Toniolo and Linder, Equation (10-)

Percentage Accurate: 33.9% → 98.7%

Time: 20.9s

Alternatives: 12

Speedup: 28.1×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

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: 33.9% 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: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k} \cdot \frac{\ell}{\tan k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{\ell \cdot \cos k}{k}}{{\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 3.2e+30)
   (* (/ l (sin k)) (/ (* (/ 2.0 k) (/ l (tan k))) (* k t)))
   (* 2.0 (* (/ (/ l k) t) (/ (/ (* l (cos k)) k) (pow (sin k) 2.0))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.2e+30) {
		tmp = (l / sin(k)) * (((2.0 / k) * (l / tan(k))) / (k * t));
	} else {
		tmp = 2.0 * (((l / k) / t) * (((l * cos(k)) / k) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 3.2d+30) then
        tmp = (l / sin(k)) * (((2.0d0 / k) * (l / tan(k))) / (k * t))
    else
        tmp = 2.0d0 * (((l / k) / t) * (((l * cos(k)) / k) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 3.2e+30) {
		tmp = (l / Math.sin(k)) * (((2.0 / k) * (l / Math.tan(k))) / (k * t));
	} else {
		tmp = 2.0 * (((l / k) / t) * (((l * Math.cos(k)) / k) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 3.2e+30:
		tmp = (l / math.sin(k)) * (((2.0 / k) * (l / math.tan(k))) / (k * t))
	else:
		tmp = 2.0 * (((l / k) / t) * (((l * math.cos(k)) / k) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 3.2e+30)
		tmp = Float64(Float64(l / sin(k)) * Float64(Float64(Float64(2.0 / k) * Float64(l / tan(k))) / Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / t) * Float64(Float64(Float64(l * cos(k)) / k) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 3.2e+30)
		tmp = (l / sin(k)) * (((2.0 / k) * (l / tan(k))) / (k * t));
	else
		tmp = 2.0 * (((l / k) / t) * (((l * cos(k)) / k) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 3.2e+30], N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision] * N[(N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.19999999999999973e30

   1. Initial program 39.2%

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

      1. associate-*l* 39.2%

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

      2. associate-*l* 39.2%

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

      3. associate-/r* 38.7%

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

      4. associate-/r/ 38.6%

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

      5. *-commutative 38.6%

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

      6. times-frac 39.6%

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

      7. +-commutative 39.6%

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

      8. associate--l+ 47.5%

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

      9. metadata-eval 47.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 47.5%

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

      11. times-frac 51.6%

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

   3. Simplified 51.6%

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

   4. Taylor expanded in t around 0 85.3%

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

      1. associate-/r* 85.1%

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

      2. unpow2 85.1%

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

      3. associate-/r* 86.0%

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

   6. Simplified 86.0%

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

   7. Taylor expanded in k around 0 85.3%

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

      1. associate-/r* 85.1%

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

      2. unpow2 85.1%

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

      3. associate-/r* 86.0%

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

      4. associate-/r* 89.0%

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

   9. Simplified 89.0%

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

       1. associate-*l/ 90.7%

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

   11. Applied egg-rr 90.7%

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

       1. associate-*l/ 89.0%

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

       2. associate-*l* 92.6%

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

       3. *-commutative 92.6%

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

       4. associate-*l* 92.6%

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

   13. Simplified 92.6%

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

       1. associate-*l/ 96.7%

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

   15. Applied egg-rr 96.7%

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

   if 3.19999999999999973e30 < k

   1. Initial program 22.3%

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

      1. associate-*l* 22.3%

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

      2. associate-*l* 22.3%

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

      3. associate-/r* 22.2%

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

      4. associate-/r/ 22.1%

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

      5. *-commutative 22.1%

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

      6. times-frac 20.6%

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

      7. +-commutative 20.6%

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

      8. associate--l+ 28.4%

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

      9. metadata-eval 28.4%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 28.4%

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

      11. times-frac 28.4%

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

   3. Simplified 28.4%

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

   4. Taylor expanded in t around 0 55.7%

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

      1. associate-/r* 56.7%

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

      2. unpow2 56.7%

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

      3. associate-/r* 56.7%

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

   6. Simplified 56.7%

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

   7. Taylor expanded in k around inf 55.7%

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

      1. associate-/r* 57.0%

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

      2. *-commutative 57.0%

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

      3. unpow2 57.0%

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

      4. unpow2 57.0%

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

      5. *-commutative 57.0%

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

   9. Simplified 57.0%

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

   10. Taylor expanded in l around 0 57.0%

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

       1. *-commutative 57.0%

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

       2. unpow2 57.0%

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

       3. associate-*l* 57.0%

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

       4. unpow2 57.0%

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

       5. times-frac 93.4%

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

   12. Simplified 93.4%

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

       1. times-frac 99.2%

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

   14. Applied egg-rr 99.2%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k} \cdot \frac{\ell}{\tan k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{\ell \cdot \cos k}{k}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 82.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{k}}{\tan k \cdot t}\right)\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{+155}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k} + \ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2e-18)
   (* (/ l k) (* (/ l k) (/ (/ 2.0 k) (* (tan k) t))))
   (if (<= k 1.45e+155)
     (* (* (/ l (sin k)) (/ l (tan k))) (/ 2.0 (* t (* k k))))
     (*
      2.0
      (+
       (* (/ -0.16666666666666666 k) (/ (* l (/ l t)) k))
       (* l (/ l (* t (pow k 4.0)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-18) {
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (tan(k) * t)));
	} else if (k <= 1.45e+155) {
		tmp = ((l / sin(k)) * (l / tan(k))) * (2.0 / (t * (k * k)));
	} else {
		tmp = 2.0 * (((-0.16666666666666666 / k) * ((l * (l / t)) / k)) + (l * (l / (t * pow(k, 4.0)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.2d-18) then
        tmp = (l / k) * ((l / k) * ((2.0d0 / k) / (tan(k) * t)))
    else if (k <= 1.45d+155) then
        tmp = ((l / sin(k)) * (l / tan(k))) * (2.0d0 / (t * (k * k)))
    else
        tmp = 2.0d0 * ((((-0.16666666666666666d0) / k) * ((l * (l / t)) / k)) + (l * (l / (t * (k ** 4.0d0)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-18) {
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (Math.tan(k) * t)));
	} else if (k <= 1.45e+155) {
		tmp = ((l / Math.sin(k)) * (l / Math.tan(k))) * (2.0 / (t * (k * k)));
	} else {
		tmp = 2.0 * (((-0.16666666666666666 / k) * ((l * (l / t)) / k)) + (l * (l / (t * Math.pow(k, 4.0)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.2e-18:
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (math.tan(k) * t)))
	elif k <= 1.45e+155:
		tmp = ((l / math.sin(k)) * (l / math.tan(k))) * (2.0 / (t * (k * k)))
	else:
		tmp = 2.0 * (((-0.16666666666666666 / k) * ((l * (l / t)) / k)) + (l * (l / (t * math.pow(k, 4.0)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2e-18)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(2.0 / k) / Float64(tan(k) * t))));
	elseif (k <= 1.45e+155)
		tmp = Float64(Float64(Float64(l / sin(k)) * Float64(l / tan(k))) * Float64(2.0 / Float64(t * Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(-0.16666666666666666 / k) * Float64(Float64(l * Float64(l / t)) / k)) + Float64(l * Float64(l / Float64(t * (k ^ 4.0))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.2e-18)
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (tan(k) * t)));
	elseif (k <= 1.45e+155)
		tmp = ((l / sin(k)) * (l / tan(k))) * (2.0 / (t * (k * k)));
	else
		tmp = 2.0 * (((-0.16666666666666666 / k) * ((l * (l / t)) / k)) + (l * (l / (t * (k ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.2e-18], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.45e+155], N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(-0.16666666666666666 / k), $MachinePrecision] * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.19999999999999997e-18

   1. Initial program 38.9%

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

      1. associate-*l* 38.9%

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

      2. associate-*l* 38.9%

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

      3. associate-/r* 38.3%

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

      4. associate-/r/ 38.3%

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

      5. *-commutative 38.3%

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

      6. times-frac 39.3%

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

      7. +-commutative 39.3%

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

      8. associate--l+ 47.5%

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

      9. metadata-eval 47.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 47.5%

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

      11. times-frac 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in t around 0 84.9%

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

      1. associate-/r* 84.6%

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

      2. unpow2 84.6%

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

      3. associate-/r* 85.6%

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

   6. Simplified 85.6%

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

   7. Taylor expanded in k around 0 84.9%

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

      1. associate-/r* 84.6%

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

      2. unpow2 84.6%

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

      3. associate-/r* 85.6%

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

      4. associate-/r* 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in k around 0 76.2%

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

       1. expm1-log1p-u 40.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)\right)} \]

       2. expm1-udef 36.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)} - 1} \]

   12. Applied egg-rr 36.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 40.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)\right)} \]

       2. expm1-log1p 76.2%

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

       3. *-commutative 76.2%

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

       4. associate-*l* 76.3%

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

       5. times-frac 76.3%

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

       6. *-commutative 76.3%

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

       7. associate-*l* 76.3%

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

       8. times-frac 76.3%

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

   14. Simplified 76.3%

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

   if 1.19999999999999997e-18 < k < 1.45e155

   1. Initial program 23.5%

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

      1. associate-*l* 23.5%

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

      2. associate-*l* 23.5%

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

      3. associate-/r* 23.5%

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

      4. associate-/r/ 23.5%

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

      5. *-commutative 23.5%

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

      6. times-frac 22.8%

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

      7. +-commutative 22.8%

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

      8. associate--l+ 39.5%

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

      9. metadata-eval 39.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 39.5%

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

      11. times-frac 39.6%

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

   3. Simplified 39.6%

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

   4. Taylor expanded in t around 0 90.4%

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

      1. unpow2 90.4%

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

   6. Simplified 90.4%

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

   if 1.45e155 < k

   1. Initial program 25.5%

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

      1. associate-*l* 25.5%

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

      2. associate-*l* 25.5%

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

      3. associate-/r* 25.3%

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

      4. associate-/r/ 25.3%

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

      5. *-commutative 25.3%

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

      6. times-frac 23.3%

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

      7. +-commutative 23.3%

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

      8. associate--l+ 23.3%

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

      9. metadata-eval 23.3%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 23.3%

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

      11. times-frac 23.3%

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

   3. Simplified 23.3%

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

   4. Taylor expanded in t around 0 36.3%

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

      1. associate-/r* 36.3%

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

      2. unpow2 36.3%

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

      3. associate-/r* 36.3%

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

   6. Simplified 36.3%

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

   7. Taylor expanded in k around inf 36.3%

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

      1. associate-/r* 36.3%

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

      2. *-commutative 36.3%

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

      3. unpow2 36.3%

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

      4. unpow2 36.3%

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

      5. *-commutative 36.3%

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

   9. Simplified 36.3%

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

   10. Taylor expanded in k around 0 36.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

   12. Simplified 45.2%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k} + \frac{\ell}{{k}^{4} \cdot t} \cdot \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{k}}{\tan k \cdot t}\right)\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{+155}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k} + \ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \]

Alternative 3: 83.4% accurate, 2.0× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* k (* k t))) (* (/ l (sin k)) (/ l (tan k)))))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = (2.0d0 / (k * (k * t))) * ((l / sin(k)) * (l / tan(k)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 / N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

   1. associate-*l* 35.0%

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

   2. associate-*l* 35.0%

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

   3. associate-/r* 34.6%

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

   4. associate-/r/ 34.5%

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

   5. *-commutative 34.5%

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

   6. times-frac 34.9%

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

   7. +-commutative 34.9%

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

   8. associate--l+ 42.8%

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

   9. metadata-eval 42.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 42.8%

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

   11. times-frac 45.8%

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

3. Simplified 45.8%

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

4. Taylor expanded in t around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

6. Simplified 78.7%

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

   1. add-cbrt-cube 67.9%

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

   2. associate-/l/ 67.9%

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

   3. associate-/l/ 67.9%

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

   4. associate-/l/ 69.0%

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

8. Applied egg-rr 69.0%

   \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{\frac{2}{k}}{t \cdot k} \cdot \frac{\frac{2}{k}}{t \cdot k}\right) \cdot \frac{\frac{2}{k}}{t \cdot k}}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
9. Step-by-step derivation

   1. add-cbrt-cube 85.1%

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

   2. associate-/l/ 84.3%

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

   3. *-commutative 84.3%

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

10. Applied egg-rr 84.3%

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

11. Final simplification 84.3%

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

Alternative 4: 93.4% accurate, 2.0× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ l (sin k)) (* (/ l k) (/ (/ 2.0 k) (* (tan k) t)))))

C

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

Fortran

NOTE: k should be positive before calling this function
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)) * ((l / k) * ((2.0d0 / k) / (tan(k) * t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

   1. associate-*l* 35.0%

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

   2. associate-*l* 35.0%

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

   3. associate-/r* 34.6%

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

   4. associate-/r/ 34.5%

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

   5. *-commutative 34.5%

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

   6. times-frac 34.9%

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

   7. +-commutative 34.9%

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

   8. associate--l+ 42.8%

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

   9. metadata-eval 42.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 42.8%

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

   11. times-frac 45.8%

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

3. Simplified 45.8%

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

4. Taylor expanded in t around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

6. Simplified 78.7%

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

7. Taylor expanded in k around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

   4. associate-/r* 85.1%

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

9. Simplified 85.1%

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

    1. associate-*l/ 88.8%

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

11. Applied egg-rr 88.8%

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

    1. associate-*l/ 85.1%

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

    2. associate-*l* 89.4%

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

    3. *-commutative 89.4%

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

    4. associate-*l* 89.4%

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

13. Simplified 89.4%

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

    1. expm1-log1p-u 65.6%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \frac{\ell}{\tan k}\right)\right)} \]

    2. expm1-udef 54.0%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \frac{\ell}{\tan k}\right)} - 1\right)} \]

15. Applied egg-rr 54.0%

    \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \frac{\ell}{\tan k}\right)} - 1\right)} \]
16. Step-by-step derivation

    1. expm1-def 65.6%

       \[\leadsto \frac{\ell}{\sin k} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \frac{\ell}{\tan k}\right)\right)} \]

    2. expm1-log1p 89.4%

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

    3. times-frac 93.6%

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

    4. *-commutative 93.6%

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

    5. associate-*l* 93.6%

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

    6. times-frac 93.9%

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

17. Simplified 93.9%

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

18. Final simplification 93.9%

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

Alternative 5: 94.3% accurate, 2.0× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ l (sin k)) (/ (* (/ 2.0 k) (/ l (tan k))) (* k t))))

C

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

Fortran

NOTE: k should be positive before calling this function
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)) * (((2.0d0 / k) * (l / tan(k))) / (k * t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

   1. associate-*l* 35.0%

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

   2. associate-*l* 35.0%

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

   3. associate-/r* 34.6%

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

   4. associate-/r/ 34.5%

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

   5. *-commutative 34.5%

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

   6. times-frac 34.9%

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

   7. +-commutative 34.9%

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

   8. associate--l+ 42.8%

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

   9. metadata-eval 42.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 42.8%

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

   11. times-frac 45.8%

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

3. Simplified 45.8%

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

4. Taylor expanded in t around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

6. Simplified 78.7%

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

7. Taylor expanded in k around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

   4. associate-/r* 85.1%

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

9. Simplified 85.1%

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

    1. associate-*l/ 88.8%

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

11. Applied egg-rr 88.8%

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

    1. associate-*l/ 85.1%

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

    2. associate-*l* 89.4%

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

    3. *-commutative 89.4%

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

    4. associate-*l* 89.4%

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

13. Simplified 89.4%

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

    1. associate-*l/ 95.4%

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

15. Applied egg-rr 95.4%

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

16. Final simplification 95.4%

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

Alternative 6: 73.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{k}}{\tan k \cdot t}\right)\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k} + \ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.8e-19)
   (* (/ l k) (* (/ l k) (/ (/ 2.0 k) (* (tan k) t))))
   (if (<= k 2.8e+78)
     (* 2.0 (/ (/ (* (cos k) (* l l)) (* k k)) (* t (* k k))))
     (*
      2.0
      (+
       (* (/ -0.16666666666666666 k) (/ (* l (/ l t)) k))
       (* l (/ l (* t (pow k 4.0)))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-19) {
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (tan(k) * t)));
	} else if (k <= 2.8e+78) {
		tmp = 2.0 * (((cos(k) * (l * l)) / (k * k)) / (t * (k * k)));
	} else {
		tmp = 2.0 * (((-0.16666666666666666 / k) * ((l * (l / t)) / k)) + (l * (l / (t * pow(k, 4.0)))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 5.8d-19) then
        tmp = (l / k) * ((l / k) * ((2.0d0 / k) / (tan(k) * t)))
    else if (k <= 2.8d+78) then
        tmp = 2.0d0 * (((cos(k) * (l * l)) / (k * k)) / (t * (k * k)))
    else
        tmp = 2.0d0 * ((((-0.16666666666666666d0) / k) * ((l * (l / t)) / k)) + (l * (l / (t * (k ** 4.0d0)))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-19) {
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (Math.tan(k) * t)));
	} else if (k <= 2.8e+78) {
		tmp = 2.0 * (((Math.cos(k) * (l * l)) / (k * k)) / (t * (k * k)));
	} else {
		tmp = 2.0 * (((-0.16666666666666666 / k) * ((l * (l / t)) / k)) + (l * (l / (t * Math.pow(k, 4.0)))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 5.8e-19:
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (math.tan(k) * t)))
	elif k <= 2.8e+78:
		tmp = 2.0 * (((math.cos(k) * (l * l)) / (k * k)) / (t * (k * k)))
	else:
		tmp = 2.0 * (((-0.16666666666666666 / k) * ((l * (l / t)) / k)) + (l * (l / (t * math.pow(k, 4.0)))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.8e-19)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(2.0 / k) / Float64(tan(k) * t))));
	elseif (k <= 2.8e+78)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l * l)) / Float64(k * k)) / Float64(t * Float64(k * k))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(-0.16666666666666666 / k) * Float64(Float64(l * Float64(l / t)) / k)) + Float64(l * Float64(l / Float64(t * (k ^ 4.0))))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 5.8e-19)
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (tan(k) * t)));
	elseif (k <= 2.8e+78)
		tmp = 2.0 * (((cos(k) * (l * l)) / (k * k)) / (t * (k * k)));
	else
		tmp = 2.0 * (((-0.16666666666666666 / k) * ((l * (l / t)) / k)) + (l * (l / (t * (k ^ 4.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.8e-19], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.8e+78], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(-0.16666666666666666 / k), $MachinePrecision] * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + N[(l * N[(l / N[(t * N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.8e-19

   1. Initial program 38.9%

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

      1. associate-*l* 38.9%

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

      2. associate-*l* 38.9%

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

      3. associate-/r* 38.3%

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

      4. associate-/r/ 38.3%

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

      5. *-commutative 38.3%

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

      6. times-frac 39.3%

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

      7. +-commutative 39.3%

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

      8. associate--l+ 47.5%

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

      9. metadata-eval 47.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 47.5%

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

      11. times-frac 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in t around 0 84.9%

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

      1. associate-/r* 84.6%

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

      2. unpow2 84.6%

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

      3. associate-/r* 85.6%

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

   6. Simplified 85.6%

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

   7. Taylor expanded in k around 0 84.9%

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

      1. associate-/r* 84.6%

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

      2. unpow2 84.6%

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

      3. associate-/r* 85.6%

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

      4. associate-/r* 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in k around 0 76.2%

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

       1. expm1-log1p-u 40.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)\right)} \]

       2. expm1-udef 36.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)} - 1} \]

   12. Applied egg-rr 36.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 40.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)\right)} \]

       2. expm1-log1p 76.2%

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

       3. *-commutative 76.2%

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

       4. associate-*l* 76.3%

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

       5. times-frac 76.3%

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

       6. *-commutative 76.3%

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

       7. associate-*l* 76.3%

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

       8. times-frac 76.3%

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

   14. Simplified 76.3%

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

   if 5.8e-19 < k < 2.8000000000000001e78

   1. Initial program 34.9%

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

      1. associate-*l* 34.9%

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

      2. associate-*l* 34.9%

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

      3. associate-/r* 34.9%

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

      4. associate-/r/ 34.9%

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

      5. *-commutative 34.9%

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

      6. times-frac 34.9%

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

      7. +-commutative 34.9%

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

      8. associate--l+ 46.6%

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

      9. metadata-eval 46.6%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 46.6%

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

      11. times-frac 46.8%

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

   3. Simplified 46.8%

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

   4. Taylor expanded in t around 0 94.3%

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

      1. associate-/r* 94.3%

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

      2. unpow2 94.3%

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

      3. associate-/r* 94.2%

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

   6. Simplified 94.2%

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

   7. Taylor expanded in k around inf 94.2%

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

      1. associate-/r* 94.2%

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

      2. *-commutative 94.2%

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

      3. unpow2 94.2%

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

      4. unpow2 94.2%

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

      5. *-commutative 94.2%

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

   9. Simplified 94.2%

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

   10. Taylor expanded in k around 0 66.8%

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

       1. *-commutative 66.8%

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

       2. unpow2 66.8%

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

   12. Simplified 66.8%

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

   if 2.8000000000000001e78 < k

   1. Initial program 21.3%

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

      1. associate-*l* 21.3%

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

      2. associate-*l* 21.3%

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

      3. associate-/r* 21.2%

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

      4. associate-/r/ 21.2%

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

      5. *-commutative 21.2%

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

      6. times-frac 19.3%

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

      7. +-commutative 19.3%

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

      8. associate--l+ 25.0%

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

      9. metadata-eval 25.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 25.0%

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

      11. times-frac 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in t around 0 48.3%

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

      1. associate-/r* 49.5%

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

      2. unpow2 49.5%

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

      3. associate-/r* 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in k around inf 48.4%

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

      1. associate-/r* 49.9%

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

      2. *-commutative 49.9%

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

      3. unpow2 49.9%

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

      4. unpow2 49.9%

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

      5. *-commutative 49.9%

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

   9. Simplified 49.9%

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

   10. Taylor expanded in k around 0 36.1%

       \[\leadsto 2 \cdot \color{blue}{\left(\left(-0.5 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t} + \frac{{\ell}^{2}}{{k}^{4} \cdot t}\right) - -0.3333333333333333 \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot t}\right)} \]

   12. Simplified 44.3%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k} + \frac{\ell}{{k}^{4} \cdot t} \cdot \ell\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-19}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{k}}{\tan k \cdot t}\right)\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+78}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{-0.16666666666666666}{k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{k} + \ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \end{array} \]

Alternative 7: 74.0% accurate, 3.5× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{k}}{\tan k \cdot t}\right)\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{k}\right)}{k \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.2e-18)
   (* (/ l k) (* (/ l k) (/ (/ 2.0 k) (* (tan k) t))))
   (if (<= k 1.8e+138)
     (* 2.0 (/ (/ (* (cos k) (* l l)) (* k k)) (* t (* k k))))
     (/ (* (/ 2.0 k) (* (/ l (tan k)) (/ l k))) (* k t)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-18) {
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (tan(k) * t)));
	} else if (k <= 1.8e+138) {
		tmp = 2.0 * (((cos(k) * (l * l)) / (k * k)) / (t * (k * k)));
	} else {
		tmp = ((2.0 / k) * ((l / tan(k)) * (l / k))) / (k * t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
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 <= 1.2d-18) then
        tmp = (l / k) * ((l / k) * ((2.0d0 / k) / (tan(k) * t)))
    else if (k <= 1.8d+138) then
        tmp = 2.0d0 * (((cos(k) * (l * l)) / (k * k)) / (t * (k * k)))
    else
        tmp = ((2.0d0 / k) * ((l / tan(k)) * (l / k))) / (k * t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.2e-18) {
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (Math.tan(k) * t)));
	} else if (k <= 1.8e+138) {
		tmp = 2.0 * (((Math.cos(k) * (l * l)) / (k * k)) / (t * (k * k)));
	} else {
		tmp = ((2.0 / k) * ((l / Math.tan(k)) * (l / k))) / (k * t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.2e-18:
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (math.tan(k) * t)))
	elif k <= 1.8e+138:
		tmp = 2.0 * (((math.cos(k) * (l * l)) / (k * k)) / (t * (k * k)))
	else:
		tmp = ((2.0 / k) * ((l / math.tan(k)) * (l / k))) / (k * t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.2e-18)
		tmp = Float64(Float64(l / k) * Float64(Float64(l / k) * Float64(Float64(2.0 / k) / Float64(tan(k) * t))));
	elseif (k <= 1.8e+138)
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) * Float64(l * l)) / Float64(k * k)) / Float64(t * Float64(k * k))));
	else
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(Float64(l / tan(k)) * Float64(l / k))) / Float64(k * t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.2e-18)
		tmp = (l / k) * ((l / k) * ((2.0 / k) / (tan(k) * t)));
	elseif (k <= 1.8e+138)
		tmp = 2.0 * (((cos(k) * (l * l)) / (k * k)) / (t * (k * k)));
	else
		tmp = ((2.0 / k) * ((l / tan(k)) * (l / k))) / (k * t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.2e-18], N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / k), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.8e+138], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.19999999999999997e-18

   1. Initial program 38.9%

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

      1. associate-*l* 38.9%

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

      2. associate-*l* 38.9%

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

      3. associate-/r* 38.3%

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

      4. associate-/r/ 38.3%

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

      5. *-commutative 38.3%

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

      6. times-frac 39.3%

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

      7. +-commutative 39.3%

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

      8. associate--l+ 47.5%

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

      9. metadata-eval 47.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 47.5%

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

      11. times-frac 51.7%

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

   3. Simplified 51.7%

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

   4. Taylor expanded in t around 0 84.9%

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

      1. associate-/r* 84.6%

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

      2. unpow2 84.6%

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

      3. associate-/r* 85.6%

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

   6. Simplified 85.6%

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

   7. Taylor expanded in k around 0 84.9%

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

      1. associate-/r* 84.6%

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

      2. unpow2 84.6%

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

      3. associate-/r* 85.6%

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

      4. associate-/r* 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in k around 0 76.2%

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

       1. expm1-log1p-u 40.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)\right)} \]

       2. expm1-udef 36.9%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)} - 1} \]

   12. Applied egg-rr 36.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)} - 1} \]
   13. Step-by-step derivation

       1. expm1-def 40.5%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\tan k}\right)\right)\right)} \]

       2. expm1-log1p 76.2%

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

       3. *-commutative 76.2%

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

       4. associate-*l* 76.3%

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

       5. times-frac 76.3%

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

       6. *-commutative 76.3%

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

       7. associate-*l* 76.3%

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

       8. times-frac 76.3%

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

   14. Simplified 76.3%

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

   if 1.19999999999999997e-18 < k < 1.8000000000000001e138

   1. Initial program 25.9%

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

      1. associate-*l* 25.9%

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

      2. associate-*l* 25.9%

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

      3. associate-/r* 25.9%

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

      4. associate-/r/ 25.9%

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

      5. *-commutative 25.9%

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

      6. times-frac 25.1%

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

      7. +-commutative 25.1%

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

      8. associate--l+ 39.9%

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

      9. metadata-eval 39.9%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 39.9%

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

      11. times-frac 40.0%

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

   3. Simplified 40.0%

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

   4. Taylor expanded in t around 0 92.9%

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

      1. associate-/r* 95.2%

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

      2. unpow2 95.2%

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

      3. associate-/r* 95.2%

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

   6. Simplified 95.2%

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

   7. Taylor expanded in k around inf 92.8%

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

      1. associate-/r* 92.4%

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

      2. *-commutative 92.4%

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

      3. unpow2 92.4%

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

      4. unpow2 92.4%

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

      5. *-commutative 92.4%

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

   9. Simplified 92.4%

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

   10. Taylor expanded in k around 0 58.1%

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

       1. *-commutative 58.1%

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

       2. unpow2 58.1%

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

   12. Simplified 58.1%

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

   if 1.8000000000000001e138 < k

   1. Initial program 23.8%

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

      1. associate-*l* 23.8%

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

      2. associate-*l* 23.8%

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

      3. associate-/r* 23.7%

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

      4. associate-/r/ 23.6%

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

      5. *-commutative 23.6%

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

      6. times-frac 21.8%

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

      7. +-commutative 21.8%

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

      8. associate--l+ 24.1%

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

      9. metadata-eval 24.1%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 24.1%

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

      11. times-frac 24.1%

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

   3. Simplified 24.1%

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

   4. Taylor expanded in t around 0 38.5%

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

      1. associate-/r* 38.5%

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

      2. unpow2 38.5%

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

      3. associate-/r* 38.5%

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

   6. Simplified 38.5%

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

   7. Taylor expanded in k around 0 38.5%

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

      1. associate-/r* 38.5%

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

      2. unpow2 38.5%

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

      3. associate-/r* 38.5%

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

      4. associate-/r* 63.3%

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

   9. Simplified 63.3%

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

   10. Taylor expanded in k around 0 41.9%

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

       1. associate-*l/ 42.0%

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

   12. Applied egg-rr 42.0%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{2}{k}}{\tan k \cdot t}\right)\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+138}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(k \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{k}\right)}{k \cdot t}\\ \end{array} \]

Alternative 8: 71.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\frac{2}{k}}{k \cdot t}\\ \mathbf{if}\;k \leq 3.9 \cdot 10^{+174}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(k \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\tan k} \cdot \frac{\ell}{k}\right) \cdot t_1\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ 2.0 k) (* k t))))
   (if (<= k 3.9e+174)
     (* t_1 (* (/ l k) (+ (/ l k) (* -0.3333333333333333 (* k l)))))
     (* (* (/ l (tan k)) (/ l k)) t_1))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (2.0 / k) / (k * t);
	double tmp;
	if (k <= 3.9e+174) {
		tmp = t_1 * ((l / k) * ((l / k) + (-0.3333333333333333 * (k * l))));
	} else {
		tmp = ((l / tan(k)) * (l / k)) * t_1;
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 / k) / (k * t)
    if (k <= 3.9d+174) then
        tmp = t_1 * ((l / k) * ((l / k) + ((-0.3333333333333333d0) * (k * l))))
    else
        tmp = ((l / tan(k)) * (l / k)) * t_1
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (2.0 / k) / (k * t);
	double tmp;
	if (k <= 3.9e+174) {
		tmp = t_1 * ((l / k) * ((l / k) + (-0.3333333333333333 * (k * l))));
	} else {
		tmp = ((l / Math.tan(k)) * (l / k)) * t_1;
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (2.0 / k) / (k * t)
	tmp = 0
	if k <= 3.9e+174:
		tmp = t_1 * ((l / k) * ((l / k) + (-0.3333333333333333 * (k * l))))
	else:
		tmp = ((l / math.tan(k)) * (l / k)) * t_1
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(2.0 / k) / Float64(k * t))
	tmp = 0.0
	if (k <= 3.9e+174)
		tmp = Float64(t_1 * Float64(Float64(l / k) * Float64(Float64(l / k) + Float64(-0.3333333333333333 * Float64(k * l)))));
	else
		tmp = Float64(Float64(Float64(l / tan(k)) * Float64(l / k)) * t_1);
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (2.0 / k) / (k * t);
	tmp = 0.0;
	if (k <= 3.9e+174)
		tmp = t_1 * ((l / k) * ((l / k) + (-0.3333333333333333 * (k * l))));
	else
		tmp = ((l / tan(k)) * (l / k)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 3.9e+174], N[(t$95$1 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] + N[(-0.3333333333333333 * N[(k * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 3.89999999999999981e174

   1. Initial program 35.6%

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

      1. associate-*l* 35.6%

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

      2. associate-*l* 35.6%

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

      3. associate-/r* 35.2%

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

      4. associate-/r/ 35.2%

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

      5. *-commutative 35.2%

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

      6. times-frac 35.9%

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

      7. +-commutative 35.9%

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

      8. associate--l+ 45.0%

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

      9. metadata-eval 45.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 45.0%

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

      11. times-frac 48.5%

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

   3. Simplified 48.5%

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

   4. Taylor expanded in t around 0 83.0%

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

      1. associate-/r* 83.1%

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

      2. unpow2 83.1%

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

      3. associate-/r* 83.9%

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

   6. Simplified 83.9%

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

   7. Taylor expanded in k around 0 83.0%

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

      1. associate-/r* 83.1%

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

      2. unpow2 83.1%

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

      3. associate-/r* 83.9%

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

      4. associate-/r* 87.8%

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

   9. Simplified 87.8%

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

   10. Taylor expanded in k around 0 70.4%

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

   11. Taylor expanded in k around 0 68.5%

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

   if 3.89999999999999981e174 < k

   1. Initial program 30.4%

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

      1. associate-*l* 30.4%

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

      2. associate-*l* 30.4%

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

      3. associate-/r* 30.3%

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

      4. associate-/r/ 30.2%

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

      5. *-commutative 30.2%

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

      6. times-frac 27.8%

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

      7. +-commutative 27.8%

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

      8. associate--l+ 27.8%

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

      9. metadata-eval 27.8%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 27.8%

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

      11. times-frac 27.8%

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

   3. Simplified 27.8%

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

   4. Taylor expanded in t around 0 43.3%

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

      1. associate-/r* 43.3%

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

      2. unpow2 43.3%

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

      3. associate-/r* 43.3%

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

   6. Simplified 43.3%

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

   7. Taylor expanded in k around 0 43.3%

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

      1. associate-/r* 43.3%

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

      2. unpow2 43.3%

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

      3. associate-/r* 43.3%

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

      4. associate-/r* 66.7%

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

   9. Simplified 66.7%

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

   10. Taylor expanded in k around 0 50.3%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.9 \cdot 10^{+174}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(k \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{\tan k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{2}{k}}{k \cdot t}\\ \end{array} \]

Alternative 9: 71.7% accurate, 3.7× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (/ (* (/ 2.0 k) (* (/ l (tan k)) (/ l k))) (* k t)))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((2.0d0 / k) * ((l / tan(k)) * (l / k))) / (k * t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(2.0 / k), $MachinePrecision] * N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

   1. associate-*l* 35.0%

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

   2. associate-*l* 35.0%

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

   3. associate-/r* 34.6%

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

   4. associate-/r/ 34.5%

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

   5. *-commutative 34.5%

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

   6. times-frac 34.9%

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

   7. +-commutative 34.9%

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

   8. associate--l+ 42.8%

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

   9. metadata-eval 42.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 42.8%

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

   11. times-frac 45.8%

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

3. Simplified 45.8%

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

4. Taylor expanded in t around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

6. Simplified 78.7%

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

7. Taylor expanded in k around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

   4. associate-/r* 85.1%

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

9. Simplified 85.1%

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

10. Taylor expanded in k around 0 67.8%

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

    1. associate-*l/ 68.2%

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

12. Applied egg-rr 68.2%

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

13. Final simplification 68.2%

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

Alternative 10: 71.8% accurate, 18.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := \frac{\frac{2}{k}}{k \cdot t}\\ \mathbf{if}\;k \leq 2.4 \cdot 10^{+187}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(k \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (/ 2.0 k) (* k t))))
   (if (<= k 2.4e+187)
     (* t_1 (* (/ l k) (+ (/ l k) (* -0.3333333333333333 (* k l)))))
     (* t_1 (* (/ l k) (/ l k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = (2.0 / k) / (k * t);
	double tmp;
	if (k <= 2.4e+187) {
		tmp = t_1 * ((l / k) * ((l / k) + (-0.3333333333333333 * (k * l))));
	} else {
		tmp = t_1 * ((l / k) * (l / k));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (2.0d0 / k) / (k * t)
    if (k <= 2.4d+187) then
        tmp = t_1 * ((l / k) * ((l / k) + ((-0.3333333333333333d0) * (k * l))))
    else
        tmp = t_1 * ((l / k) * (l / k))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = (2.0 / k) / (k * t);
	double tmp;
	if (k <= 2.4e+187) {
		tmp = t_1 * ((l / k) * ((l / k) + (-0.3333333333333333 * (k * l))));
	} else {
		tmp = t_1 * ((l / k) * (l / k));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	t_1 = (2.0 / k) / (k * t)
	tmp = 0
	if k <= 2.4e+187:
		tmp = t_1 * ((l / k) * ((l / k) + (-0.3333333333333333 * (k * l))))
	else:
		tmp = t_1 * ((l / k) * (l / k))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(Float64(2.0 / k) / Float64(k * t))
	tmp = 0.0
	if (k <= 2.4e+187)
		tmp = Float64(t_1 * Float64(Float64(l / k) * Float64(Float64(l / k) + Float64(-0.3333333333333333 * Float64(k * l)))));
	else
		tmp = Float64(t_1 * Float64(Float64(l / k) * Float64(l / k)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	t_1 = (2.0 / k) / (k * t);
	tmp = 0.0;
	if (k <= 2.4e+187)
		tmp = t_1 * ((l / k) * ((l / k) + (-0.3333333333333333 * (k * l))));
	else
		tmp = t_1 * ((l / k) * (l / k));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.4e+187], N[(t$95$1 * N[(N[(l / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] + N[(-0.3333333333333333 * N[(k * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999985e187

   1. Initial program 35.3%

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

      1. associate-*l* 35.3%

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

      2. associate-*l* 35.3%

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

      3. associate-/r* 34.8%

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

      4. associate-/r/ 34.8%

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

      5. *-commutative 34.8%

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

      6. times-frac 35.2%

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

      7. +-commutative 35.2%

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

      8. associate--l+ 44.0%

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

      9. metadata-eval 44.0%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 44.0%

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

      11. times-frac 47.4%

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

   3. Simplified 47.4%

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

   4. Taylor expanded in t around 0 81.7%

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

      1. associate-/r* 81.8%

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

      2. unpow2 81.8%

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

      3. associate-/r* 82.6%

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

   6. Simplified 82.6%

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

   7. Taylor expanded in k around 0 81.7%

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

      1. associate-/r* 81.8%

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

      2. unpow2 81.8%

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

      3. associate-/r* 82.6%

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

      4. associate-/r* 87.2%

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

   9. Simplified 87.2%

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

   10. Taylor expanded in k around 0 69.8%

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

   11. Taylor expanded in k around 0 67.9%

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

   if 2.39999999999999985e187 < k

   1. Initial program 32.4%

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

      1. associate-*l* 32.4%

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

      2. associate-*l* 32.4%

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

      3. associate-/r* 32.1%

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

      4. associate-/r/ 32.1%

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

      5. *-commutative 32.1%

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

      6. times-frac 32.5%

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

      7. +-commutative 32.5%

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

      8. associate--l+ 32.5%

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

      9. metadata-eval 32.5%

         \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

      10. +-rgt-identity 32.5%

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

      11. times-frac 32.5%

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

   3. Simplified 32.5%

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

   4. Taylor expanded in t around 0 47.2%

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

      1. associate-/r* 47.2%

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

      2. unpow2 47.2%

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

      3. associate-/r* 47.2%

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

   6. Simplified 47.2%

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

   7. Taylor expanded in k around 0 47.2%

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

      1. associate-/r* 47.2%

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

      2. unpow2 47.2%

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

      3. associate-/r* 47.2%

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

      4. associate-/r* 67.8%

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

   9. Simplified 67.8%

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

   10. Taylor expanded in k around 0 47.2%

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

       1. unpow2 47.2%

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

       2. unpow2 47.2%

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

   12. Simplified 47.2%

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

       1. frac-times 51.8%

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

   14. Applied egg-rr 51.8%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} + -0.3333333333333333 \cdot \left(k \cdot \ell\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \end{array} \]

Alternative 11: 62.6% accurate, 28.1× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ 2.0 (* t (* k k))) (/ (* l l) (* k k))))

C

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

Fortran

NOTE: k should be positive before calling this function
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 * (k * k))) * ((l * l) / (k * k))
end function

Java

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

Python

k = abs(k)
def code(t, l, k):
	return (2.0 / (t * (k * k))) * ((l * l) / (k * k))

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

   1. associate-*l* 35.0%

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

   2. associate-*l* 35.0%

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

   3. associate-/r* 34.6%

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

   4. associate-/r/ 34.5%

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

   5. *-commutative 34.5%

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

   6. times-frac 34.9%

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

   7. +-commutative 34.9%

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

   8. associate--l+ 42.8%

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

   9. metadata-eval 42.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 42.8%

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

   11. times-frac 45.8%

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

3. Simplified 45.8%

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

4. Taylor expanded in t around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

6. Simplified 78.7%

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

7. Taylor expanded in k around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

   4. associate-/r* 85.1%

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

9. Simplified 85.1%

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

10. Taylor expanded in k around 0 58.9%

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

    1. unpow2 58.9%

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

    2. unpow2 58.9%

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

12. Simplified 58.9%

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

13. Taylor expanded in k around 0 58.9%

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

    1. *-commutative 58.9%

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

    2. unpow2 58.9%

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

15. Simplified 58.9%

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

16. Final simplification 58.9%

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

Alternative 12: 70.7% accurate, 28.1× speedup

Math

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

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (* (/ (/ 2.0 k) (* k t)) (* (/ l k) (/ l k))))

C

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

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = ((2.0d0 / k) / (k * t)) * ((l / k) * (l / k))
end function

Java

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

Python

k = abs(k)
def code(t, l, k):
	return ((2.0 / k) / (k * t)) * ((l / k) * (l / k))

Julia

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

MATLAB

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

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 35.0%

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

   1. associate-*l* 35.0%

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

   2. associate-*l* 35.0%

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

   3. associate-/r* 34.6%

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

   4. associate-/r/ 34.5%

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

   5. *-commutative 34.5%

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

   6. times-frac 34.9%

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

   7. +-commutative 34.9%

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

   8. associate--l+ 42.8%

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

   9. metadata-eval 42.8%

      \[\leadsto \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]

   10. +-rgt-identity 42.8%

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

   11. times-frac 45.8%

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

3. Simplified 45.8%

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

4. Taylor expanded in t around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

6. Simplified 78.7%

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

7. Taylor expanded in k around 0 77.9%

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

   1. associate-/r* 78.0%

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

   2. unpow2 78.0%

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

   3. associate-/r* 78.7%

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

   4. associate-/r* 85.1%

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

9. Simplified 85.1%

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

10. Taylor expanded in k around 0 58.9%

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

    1. unpow2 58.9%

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

    2. unpow2 58.9%

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

12. Simplified 58.9%

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

    1. frac-times 66.9%

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

14. Applied egg-rr 66.9%

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

15. Final simplification 66.9%

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

Reproduce

herbie shell --seed 2023258 
(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))))