (l * (2.0d0 / (t * ((k * k) * (k * k)))))
    else
        tmp = (l / (k / (l / t))) / (k / (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = l * (l * (2.0 / (t * ((k * k) * (k * k)))));
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 450.0:
		tmp = l * (l * (2.0 / (t * ((k * k) * (k * k)))))
	else:
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 450.0)
		tmp = Float64(l * Float64(l * Float64(2.0 / Float64(t * Float64(Float64(k * k) * Float64(k * k))))));
	else
		tmp = Float64(Float64(l / Float64(k / Float64(l / t))) / Float64(k / -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 450.0)
		tmp = l * (l * (2.0 / (t * ((k * k) * (k * k)))));
	else
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 450.0], N[(l * N[(l * N[(2.0 / N[(t * N[(N[(k * k), $MachinePrecision] * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 450

   1. Initial program 39.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left(\frac{{k}^{4} \cdot t}{{\ell}^{2}}\right)}\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left({k}^{4} \cdot t\right), \color{blue}{\left({\ell}^{2}\right)}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\left(t \cdot {k}^{4}\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left({k}^{4}\right)\right), \left({\color{blue}{\ell}}^{2}\right)\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left({\ell}^{2}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \left(\ell \cdot \color{blue}{\ell}\right)\right)\right) \]

      6. *-lowering-*.f64 65.1%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{pow.f64}\left(k, 4\right)\right), \mathsf{*.f64}\left(\ell, \color{blue}{\ell}\right)\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{t \cdot {k}^{4}}{\ell \cdot \ell}}} \]
   6. Step-by-step derivation

      1. associate-/r/ N/A

         \[\leadsto \frac{2}{t \cdot {k}^{4}} \cdot \color{blue}{\left(\ell \cdot \ell\right)} \]

      2. associate-*r* N/A

         \[\leadsto \left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right) \cdot \color{blue}{\ell} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}} \cdot \ell\right), \color{blue}{\ell}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{t \cdot {k}^{4}}\right), \ell\right), \ell\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \left(t \cdot {k}^{4}\right)\right), \ell\right), \ell\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{4}\right)\right)\right), \ell\right), \ell\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{\left(2 + 2\right)}\right)\right)\right), \ell\right), \ell\right) \]

      8. pow-prod-up N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left({k}^{2} \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      9. pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot {k}^{2}\right)\right)\right), \ell\right), \ell\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\left(k \cdot k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(k \cdot k\right)\right)\right)\right), \ell\right), \ell\right) \]

      13. *-lowering-*.f64 74.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, \mathsf{*.f64}\left(t, \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{*.f64}\left(k, k\right)\right)\right)\right), \ell\right), \ell\right) \]

   7. Applied egg-rr 74.0%

      \[\leadsto \color{blue}{\left(\frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)} \cdot \ell\right) \cdot \ell} \]

   if 450 < k

   1. Initial program 24.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 86.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 86.0%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

       4. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       11. *-lowering-*.f64 52.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 52.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{\color{blue}{k} \cdot k} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{\frac{-1}{3}}{k}} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       5. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       6. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{\ell}{\frac{t}{\ell}}}{k}}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k}\right), \color{blue}{\left(\frac{k}{\frac{-1}{3}}\right)}\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{t}{\ell}}\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{1}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{\frac{\ell}{t}}\right)\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       14. /-lowering-/.f64 55.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \mathsf{/.f64}\left(k, \color{blue}{\frac{-1}{3}}\right)\right) \]

   15. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{2}{t \cdot \left(\left(k \cdot k\right) \cdot \left(k \cdot k\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 37.1% accurate, 23.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 450:\\ \;\;\;\;\frac{\frac{0.3333333333333333}{0 - \frac{t}{\ell \cdot \ell}}}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 450.0)
   (/ (/ 0.3333333333333333 (- 0.0 (/ t (* l l)))) (* k k))
   (/ (/ l (/ k (/ l t))) (/ k -0.3333333333333333))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = (0.3333333333333333 / (0.0 - (t / (l * l)))) / (k * k);
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	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 <= 450.0d0) then
        tmp = (0.3333333333333333d0 / (0.0d0 - (t / (l * l)))) / (k * k)
    else
        tmp = (l / (k / (l / t))) / (k / (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 450.0) {
		tmp = (0.3333333333333333 / (0.0 - (t / (l * l)))) / (k * k);
	} else {
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 450.0:
		tmp = (0.3333333333333333 / (0.0 - (t / (l * l)))) / (k * k)
	else:
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333)
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 450.0)
		tmp = Float64(Float64(0.3333333333333333 / Float64(0.0 - Float64(t / Float64(l * l)))) / Float64(k * k));
	else
		tmp = Float64(Float64(l / Float64(k / Float64(l / t))) / Float64(k / -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 450.0)
		tmp = (0.3333333333333333 / (0.0 - (t / (l * l)))) / (k * k);
	else
		tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 450.0], N[(N[(0.3333333333333333 / N[(0.0 - N[(t / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 450

   1. Initial program 39.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 31.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 93.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 93.3%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 43.4%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

       4. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       11. *-lowering-*.f64 20.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 20.8%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{1}{\frac{t}{\ell \cdot \ell}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       2. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{3}}{\frac{t}{\ell \cdot \ell}}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{3}}{\frac{\frac{t}{\ell}}{\ell}}\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       4. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\frac{-1}{3}\right)}{\mathsf{neg}\left(\frac{\frac{t}{\ell}}{\ell}\right)}\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{-1}{3}\right)\right), \left(\mathsf{neg}\left(\frac{\frac{t}{\ell}}{\ell}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{k}, k\right)\right) \]

       6. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \left(\mathsf{neg}\left(\frac{\frac{t}{\ell}}{\ell}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       7. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \left(0 - \frac{\frac{t}{\ell}}{\ell}\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       8. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{\_.f64}\left(0, \left(\frac{\frac{t}{\ell}}{\ell}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       9. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{\_.f64}\left(0, \left(\frac{t}{\ell \cdot \ell}\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       10. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \left(\ell \cdot \ell\right)\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

       11. *-lowering-*.f64 28.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{1}{3}, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(t, \mathsf{*.f64}\left(\ell, \ell\right)\right)\right)\right), \mathsf{*.f64}\left(k, k\right)\right) \]

   15. Applied egg-rr 28.5%

       \[\leadsto \frac{\color{blue}{\frac{0.3333333333333333}{0 - \frac{t}{\ell \cdot \ell}}}}{k \cdot k} \]

   if 450 < k

   1. Initial program 24.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
   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-/l/ N/A

         \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      6. tan-lowering-tan.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

   3. Simplified 23.1%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
   4. Add Preprocessing

   5. Taylor expanded in k around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

      6. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

      12. sin-lowering-sin.f64 86.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

   7. Simplified 86.0%

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

   8. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

      5. *-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

   10. Simplified 38.6%

       \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

   11. Taylor expanded in k around inf

       \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
   12. Step-by-step derivation

       1. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

       2. *-commutative N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

       3. associate-/r* N/A

          \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

       4. associate-*r/ N/A

          \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

       11. *-lowering-*.f64 52.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

   13. Simplified 52.3%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{\color{blue}{k} \cdot k} \]

       2. times-frac N/A

          \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{\frac{-1}{3}}{k}} \]

       3. associate-*l/ N/A

          \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       4. associate-/r/ N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{\frac{-1}{3}}{k} \]

       5. clear-num N/A

          \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       6. un-div-inv N/A

          \[\leadsto \frac{\frac{\frac{\ell}{\frac{t}{\ell}}}{k}}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

       7. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k}\right), \color{blue}{\left(\frac{k}{\frac{-1}{3}}\right)}\right) \]

       8. associate-/l/ N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{t}{\ell}}\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       9. clear-num N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{1}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       10. div-inv N/A

           \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{\frac{\ell}{t}}\right)\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

       14. /-lowering-/.f64 55.1%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \mathsf{/.f64}\left(k, \color{blue}{\frac{-1}{3}}\right)\right) \]

   15. Applied egg-rr 55.1%

       \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 30.7% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ l (/ k (/ l t))) (/ k -0.3333333333333333)))

C

double code(double t, double l, double k) {
	return (l / (k / (l / t))) / (k / -0.3333333333333333);
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (l / (k / (l / t))) / (k / (-0.3333333333333333d0))
end function

Java

public static double code(double t, double l, double k) {
	return (l / (k / (l / t))) / (k / -0.3333333333333333);
}

Python

def code(t, l, k):
	return (l / (k / (l / t))) / (k / -0.3333333333333333)

Julia

function code(t, l, k)
	return Float64(Float64(l / Float64(k / Float64(l / t))) / Float64(k / -0.3333333333333333))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (l / (k / (l / t))) / (k / -0.3333333333333333);
end

Wolfram

code[t_, l_, k_] := N[(N[(l / N[(k / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k / -0.3333333333333333), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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-/l/ N/A

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

   6. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

3. Simplified 28.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
4. Add Preprocessing

5. Taylor expanded in k around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

   12. sin-lowering-sin.f64 91.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

7. Simplified 91.2%

   \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

10. Simplified 42.0%

    \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

    4. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

    11. *-lowering-*.f64 29.8%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

13. Simplified 29.8%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
14. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t} \cdot \frac{-1}{3}}{\color{blue}{k} \cdot k} \]

    2. times-frac N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k} \cdot \color{blue}{\frac{\frac{-1}{3}}{k}} \]

    3. associate-*l/ N/A

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k} \cdot \frac{\frac{-1}{3}}{k} \]

    4. associate-/r/ N/A

       \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{\frac{-1}{3}}{k} \]

    5. clear-num N/A

       \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

    6. un-div-inv N/A

       \[\leadsto \frac{\frac{\frac{\ell}{\frac{t}{\ell}}}{k}}{\color{blue}{\frac{k}{\frac{-1}{3}}}} \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k}\right), \color{blue}{\left(\frac{k}{\frac{-1}{3}}\right)}\right) \]

    8. associate-/l/ N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{t}{\ell}}\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

    9. clear-num N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{k \cdot \frac{1}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

    10. div-inv N/A

        \[\leadsto \mathsf{/.f64}\left(\left(\frac{\ell}{\frac{k}{\frac{\ell}{t}}}\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

    11. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \left(\frac{k}{\frac{\ell}{t}}\right)\right), \left(\frac{\color{blue}{k}}{\frac{-1}{3}}\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \left(\frac{\ell}{t}\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \left(\frac{k}{\frac{-1}{3}}\right)\right) \]

    14. /-lowering-/.f64 31.1%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, t\right)\right)\right), \mathsf{/.f64}\left(k, \color{blue}{\frac{-1}{3}}\right)\right) \]

15. Applied egg-rr 31.1%

    \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{k}{\frac{\ell}{t}}}}{\frac{k}{-0.3333333333333333}}} \]
16. Add Preprocessing

Alternative 16: 30.4% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-0.3333333333333333}{k}}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ (/ -0.3333333333333333 k) (/ k (/ l (/ t l)))))

C

double code(double t, double l, double k) {
	return (-0.3333333333333333 / k) / (k / (l / (t / l)));
}

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) / k) / (k / (l / (t / l)))
end function

Java

public static double code(double t, double l, double k) {
	return (-0.3333333333333333 / k) / (k / (l / (t / l)));
}

Python

def code(t, l, k):
	return (-0.3333333333333333 / k) / (k / (l / (t / l)))

Julia

function code(t, l, k)
	return Float64(Float64(-0.3333333333333333 / k) / Float64(k / Float64(l / Float64(t / l))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = (-0.3333333333333333 / k) / (k / (l / (t / l)));
end

Wolfram

code[t_, l_, k_] := N[(N[(-0.3333333333333333 / k), $MachinePrecision] / N[(k / N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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-/l/ N/A

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

   6. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

3. Simplified 28.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
4. Add Preprocessing

5. Taylor expanded in k around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

   12. sin-lowering-sin.f64 91.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

7. Simplified 91.2%

   \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

10. Simplified 42.0%

    \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

    4. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

    11. *-lowering-*.f64 29.8%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

13. Simplified 29.8%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
14. Step-by-step derivation

    1. times-frac N/A

       \[\leadsto \frac{\frac{-1}{3}}{k} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k}} \]

    2. clear-num N/A

       \[\leadsto \frac{\frac{-1}{3}}{k} \cdot \frac{1}{\color{blue}{\frac{k}{\frac{\ell \cdot \ell}{t}}}} \]

    3. un-div-inv N/A

       \[\leadsto \frac{\frac{\frac{-1}{3}}{k}}{\color{blue}{\frac{k}{\frac{\ell \cdot \ell}{t}}}} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{\frac{-1}{3}}{k}\right), \color{blue}{\left(\frac{k}{\frac{\ell \cdot \ell}{t}}\right)}\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, k\right), \left(\frac{\color{blue}{k}}{\frac{\ell \cdot \ell}{t}}\right)\right) \]

    6. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, k\right), \left(\frac{k}{\frac{\ell}{t} \cdot \color{blue}{\ell}}\right)\right) \]

    7. associate-/r/ N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, k\right), \left(\frac{k}{\frac{\ell}{\color{blue}{\frac{t}{\ell}}}}\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, k\right), \mathsf{/.f64}\left(k, \color{blue}{\left(\frac{\ell}{\frac{t}{\ell}}\right)}\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

    10. /-lowering-/.f64 30.9%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\frac{-1}{3}, k\right), \mathsf{/.f64}\left(k, \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

15. Applied egg-rr 30.9%

    \[\leadsto \color{blue}{\frac{\frac{-0.3333333333333333}{k}}{\frac{k}{\frac{\ell}{\frac{t}{\ell}}}}} \]
16. Add Preprocessing

Alternative 17: 29.9% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ \frac{-0.3333333333333333}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ -0.3333333333333333 (/ (* k k) (/ l (/ t l)))))

C

double code(double t, double l, double k) {
	return -0.3333333333333333 / ((k * k) / (l / (t / l)));
}

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) / ((k * k) / (l / (t / l)))
end function

Java

public static double code(double t, double l, double k) {
	return -0.3333333333333333 / ((k * k) / (l / (t / l)));
}

Python

def code(t, l, k):
	return -0.3333333333333333 / ((k * k) / (l / (t / l)))

Julia

function code(t, l, k)
	return Float64(-0.3333333333333333 / Float64(Float64(k * k) / Float64(l / Float64(t / l))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.3333333333333333 / ((k * k) / (l / (t / l)));
end

Wolfram

code[t_, l_, k_] := N[(-0.3333333333333333 / N[(N[(k * k), $MachinePrecision] / N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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-/l/ N/A

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

   6. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

3. Simplified 28.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
4. Add Preprocessing

5. Taylor expanded in k around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

   12. sin-lowering-sin.f64 91.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

7. Simplified 91.2%

   \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

10. Simplified 42.0%

    \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

    4. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

    11. *-lowering-*.f64 29.8%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

13. Simplified 29.8%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
14. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

    2. clear-num N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{1}{\color{blue}{\frac{k \cdot k}{\frac{\ell \cdot \ell}{t}}}} \]

    3. associate-*l/ N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{1}{\frac{k \cdot k}{\frac{\ell}{t} \cdot \color{blue}{\ell}}} \]

    4. associate-/r/ N/A

       \[\leadsto \frac{-1}{3} \cdot \frac{1}{\frac{k \cdot k}{\frac{\ell}{\color{blue}{\frac{t}{\ell}}}}} \]

    5. un-div-inv N/A

       \[\leadsto \frac{\frac{-1}{3}}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}}} \]

    6. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \color{blue}{\left(\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}\right)}\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(k \cdot k\right), \color{blue}{\left(\frac{\ell}{\frac{t}{\ell}}\right)}\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \left(\frac{\color{blue}{\ell}}{\frac{t}{\ell}}\right)\right)\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \color{blue}{\left(\frac{t}{\ell}\right)}\right)\right)\right) \]

    10. /-lowering-/.f64 30.4%

        \[\leadsto \mathsf{/.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, k\right), \mathsf{/.f64}\left(\ell, \mathsf{/.f64}\left(t, \color{blue}{\ell}\right)\right)\right)\right) \]

15. Applied egg-rr 30.4%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333}{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}}} \]
16. Add Preprocessing

Alternative 18: 29.4% accurate, 38.3× speedup

Math

\[\begin{array}{l} \\ -0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (* -0.3333333333333333 (/ (* l l) (* t (* k k)))))

C

double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l * l) / (t * (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 = (-0.3333333333333333d0) * ((l * l) / (t * (k * k)))
end function

Java

public static double code(double t, double l, double k) {
	return -0.3333333333333333 * ((l * l) / (t * (k * k)));
}

Python

def code(t, l, k):
	return -0.3333333333333333 * ((l * l) / (t * (k * k)))

Julia

function code(t, l, k)
	return Float64(-0.3333333333333333 * Float64(Float64(l * l) / Float64(t * Float64(k * k))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = -0.3333333333333333 * ((l * l) / (t * (k * k)));
end

Wolfram

code[t_, l_, k_] := N[(-0.3333333333333333 * N[(N[(l * l), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.2%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
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-/l/ N/A

      \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{2}{\tan k}\right), \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \tan k\right), \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

   6. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - \color{blue}{1}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right)\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell}}\right)\right) \]

   9. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\frac{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}}{\color{blue}{\ell \cdot \ell}}\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{/.f64}\left(\left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \sin k\right) \cdot {t}^{3}\right), \color{blue}{\left(\ell \cdot \ell\right)}\right)\right) \]

3. Simplified 28.8%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{\left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right) \cdot \left(t \cdot \left(t \cdot t\right)\right)}{\ell \cdot \ell}}} \]
4. Add Preprocessing

5. Taylor expanded in k around inf

   \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \color{blue}{\left(\frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{{\ell}^{2}}\right)}\right) \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left({k}^{2} \cdot \color{blue}{\frac{t \cdot \sin k}{{\ell}^{2}}}\right)\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(\left(k \cdot k\right) \cdot \frac{\color{blue}{t \cdot \sin k}}{{\ell}^{2}}\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \left(k \cdot \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \color{blue}{\left(k \cdot \frac{t \cdot \sin k}{{\ell}^{2}}\right)}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{t \cdot \sin k}{\ell \cdot \color{blue}{\ell}}\right)\right)\right) \]

   6. associate-/r* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(k \cdot \frac{\frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \left(\frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\color{blue}{\ell}}\right)\right)\right) \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(k \cdot \frac{t \cdot \sin k}{\ell}\right), \color{blue}{\ell}\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \left(\frac{t \cdot \sin k}{\ell}\right)\right), \ell\right)\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\left(t \cdot \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \sin k\right), \ell\right)\right), \ell\right)\right)\right) \]

   12. sin-lowering-sin.f64 91.2%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(2, \mathsf{tan.f64}\left(k\right)\right), \mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(k, \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{sin.f64}\left(k\right)\right), \ell\right)\right), \ell\right)\right)\right) \]

7. Simplified 91.2%

   \[\leadsto \frac{\frac{2}{\tan k}}{\color{blue}{k \cdot \frac{k \cdot \frac{t \cdot \sin k}{\ell}}{\ell}}} \]

8. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t} + 2 \cdot \frac{{\ell}^{2}}{t}}{{k}^{4}}} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{-1}{3} \cdot \frac{{k}^{2} \cdot {\ell}^{2}}{t}}{{\color{blue}{k}}^{4}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \frac{{k}^{2} \cdot {\ell}^{2}}{t} \cdot \frac{-1}{3}}{{k}^{4}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + \left({k}^{2} \cdot \frac{{\ell}^{2}}{t}\right) \cdot \frac{-1}{3}}{{k}^{4}} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{{\ell}^{2}}{t} \cdot \frac{-1}{3}\right)}{{k}^{4}} \]

   5. *-commutative N/A

      \[\leadsto \frac{2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)}{{k}^{4}} \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(2 \cdot \frac{{\ell}^{2}}{t} + {k}^{2} \cdot \left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right)\right), \color{blue}{\left({k}^{4}\right)}\right) \]

10. Simplified 42.0%

    \[\leadsto \color{blue}{\frac{\frac{2 \cdot \left(\ell \cdot \ell\right)}{t} + k \cdot \left(k \cdot \frac{\left(\ell \cdot \ell\right) \cdot -0.3333333333333333}{t}\right)}{{k}^{4}}} \]

11. Taylor expanded in k around inf

    \[\leadsto \color{blue}{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}} \]
12. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{\color{blue}{{k}^{2} \cdot t}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot {\ell}^{2}}{t \cdot \color{blue}{{k}^{2}}} \]

    3. associate-/r* N/A

       \[\leadsto \frac{\frac{\frac{-1}{3} \cdot {\ell}^{2}}{t}}{\color{blue}{{k}^{2}}} \]

    4. associate-*r/ N/A

       \[\leadsto \frac{\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}}{{\color{blue}{k}}^{2}} \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\left(\frac{-1}{3} \cdot \frac{{\ell}^{2}}{t}\right), \color{blue}{\left({k}^{2}\right)}\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \left(\frac{{\ell}^{2}}{t}\right)\right), \left({\color{blue}{k}}^{2}\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left({\ell}^{2}\right), t\right)\right), \left({k}^{2}\right)\right) \]

    8. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\left(\ell \cdot \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

    9. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left({k}^{2}\right)\right) \]

    10. unpow2 N/A

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \left(k \cdot \color{blue}{k}\right)\right) \]

    11. *-lowering-*.f64 29.8%

        \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{-1}{3}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), t\right)\right), \mathsf{*.f64}\left(k, \color{blue}{k}\right)\right) \]

13. Simplified 29.8%

    \[\leadsto \color{blue}{\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k}} \]
14. Step-by-step derivation

    1. associate-/l* N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}} \]

    2. *-commutative N/A

       \[\leadsto \frac{\frac{\ell \cdot \ell}{t}}{k \cdot k} \cdot \color{blue}{\frac{-1}{3}} \]

    3. associate-*l/ N/A

       \[\leadsto \frac{\frac{\ell}{t} \cdot \ell}{k \cdot k} \cdot \frac{-1}{3} \]

    4. associate-/r/ N/A

       \[\leadsto \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k} \cdot \frac{-1}{3} \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\right), \color{blue}{\frac{-1}{3}}\right) \]

    6. associate-/r/ N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell}{t} \cdot \ell}{k \cdot k}\right), \frac{-1}{3}\right) \]

    7. associate-*l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right), \frac{-1}{3}\right) \]

    8. associate-/l/ N/A

       \[\leadsto \mathsf{*.f64}\left(\left(\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t}\right), \frac{-1}{3}\right) \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\ell \cdot \ell\right), \left(\left(k \cdot k\right) \cdot t\right)\right), \frac{-1}{3}\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \left(\left(k \cdot k\right) \cdot t\right)\right), \frac{-1}{3}\right) \]

    11. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\left(k \cdot k\right), t\right)\right), \frac{-1}{3}\right) \]

    12. *-lowering-*.f64 29.8%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\ell, \ell\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(k, k\right), t\right)\right), \frac{-1}{3}\right) \]

15. Applied egg-rr 29.8%

    \[\leadsto \color{blue}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot t} \cdot -0.3333333333333333} \]

16. Final simplification 29.8%

    \[\leadsto -0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t \cdot \left(k \cdot k\right)} \]
17. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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))))