Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.6% → 85.5%

Time: 38.2s

Alternatives: 22

Speedup: 28.1×

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)}{t_1}\\ \mathbf{elif}\;t \leq 7200:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -3.9e-8)
     (/
      (* (/ 2.0 (pow (* (* t (cbrt (sin k))) (cbrt (tan k))) 3.0)) (* l l))
      t_1)
     (if (<= t 7200.0)
       (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))
       (/ (/ (/ 2.0 (tan k)) (* (sin k) (pow (/ (pow t 1.5) l) 2.0))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -3.9e-8) {
		tmp = ((2.0 / pow(((t * cbrt(sin(k))) * cbrt(tan(k))), 3.0)) * (l * l)) / t_1;
	} else if (t <= 7200.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else {
		tmp = ((2.0 / tan(k)) / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -3.9e-8) {
		tmp = ((2.0 / Math.pow(((t * Math.cbrt(Math.sin(k))) * Math.cbrt(Math.tan(k))), 3.0)) * (l * l)) / t_1;
	} else if (t <= 7200.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	} else {
		tmp = ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -3.9e-8)
		tmp = Float64(Float64(Float64(2.0 / (Float64(Float64(t * cbrt(sin(k))) * cbrt(tan(k))) ^ 3.0)) * Float64(l * l)) / t_1);
	elseif (t <= 7200.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.9e-8], N[(N[(N[(2.0 / N[Power[N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 7200.0], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.89999999999999985e-8

   1. Initial program 60.1%

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

      1. associate-/r* 60.1%

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

      2. associate-*l* 55.1%

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

      3. sqr-neg 55.1%

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

      4. associate-*l* 60.1%

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

      5. *-commutative 60.1%

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

      6. sqr-neg 60.1%

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

      7. associate-*l/ 60.0%

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

      8. associate-*r/ 60.1%

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

      9. associate-/r/ 60.1%

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

   3. Simplified 60.1%

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

      1. add-cube-cbrt 60.0%

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

      2. pow3 60.0%

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

      3. *-commutative 60.0%

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

      4. cbrt-prod 60.0%

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

      5. cbrt-prod 59.9%

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

      6. rem-cbrt-cube 73.9%

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

   5. Applied egg-rr 73.9%

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

   if -3.89999999999999985e-8 < t < 7200

   1. Initial program 40.2%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*l* 75.9%

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

      4. times-frac 84.4%

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

      5. unpow2 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in k around inf 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*r* 81.6%

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

      4. unpow2 81.6%

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

      5. associate-*r* 81.5%

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

      6. times-frac 90.7%

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

      7. associate-*r* 84.4%

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

      8. unpow2 84.4%

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

      9. associate-/l* 87.3%

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

      10. unpow2 87.3%

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

      11. associate-/r* 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 84.4%

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

      1. associate-*l/ 86.7%

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

      2. unpow2 86.7%

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

      3. associate-*l/ 91.0%

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

      4. associate-/r/ 91.0%

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

      5. *-commutative 91.0%

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

      6. associate-/r/ 91.0%

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

   10. Simplified 91.0%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. associate-*l/ 86.7%

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

       2. unpow2 86.7%

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

       3. associate-*r/ 91.0%

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

       4. associate-*l* 96.5%

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

   13. Simplified 96.5%

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

   if 7200 < t

   1. Initial program 59.0%

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

      1. associate-/r* 59.2%

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

      2. associate-*l* 52.9%

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

      3. sqr-neg 52.9%

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

      4. associate-*l* 59.2%

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

      5. *-commutative 59.2%

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

      6. sqr-neg 59.2%

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

      7. associate-/r* 59.2%

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

   3. Simplified 61.5%

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

      1. associate-/r/ 59.2%

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

      2. add-sqr-sqrt 59.1%

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

      3. pow2 59.1%

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

      4. sqrt-div 59.1%

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

      5. sqrt-pow1 65.1%

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

      6. metadata-eval 65.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. sqrt-prod 34.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. add-sqr-sqrt 78.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 78.1%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{2}{{\left(\left(t \cdot \sqrt[3]{\sin k}\right) \cdot \sqrt[3]{\tan k}\right)}^{3}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 7200:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 2: 84.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(1 + \left(t_1 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k \cdot {t}^{3}}}{\frac{\sin k}{\ell}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (+ 1.0 (+ t_1 1.0))
          (* (tan k) (* (sin k) (/ (pow t 3.0) (* l l))))))
        5e+307)
     (/ (/ (* l (/ 2.0 (* (tan k) (pow t 3.0)))) (/ (sin k) l)) (+ 2.0 t_1))
     (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k)))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (pow(t, 3.0) / (l * l)))))) <= 5e+307) {
		tmp = ((l * (2.0 / (tan(k) * pow(t, 3.0)))) / (sin(k) / l)) / (2.0 + t_1);
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if ((2.0d0 / ((1.0d0 + (t_1 + 1.0d0)) * (tan(k) * (sin(k) * ((t ** 3.0d0) / (l * l)))))) <= 5d+307) then
        tmp = ((l * (2.0d0 / (tan(k) * (t ** 3.0d0)))) / (sin(k) / l)) / (2.0d0 + t_1)
    else
        tmp = 2.0d0 / ((k * (t * (k / l))) * (((sin(k) ** 2.0d0) / l) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t, 3.0) / (l * l)))))) <= 5e+307) {
		tmp = ((l * (2.0 / (Math.tan(k) * Math.pow(t, 3.0)))) / (Math.sin(k) / l)) / (2.0 + t_1);
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((1.0 + (t_1 + 1.0)) * (math.tan(k) * (math.sin(k) * (math.pow(t, 3.0) / (l * l)))))) <= 5e+307:
		tmp = ((l * (2.0 / (math.tan(k) * math.pow(t, 3.0)))) / (math.sin(k) / l)) / (2.0 + t_1)
	else:
		tmp = 2.0 / ((k * (t * (k / l))) * ((math.pow(math.sin(k), 2.0) / l) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(1.0 + Float64(t_1 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64((t ^ 3.0) / Float64(l * l)))))) <= 5e+307)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(tan(k) * (t ^ 3.0)))) / Float64(sin(k) / l)) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * ((t ^ 3.0) / (l * l)))))) <= 5e+307)
		tmp = ((l * (2.0 / (tan(k) * (t ^ 3.0)))) / (sin(k) / l)) / (2.0 + t_1);
	else
		tmp = 2.0 / ((k * (t * (k / l))) * (((sin(k) ^ 2.0) / l) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+307], N[(N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 5e307

   1. Initial program 76.0%

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

      1. associate-/r* 76.1%

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

      2. associate-*l* 70.7%

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

      3. sqr-neg 70.7%

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

      4. associate-*l* 76.1%

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

      5. *-commutative 76.1%

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

      6. sqr-neg 76.1%

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

      7. associate-/r* 76.1%

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

   3. Simplified 76.5%

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

      1. div-inv 76.5%

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

      2. associate-/r/ 75.6%

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

   5. Applied egg-rr 75.6%

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

      1. associate-*r/ 75.6%

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

      2. *-rgt-identity 75.6%

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

      3. *-commutative 75.6%

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

      4. associate-/l* 77.8%

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

      5. associate-/l/ 77.8%

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

   7. Simplified 77.8%

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

      1. associate-*l/ 80.5%

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

   9. Applied egg-rr 80.5%

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

   if 5e307 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 22.2%

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

   2. Taylor expanded in t around 0 59.1%

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

      1. associate-*r* 59.2%

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

      2. unpow2 59.2%

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

      3. associate-*l* 59.1%

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

      4. times-frac 69.1%

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

      5. unpow2 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in k around inf 59.1%

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

      1. associate-*r* 59.2%

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

      2. unpow2 59.2%

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

      3. associate-*r* 65.2%

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

      4. unpow2 65.2%

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

      5. associate-*r* 65.2%

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

      6. times-frac 75.9%

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

      7. associate-*r* 69.1%

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

      8. unpow2 69.1%

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

      9. associate-/l* 72.1%

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

      10. unpow2 72.1%

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

      11. associate-/r* 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in k around 0 69.1%

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

      1. associate-*l/ 72.2%

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

      2. unpow2 72.2%

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

      3. associate-*l/ 78.3%

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

      4. associate-/r/ 78.3%

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

      5. *-commutative 78.3%

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

      6. associate-/r/ 78.3%

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

   10. Simplified 78.3%

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

   11. Taylor expanded in t around 0 69.1%

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

       1. associate-*l/ 72.2%

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

       2. unpow2 72.2%

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

       3. associate-*r/ 78.3%

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

       4. associate-*l* 83.5%

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

   13. Simplified 83.5%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k \cdot {t}^{3}}}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 3: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ \mathbf{if}\;t \leq -2.95 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t_1 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)\right)}\\ \mathbf{elif}\;t \leq 10800:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k \cdot {t}^{3}}}{\frac{\sin k}{\ell}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (+ 2.0 t_1)))
   (if (<= t -2.95e-10)
     (/
      2.0
      (*
       (+ 1.0 (+ t_1 1.0))
       (* (tan k) (* (sin k) (/ 1.0 (/ l (/ (pow t 3.0) l)))))))
     (if (<= t 10800.0)
       (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))
       (if (<= t 5.5e+102)
         (/ (/ (* l (/ 2.0 (* (tan k) (pow t 3.0)))) (/ (sin k) l)) t_2)
         (/
          2.0
          (* t_2 (* (sin k) (* (tan k) (pow (/ (pow t 1.5) l) 2.0))))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (t <= -2.95e-10) {
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (1.0 / (l / (pow(t, 3.0) / l))))));
	} else if (t <= 10800.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else if (t <= 5.5e+102) {
		tmp = ((l * (2.0 / (tan(k) * pow(t, 3.0)))) / (sin(k) / l)) / t_2;
	} else {
		tmp = 2.0 / (t_2 * (sin(k) * (tan(k) * pow((pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    t_2 = 2.0d0 + t_1
    if (t <= (-2.95d-10)) then
        tmp = 2.0d0 / ((1.0d0 + (t_1 + 1.0d0)) * (tan(k) * (sin(k) * (1.0d0 / (l / ((t ** 3.0d0) / l))))))
    else if (t <= 10800.0d0) then
        tmp = 2.0d0 / ((k * (t * (k / l))) * (((sin(k) ** 2.0d0) / l) / cos(k)))
    else if (t <= 5.5d+102) then
        tmp = ((l * (2.0d0 / (tan(k) * (t ** 3.0d0)))) / (sin(k) / l)) / t_2
    else
        tmp = 2.0d0 / (t_2 * (sin(k) * (tan(k) * (((t ** 1.5d0) / l) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = 2.0 + t_1;
	double tmp;
	if (t <= -2.95e-10) {
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (1.0 / (l / (Math.pow(t, 3.0) / l))))));
	} else if (t <= 10800.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	} else if (t <= 5.5e+102) {
		tmp = ((l * (2.0 / (Math.tan(k) * Math.pow(t, 3.0)))) / (Math.sin(k) / l)) / t_2;
	} else {
		tmp = 2.0 / (t_2 * (Math.sin(k) * (Math.tan(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	t_2 = 2.0 + t_1
	tmp = 0
	if t <= -2.95e-10:
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (math.tan(k) * (math.sin(k) * (1.0 / (l / (math.pow(t, 3.0) / l))))))
	elif t <= 10800.0:
		tmp = 2.0 / ((k * (t * (k / l))) * ((math.pow(math.sin(k), 2.0) / l) / math.cos(k)))
	elif t <= 5.5e+102:
		tmp = ((l * (2.0 / (math.tan(k) * math.pow(t, 3.0)))) / (math.sin(k) / l)) / t_2
	else:
		tmp = 2.0 / (t_2 * (math.sin(k) * (math.tan(k) * math.pow((math.pow(t, 1.5) / l), 2.0))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(2.0 + t_1)
	tmp = 0.0
	if (t <= -2.95e-10)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_1 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64(1.0 / Float64(l / Float64((t ^ 3.0) / l)))))));
	elseif (t <= 10800.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	elseif (t <= 5.5e+102)
		tmp = Float64(Float64(Float64(l * Float64(2.0 / Float64(tan(k) * (t ^ 3.0)))) / Float64(sin(k) / l)) / t_2);
	else
		tmp = Float64(2.0 / Float64(t_2 * Float64(sin(k) * Float64(tan(k) * (Float64((t ^ 1.5) / l) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	t_2 = 2.0 + t_1;
	tmp = 0.0;
	if (t <= -2.95e-10)
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (1.0 / (l / ((t ^ 3.0) / l))))));
	elseif (t <= 10800.0)
		tmp = 2.0 / ((k * (t * (k / l))) * (((sin(k) ^ 2.0) / l) / cos(k)));
	elseif (t <= 5.5e+102)
		tmp = ((l * (2.0 / (tan(k) * (t ^ 3.0)))) / (sin(k) / l)) / t_2;
	else
		tmp = 2.0 / (t_2 * (sin(k) * (tan(k) * (((t ^ 1.5) / l) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[t, -2.95e-10], N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(1.0 / N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 10800.0], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.5e+102], N[(N[(N[(l * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(2.0 / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -2.9500000000000002e-10

   1. Initial program 60.1%

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

      1. clear-num 60.1%

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

      2. inv-pow 60.1%

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

   3. Applied egg-rr 60.1%

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

      1. unpow-1 60.1%

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

      2. associate-/l* 63.0%

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

   5. Simplified 63.0%

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

   if -2.9500000000000002e-10 < t < 10800

   1. Initial program 40.2%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*l* 75.9%

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

      4. times-frac 84.4%

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

      5. unpow2 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in k around inf 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*r* 81.6%

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

      4. unpow2 81.6%

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

      5. associate-*r* 81.5%

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

      6. times-frac 90.7%

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

      7. associate-*r* 84.4%

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

      8. unpow2 84.4%

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

      9. associate-/l* 87.3%

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

      10. unpow2 87.3%

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

      11. associate-/r* 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 84.4%

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

      1. associate-*l/ 86.7%

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

      2. unpow2 86.7%

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

      3. associate-*l/ 91.0%

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

      4. associate-/r/ 91.0%

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

      5. *-commutative 91.0%

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

      6. associate-/r/ 91.0%

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

   10. Simplified 91.0%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. associate-*l/ 86.7%

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

       2. unpow2 86.7%

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

       3. associate-*r/ 91.0%

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

       4. associate-*l* 96.5%

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

   13. Simplified 96.5%

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

   if 10800 < t < 5.49999999999999981e102

   1. Initial program 69.8%

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

      1. associate-/r* 70.3%

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

      2. associate-*l* 70.2%

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

      3. sqr-neg 70.2%

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

      4. associate-*l* 70.3%

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

      5. *-commutative 70.3%

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

      6. sqr-neg 70.3%

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

      7. associate-/r* 70.3%

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

   3. Simplified 76.6%

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

      1. div-inv 76.6%

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

      2. associate-/r/ 76.6%

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

   5. Applied egg-rr 76.6%

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

      1. associate-*r/ 76.6%

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

      2. *-rgt-identity 76.6%

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

      3. *-commutative 76.6%

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

      4. associate-/l* 79.3%

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

      5. associate-/l/ 79.4%

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

   7. Simplified 79.4%

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

      1. associate-*l/ 92.2%

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

   9. Applied egg-rr 92.2%

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

   if 5.49999999999999981e102 < t

   1. Initial program 52.1%

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

      1. associate-/r* 52.1%

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

      2. associate-*l* 41.9%

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

      3. sqr-neg 41.9%

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

      4. associate-*l* 52.1%

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

      5. *-commutative 52.1%

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

      6. sqr-neg 52.1%

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

      7. associate-/r* 52.1%

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

   3. Simplified 51.9%

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

      1. associate-/r/ 52.1%

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

      2. add-sqr-sqrt 52.1%

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

      3. pow2 52.1%

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

      4. sqrt-div 52.1%

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

      5. sqrt-pow1 61.8%

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

      6. metadata-eval 61.8%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. sqrt-prod 34.5%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. add-sqr-sqrt 78.6%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 78.6%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   6. Step-by-step derivation

      1. div-inv 78.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. associate-/l/ 78.7%

         \[\leadsto \color{blue}{\frac{2}{\left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k\right) \cdot \tan k}} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. *-commutative 78.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)} \cdot \tan k} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Applied egg-rr 78.7%

      \[\leadsto \color{blue}{\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 78.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k} \cdot 1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. *-rgt-identity 78.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. associate-/l/ 78.7%

         \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \tan k\right)}} \]

      4. associate-*l* 78.7%

         \[\leadsto \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)}} \]

   9. Simplified 78.7%

      \[\leadsto \color{blue}{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left({\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \tan k\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.95 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)\right)}\\ \mathbf{elif}\;t \leq 10800:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{2}{\tan k \cdot {t}^{3}}}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 4: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}\right) \cdot \left(1 + \left(t_1 + 1\right)\right)}\\ \mathbf{elif}\;t \leq 5800:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -2.2e-12)
     (/
      2.0
      (*
       (* (tan k) (pow (/ t (cbrt (/ (* l l) (sin k)))) 3.0))
       (+ 1.0 (+ t_1 1.0))))
     (if (<= t 5800.0)
       (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))
       (/
        (/ (/ 2.0 (tan k)) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))
        (+ 2.0 t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -2.2e-12) {
		tmp = 2.0 / ((tan(k) * pow((t / cbrt(((l * l) / sin(k)))), 3.0)) * (1.0 + (t_1 + 1.0)));
	} else if (t <= 5800.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else {
		tmp = ((2.0 / tan(k)) / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -2.2e-12) {
		tmp = 2.0 / ((Math.tan(k) * Math.pow((t / Math.cbrt(((l * l) / Math.sin(k)))), 3.0)) * (1.0 + (t_1 + 1.0)));
	} else if (t <= 5800.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	} else {
		tmp = ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -2.2e-12)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(t / cbrt(Float64(Float64(l * l) / sin(k)))) ^ 3.0)) * Float64(1.0 + Float64(t_1 + 1.0))));
	elseif (t <= 5800.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / Float64(2.0 + t_1));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -2.2e-12], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[(N[(l * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5800.0], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.19999999999999992e-12

   1. Initial program 60.1%

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

      1. associate-/r/ 58.2%

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

      2. add-cube-cbrt 58.1%

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

      3. pow3 58.1%

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

      4. cbrt-div 58.0%

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

      5. rem-cbrt-cube 65.7%

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

   3. Applied egg-rr 65.7%

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

   if -2.19999999999999992e-12 < t < 5800

   1. Initial program 40.2%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*l* 75.9%

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

      4. times-frac 84.4%

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

      5. unpow2 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in k around inf 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*r* 81.6%

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

      4. unpow2 81.6%

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

      5. associate-*r* 81.5%

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

      6. times-frac 90.7%

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

      7. associate-*r* 84.4%

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

      8. unpow2 84.4%

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

      9. associate-/l* 87.3%

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

      10. unpow2 87.3%

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

      11. associate-/r* 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 84.4%

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

      1. associate-*l/ 86.7%

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

      2. unpow2 86.7%

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

      3. associate-*l/ 91.0%

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

      4. associate-/r/ 91.0%

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

      5. *-commutative 91.0%

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

      6. associate-/r/ 91.0%

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

   10. Simplified 91.0%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. associate-*l/ 86.7%

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

       2. unpow2 86.7%

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

       3. associate-*r/ 91.0%

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

       4. associate-*l* 96.5%

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

   13. Simplified 96.5%

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

   if 5800 < t

   1. Initial program 59.0%

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

      1. associate-/r* 59.2%

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

      2. associate-*l* 52.9%

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

      3. sqr-neg 52.9%

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

      4. associate-*l* 59.2%

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

      5. *-commutative 59.2%

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

      6. sqr-neg 59.2%

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

      7. associate-/r* 59.2%

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

   3. Simplified 61.5%

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

      1. associate-/r/ 59.2%

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

      2. add-sqr-sqrt 59.1%

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

      3. pow2 59.1%

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

      4. sqrt-div 59.1%

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

      5. sqrt-pow1 65.1%

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

      6. metadata-eval 65.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. sqrt-prod 34.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. add-sqr-sqrt 78.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 78.1%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot {\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3}\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \mathbf{elif}\;t \leq 5800:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 5: 84.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4800:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -1.6e-11)
     (/
      2.0
      (*
       (pow (/ t (cbrt (/ (* l l) (sin k)))) 3.0)
       (* (tan k) (+ 1.0 (+ t_1 1.0)))))
     (if (<= t 4800.0)
       (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))
       (/
        (/ (/ 2.0 (tan k)) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))
        (+ 2.0 t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -1.6e-11) {
		tmp = 2.0 / (pow((t / cbrt(((l * l) / sin(k)))), 3.0) * (tan(k) * (1.0 + (t_1 + 1.0))));
	} else if (t <= 4800.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else {
		tmp = ((2.0 / tan(k)) / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -1.6e-11) {
		tmp = 2.0 / (Math.pow((t / Math.cbrt(((l * l) / Math.sin(k)))), 3.0) * (Math.tan(k) * (1.0 + (t_1 + 1.0))));
	} else if (t <= 4800.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	} else {
		tmp = ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -1.6e-11)
		tmp = Float64(2.0 / Float64((Float64(t / cbrt(Float64(Float64(l * l) / sin(k)))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(t_1 + 1.0)))));
	elseif (t <= 4800.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / Float64(2.0 + t_1));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -1.6e-11], N[(2.0 / N[(N[Power[N[(t / N[Power[N[(N[(l * l), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4800.0], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.59999999999999997e-11

   1. Initial program 60.1%

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

      1. associate-/r/ 58.2%

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

      2. add-cube-cbrt 58.1%

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

      3. pow3 58.1%

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

      4. cbrt-div 58.0%

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

      5. rem-cbrt-cube 65.7%

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

   3. Applied egg-rr 65.7%

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

      1. distribute-lft-in 65.7%

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

      2. *-commutative 65.7%

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

      3. associate-/l* 65.7%

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

      4. associate-/r/ 65.7%

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

      5. +-commutative 65.7%

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

   5. Applied egg-rr 65.7%

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

      1. distribute-lft-out 65.7%

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

      2. *-commutative 65.7%

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

      3. +-commutative 65.7%

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

      4. associate-*r* 65.7%

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

      5. associate-*l/ 65.7%

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

      6. +-commutative 65.7%

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

   7. Simplified 65.7%

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

   if -1.59999999999999997e-11 < t < 4800

   1. Initial program 40.2%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*l* 75.9%

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

      4. times-frac 84.4%

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

      5. unpow2 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in k around inf 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*r* 81.6%

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

      4. unpow2 81.6%

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

      5. associate-*r* 81.5%

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

      6. times-frac 90.7%

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

      7. associate-*r* 84.4%

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

      8. unpow2 84.4%

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

      9. associate-/l* 87.3%

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

      10. unpow2 87.3%

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

      11. associate-/r* 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 84.4%

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

      1. associate-*l/ 86.7%

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

      2. unpow2 86.7%

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

      3. associate-*l/ 91.0%

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

      4. associate-/r/ 91.0%

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

      5. *-commutative 91.0%

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

      6. associate-/r/ 91.0%

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

   10. Simplified 91.0%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. associate-*l/ 86.7%

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

       2. unpow2 86.7%

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

       3. associate-*r/ 91.0%

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

       4. associate-*l* 96.5%

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

   13. Simplified 96.5%

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

   if 4800 < t

   1. Initial program 59.0%

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

      1. associate-/r* 59.2%

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

      2. associate-*l* 52.9%

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

      3. sqr-neg 52.9%

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

      4. associate-*l* 59.2%

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

      5. *-commutative 59.2%

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

      6. sqr-neg 59.2%

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

      7. associate-/r* 59.2%

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

   3. Simplified 61.5%

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

      1. associate-/r/ 59.2%

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

      2. add-sqr-sqrt 59.1%

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

      3. pow2 59.1%

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

      4. sqrt-div 59.1%

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

      5. sqrt-pow1 65.1%

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

      6. metadata-eval 65.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. sqrt-prod 34.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. add-sqr-sqrt 78.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 78.1%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\frac{\ell \cdot \ell}{\sin k}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4800:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 6: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t_1 + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)\right)}\\ \mathbf{elif}\;t \leq 24000:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<= t -1.3e-12)
     (/
      2.0
      (*
       (+ 1.0 (+ t_1 1.0))
       (* (tan k) (* (sin k) (/ 1.0 (/ l (/ (pow t 3.0) l)))))))
     (if (<= t 24000.0)
       (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))
       (/
        (/ (/ 2.0 (tan k)) (* (sin k) (pow (/ (pow t 1.5) l) 2.0)))
        (+ 2.0 t_1))))))

C

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if (t <= -1.3e-12) {
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (1.0 / (l / (pow(t, 3.0) / l))))));
	} else if (t <= 24000.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else {
		tmp = ((2.0 / tan(k)) / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if (t <= (-1.3d-12)) then
        tmp = 2.0d0 / ((1.0d0 + (t_1 + 1.0d0)) * (tan(k) * (sin(k) * (1.0d0 / (l / ((t ** 3.0d0) / l))))))
    else if (t <= 24000.0d0) then
        tmp = 2.0d0 / ((k * (t * (k / l))) * (((sin(k) ** 2.0d0) / l) / cos(k)))
    else
        tmp = ((2.0d0 / tan(k)) / (sin(k) * (((t ** 1.5d0) / l) ** 2.0d0))) / (2.0d0 + t_1)
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -1.3e-12) {
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (1.0 / (l / (Math.pow(t, 3.0) / l))))));
	} else if (t <= 24000.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	} else {
		tmp = ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / (2.0 + t_1);
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if t <= -1.3e-12:
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (math.tan(k) * (math.sin(k) * (1.0 / (l / (math.pow(t, 3.0) / l))))))
	elif t <= 24000.0:
		tmp = 2.0 / ((k * (t * (k / l))) * ((math.pow(math.sin(k), 2.0) / l) / math.cos(k)))
	else:
		tmp = ((2.0 / math.tan(k)) / (math.sin(k) * math.pow((math.pow(t, 1.5) / l), 2.0))) / (2.0 + t_1)
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (t <= -1.3e-12)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_1 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64(1.0 / Float64(l / Float64((t ^ 3.0) / l)))))));
	elseif (t <= 24000.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / Float64(2.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if (t <= -1.3e-12)
		tmp = 2.0 / ((1.0 + (t_1 + 1.0)) * (tan(k) * (sin(k) * (1.0 / (l / ((t ^ 3.0) / l))))));
	elseif (t <= 24000.0)
		tmp = 2.0 / ((k * (t * (k / l))) * (((sin(k) ^ 2.0) / l) / cos(k)));
	else
		tmp = ((2.0 / tan(k)) / (sin(k) * (((t ^ 1.5) / l) ^ 2.0))) / (2.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t, -1.3e-12], N[(2.0 / N[(N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(1.0 / N[(l / N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 24000.0], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.29999999999999991e-12

   1. Initial program 60.1%

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

      1. clear-num 60.1%

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

      2. inv-pow 60.1%

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

   3. Applied egg-rr 60.1%

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

      1. unpow-1 60.1%

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

      2. associate-/l* 63.0%

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

   5. Simplified 63.0%

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

   if -1.29999999999999991e-12 < t < 24000

   1. Initial program 40.2%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*l* 75.9%

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

      4. times-frac 84.4%

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

      5. unpow2 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in k around inf 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*r* 81.6%

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

      4. unpow2 81.6%

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

      5. associate-*r* 81.5%

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

      6. times-frac 90.7%

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

      7. associate-*r* 84.4%

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

      8. unpow2 84.4%

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

      9. associate-/l* 87.3%

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

      10. unpow2 87.3%

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

      11. associate-/r* 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 84.4%

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

      1. associate-*l/ 86.7%

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

      2. unpow2 86.7%

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

      3. associate-*l/ 91.0%

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

      4. associate-/r/ 91.0%

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

      5. *-commutative 91.0%

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

      6. associate-/r/ 91.0%

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

   10. Simplified 91.0%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. associate-*l/ 86.7%

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

       2. unpow2 86.7%

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

       3. associate-*r/ 91.0%

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

       4. associate-*l* 96.5%

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

   13. Simplified 96.5%

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

   if 24000 < t

   1. Initial program 59.0%

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

      1. associate-/r* 59.2%

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

      2. associate-*l* 52.9%

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

      3. sqr-neg 52.9%

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

      4. associate-*l* 59.2%

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

      5. *-commutative 59.2%

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

      6. sqr-neg 59.2%

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

      7. associate-/r* 59.2%

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

   3. Simplified 61.5%

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

      1. associate-/r/ 59.2%

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

      2. add-sqr-sqrt 59.1%

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

      3. pow2 59.1%

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

      4. sqrt-div 59.1%

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

      5. sqrt-pow1 65.1%

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

      6. metadata-eval 65.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. sqrt-prod 34.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. add-sqr-sqrt 78.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 78.1%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{1}{\frac{\ell}{\frac{{t}^{3}}{\ell}}}\right)\right)}\\ \mathbf{elif}\;t \leq 24000:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 7: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\tan k}\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{t_2 \cdot \frac{\ell \cdot \ell}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{t_1}\\ \mathbf{elif}\;t \leq 4500:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_2}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))) (t_2 (/ 2.0 (tan k))))
   (if (<= t -4.4e-13)
     (/ (* t_2 (/ (* l l) (pow (* t (cbrt (sin k))) 3.0))) t_1)
     (if (<= t 4500.0)
       (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))
       (/ (/ t_2 (* (sin k) (pow (/ (pow t 1.5) l) 2.0))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = 2.0 / tan(k);
	double tmp;
	if (t <= -4.4e-13) {
		tmp = (t_2 * ((l * l) / pow((t * cbrt(sin(k))), 3.0))) / t_1;
	} else if (t <= 4500.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else {
		tmp = (t_2 / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = 2.0 / Math.tan(k);
	double tmp;
	if (t <= -4.4e-13) {
		tmp = (t_2 * ((l * l) / Math.pow((t * Math.cbrt(Math.sin(k))), 3.0))) / t_1;
	} else if (t <= 4500.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	} else {
		tmp = (t_2 / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(2.0 / tan(k))
	tmp = 0.0
	if (t <= -4.4e-13)
		tmp = Float64(Float64(t_2 * Float64(Float64(l * l) / (Float64(t * cbrt(sin(k))) ^ 3.0))) / t_1);
	elseif (t <= 4500.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	else
		tmp = Float64(Float64(t_2 / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.4e-13], N[(N[(t$95$2 * N[(N[(l * l), $MachinePrecision] / N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 4500.0], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$2 / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.39999999999999993e-13

   1. Initial program 60.1%

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

      1. associate-/r* 60.1%

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

      2. associate-*l* 55.1%

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

      3. sqr-neg 55.1%

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

      4. associate-*l* 60.1%

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

      5. *-commutative 60.1%

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

      6. sqr-neg 60.1%

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

      7. associate-*l/ 60.0%

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

      8. associate-*r/ 60.1%

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

      9. associate-/r/ 60.1%

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

   3. Simplified 60.1%

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

      1. add-cube-cbrt 60.0%

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

      2. pow3 60.0%

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

      3. cbrt-prod 60.0%

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

      4. rem-cbrt-cube 65.5%

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

   5. Applied egg-rr 65.5%

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

      1. associate-*l/ 65.5%

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

   7. Applied egg-rr 65.5%

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

      1. unpow2 65.5%

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

      2. times-frac 65.5%

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

      3. unpow2 65.5%

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

   9. Simplified 65.5%

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

   if -4.39999999999999993e-13 < t < 4500

   1. Initial program 40.2%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*l* 75.9%

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

      4. times-frac 84.4%

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

      5. unpow2 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in k around inf 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*r* 81.6%

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

      4. unpow2 81.6%

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

      5. associate-*r* 81.5%

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

      6. times-frac 90.7%

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

      7. associate-*r* 84.4%

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

      8. unpow2 84.4%

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

      9. associate-/l* 87.3%

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

      10. unpow2 87.3%

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

      11. associate-/r* 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 84.4%

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

      1. associate-*l/ 86.7%

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

      2. unpow2 86.7%

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

      3. associate-*l/ 91.0%

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

      4. associate-/r/ 91.0%

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

      5. *-commutative 91.0%

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

      6. associate-/r/ 91.0%

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

   10. Simplified 91.0%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. associate-*l/ 86.7%

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

       2. unpow2 86.7%

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

       3. associate-*r/ 91.0%

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

       4. associate-*l* 96.5%

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

   13. Simplified 96.5%

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

   if 4500 < t

   1. Initial program 59.0%

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

      1. associate-/r* 59.2%

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

      2. associate-*l* 52.9%

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

      3. sqr-neg 52.9%

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

      4. associate-*l* 59.2%

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

      5. *-commutative 59.2%

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

      6. sqr-neg 59.2%

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

      7. associate-/r* 59.2%

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

   3. Simplified 61.5%

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

      1. associate-/r/ 59.2%

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

      2. add-sqr-sqrt 59.1%

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

      3. pow2 59.1%

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

      4. sqrt-div 59.1%

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

      5. sqrt-pow1 65.1%

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

      6. metadata-eval 65.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. sqrt-prod 34.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. add-sqr-sqrt 78.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 78.1%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{2}{\tan k} \cdot \frac{\ell \cdot \ell}{{\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 4500:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 8: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -3.45 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{t_1}\\ \mathbf{elif}\;t \leq 6400:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -3.45e-13)
     (/ (* (* l l) (/ 2.0 (* (tan k) (pow (* t (cbrt (sin k))) 3.0)))) t_1)
     (if (<= t 6400.0)
       (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))
       (/ (/ (/ 2.0 (tan k)) (* (sin k) (pow (/ (pow t 1.5) l) 2.0))) t_1)))))

C

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -3.45e-13) {
		tmp = ((l * l) * (2.0 / (tan(k) * pow((t * cbrt(sin(k))), 3.0)))) / t_1;
	} else if (t <= 6400.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	} else {
		tmp = ((2.0 / tan(k)) / (sin(k) * pow((pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -3.45e-13) {
		tmp = ((l * l) * (2.0 / (Math.tan(k) * Math.pow((t * Math.cbrt(Math.sin(k))), 3.0)))) / t_1;
	} else if (t <= 6400.0) {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	} else {
		tmp = ((2.0 / Math.tan(k)) / (Math.sin(k) * Math.pow((Math.pow(t, 1.5) / l), 2.0))) / t_1;
	}
	return tmp;
}

Julia

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -3.45e-13)
		tmp = Float64(Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * (Float64(t * cbrt(sin(k))) ^ 3.0)))) / t_1);
	elseif (t <= 6400.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	else
		tmp = Float64(Float64(Float64(2.0 / tan(k)) / Float64(sin(k) * (Float64((t ^ 1.5) / l) ^ 2.0))) / t_1);
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.45e-13], N[(N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t, 6400.0], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.44999999999999994e-13

   1. Initial program 60.1%

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

      1. associate-/r* 60.1%

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

      2. associate-*l* 55.1%

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

      3. sqr-neg 55.1%

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

      4. associate-*l* 60.1%

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

      5. *-commutative 60.1%

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

      6. sqr-neg 60.1%

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

      7. associate-*l/ 60.0%

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

      8. associate-*r/ 60.1%

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

      9. associate-/r/ 60.1%

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

   3. Simplified 60.1%

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

      1. add-cube-cbrt 60.0%

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

      2. pow3 60.0%

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

      3. cbrt-prod 60.0%

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

      4. rem-cbrt-cube 65.5%

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

   5. Applied egg-rr 65.5%

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

   if -3.44999999999999994e-13 < t < 6400

   1. Initial program 40.2%

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

   2. Taylor expanded in t around 0 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*l* 75.9%

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

      4. times-frac 84.4%

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

      5. unpow2 84.4%

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

   4. Simplified 84.4%

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

   5. Taylor expanded in k around inf 75.9%

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

      1. associate-*r* 75.9%

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

      2. unpow2 75.9%

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

      3. associate-*r* 81.6%

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

      4. unpow2 81.6%

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

      5. associate-*r* 81.5%

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

      6. times-frac 90.7%

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

      7. associate-*r* 84.4%

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

      8. unpow2 84.4%

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

      9. associate-/l* 87.3%

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

      10. unpow2 87.3%

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

      11. associate-/r* 87.3%

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

   7. Simplified 87.3%

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

   8. Taylor expanded in k around 0 84.4%

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

      1. associate-*l/ 86.7%

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

      2. unpow2 86.7%

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

      3. associate-*l/ 91.0%

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

      4. associate-/r/ 91.0%

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

      5. *-commutative 91.0%

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

      6. associate-/r/ 91.0%

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

   10. Simplified 91.0%

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

   11. Taylor expanded in t around 0 84.4%

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

       1. associate-*l/ 86.7%

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

       2. unpow2 86.7%

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

       3. associate-*r/ 91.0%

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

       4. associate-*l* 96.5%

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

   13. Simplified 96.5%

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

   if 6400 < t

   1. Initial program 59.0%

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

      1. associate-/r* 59.2%

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

      2. associate-*l* 52.9%

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

      3. sqr-neg 52.9%

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

      4. associate-*l* 59.2%

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

      5. *-commutative 59.2%

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

      6. sqr-neg 59.2%

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

      7. associate-/r* 59.2%

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

   3. Simplified 61.5%

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

      1. associate-/r/ 59.2%

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

      2. add-sqr-sqrt 59.1%

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

      3. pow2 59.1%

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

      4. sqrt-div 59.1%

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

      5. sqrt-pow1 65.1%

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

      6. metadata-eval 65.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      7. sqrt-prod 34.7%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      8. add-sqr-sqrt 78.1%

         \[\leadsto \frac{\frac{\frac{2}{\tan k}}{{\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 78.1%

      \[\leadsto \frac{\frac{\frac{2}{\tan k}}{\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.45 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot {\left(t \cdot \sqrt[3]{\sin k}\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 6400:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{\tan k}}{\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 9: 70.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+134}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{\frac{2}{t_1}}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell \cdot \cos k}{t \cdot t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0)))
   (if (<= (* l l) 0.0)
     (/ 2.0 (* (/ (* k k) l) (* (* k k) (/ 1.0 (/ l t)))))
     (if (<= (* l l) 1e+134)
       (* (* l l) (* (cos k) (/ (/ 2.0 t_1) (* k (* t k)))))
       (* 2.0 (* (/ (/ l k) k) (/ (* l (cos k)) (* t t_1))))))))

C

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
	} else if ((l * l) <= 1e+134) {
		tmp = (l * l) * (cos(k) * ((2.0 / t_1) / (k * (t * k))));
	} else {
		tmp = 2.0 * (((l / k) / k) * ((l * cos(k)) / (t * t_1)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    if ((l * l) <= 0.0d0) then
        tmp = 2.0d0 / (((k * k) / l) * ((k * k) * (1.0d0 / (l / t))))
    else if ((l * l) <= 1d+134) then
        tmp = (l * l) * (cos(k) * ((2.0d0 / t_1) / (k * (t * k))))
    else
        tmp = 2.0d0 * (((l / k) / k) * ((l * cos(k)) / (t * t_1)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if ((l * l) <= 0.0) {
		tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
	} else if ((l * l) <= 1e+134) {
		tmp = (l * l) * (Math.cos(k) * ((2.0 / t_1) / (k * (t * k))));
	} else {
		tmp = 2.0 * (((l / k) / k) * ((l * Math.cos(k)) / (t * t_1)));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if (l * l) <= 0.0:
		tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))))
	elif (l * l) <= 1e+134:
		tmp = (l * l) * (math.cos(k) * ((2.0 / t_1) / (k * (t * k))))
	else:
		tmp = 2.0 * (((l / k) / k) * ((l * math.cos(k)) / (t * t_1)))
	return tmp

Julia

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	tmp = 0.0
	if (Float64(l * l) <= 0.0)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * k) * Float64(1.0 / Float64(l / t)))));
	elseif (Float64(l * l) <= 1e+134)
		tmp = Float64(Float64(l * l) * Float64(cos(k) * Float64(Float64(2.0 / t_1) / Float64(k * Float64(t * k)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / k) * Float64(Float64(l * cos(k)) / Float64(t * t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	tmp = 0.0;
	if ((l * l) <= 0.0)
		tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
	elseif ((l * l) <= 1e+134)
		tmp = (l * l) * (cos(k) * ((2.0 / t_1) / (k * (t * k))));
	else
		tmp = 2.0 * (((l / k) / k) * ((l * cos(k)) / (t * t_1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 0.0], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(1.0 / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(l * l), $MachinePrecision], 1e+134], N[(N[(l * l), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] * N[(N[(2.0 / t$95$1), $MachinePrecision] / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 l l) < 0.0

   1. Initial program 44.3%

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

   2. Taylor expanded in t around 0 50.4%

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

      1. associate-*r* 50.4%

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

      2. unpow2 50.4%

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

      3. associate-*l* 50.4%

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

      4. times-frac 69.6%

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

      5. unpow2 69.6%

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

   4. Simplified 69.6%

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

   5. Taylor expanded in k around 0 67.0%

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

      1. unpow2 67.0%

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

   7. Simplified 67.0%

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

   8. Taylor expanded in k around 0 67.0%

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

      1. associate-/l* 81.2%

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

      2. unpow2 81.2%

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

   10. Simplified 81.2%

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

       1. div-inv 83.2%

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

   12. Applied egg-rr 83.2%

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

   if 0.0 < (*.f64 l l) < 9.99999999999999921e133

   1. Initial program 55.8%

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

   2. Taylor expanded in t around 0 72.2%

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

      1. associate-*r* 72.1%

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

      2. unpow2 72.1%

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

      3. associate-*l* 72.1%

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

      4. times-frac 73.4%

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

      5. unpow2 73.4%

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

   4. Simplified 73.4%

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

   5. Taylor expanded in k around inf 72.2%

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

      1. associate-*r* 72.1%

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

      2. unpow2 72.1%

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

      3. associate-*r* 76.3%

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

      4. unpow2 76.3%

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

      5. associate-*r* 76.2%

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

      6. times-frac 77.6%

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

      7. associate-*r* 73.4%

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

      8. unpow2 73.4%

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

      9. associate-/l* 73.4%

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

      10. unpow2 73.4%

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

      11. associate-/r* 73.4%

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

   7. Simplified 73.4%

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

   8. Taylor expanded in k around 0 73.4%

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

      1. associate-*l/ 72.6%

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

      2. unpow2 72.6%

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

      3. associate-*l/ 73.5%

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

      4. associate-/r/ 73.4%

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

      5. *-commutative 73.4%

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

      6. associate-/r/ 73.5%

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

   10. Simplified 73.5%

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

   11. Taylor expanded in t around 0 71.9%

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

       1. *-commutative 71.9%

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

       2. associate-*r* 71.9%

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

       3. unpow2 71.9%

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

       4. associate-*r* 76.0%

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

       5. *-commutative 76.0%

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

       6. *-commutative 76.0%

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

       7. associate-*l/ 76.0%

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

       8. associate-*r/ 76.0%

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

       9. associate-*l* 76.0%

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

       10. unpow2 76.0%

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

       11. associate-/r* 76.0%

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

       12. *-commutative 76.0%

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

   13. Simplified 76.0%

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

   if 9.99999999999999921e133 < (*.f64 l l)

   1. Initial program 44.0%

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

      1. associate-/r* 44.0%

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

      2. associate-*l* 43.9%

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

      3. sqr-neg 43.9%

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

      4. associate-*l* 44.0%

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

      5. *-commutative 44.0%

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

      6. sqr-neg 44.0%

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

      7. associate-/r* 44.0%

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

   3. Simplified 42.0%

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

      1. div-inv 42.0%

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

      2. associate-/r/ 42.0%

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

   5. Applied egg-rr 42.0%

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

      1. associate-*r/ 42.0%

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

      2. *-rgt-identity 42.0%

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

      3. *-commutative 42.0%

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

      4. associate-/l* 42.0%

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

      5. associate-/l/ 42.0%

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

   7. Simplified 42.0%

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

   8. Taylor expanded in k around inf 52.3%

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

      1. unpow2 52.3%

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

      2. associate-*r* 52.3%

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

      3. times-frac 58.6%

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

      4. unpow2 58.6%

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

      5. associate-/r* 68.4%

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

   10. Simplified 68.4%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+134}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \left(\cos k \cdot \frac{\frac{2}{{\sin k}^{2}}}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 10: 66.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-224}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{\frac{2}{{t}^{3}}}{k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 1.35e-224)
   (/ 2.0 (* (* t (* k (/ k l))) (/ (* k k) l)))
   (if (<= k 2.05e-135)
     (/
      (* (/ l (/ (sin k) l)) (/ (/ 2.0 (pow t 3.0)) k))
      (+ 2.0 (pow (/ k t) 2.0)))
     (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e-224) {
		tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
	} else if (k <= 2.05e-135) {
		tmp = ((l / (sin(k) / l)) * ((2.0 / pow(t, 3.0)) / k)) / (2.0 + pow((k / t), 2.0));
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.35d-224) then
        tmp = 2.0d0 / ((t * (k * (k / l))) * ((k * k) / l))
    else if (k <= 2.05d-135) then
        tmp = ((l / (sin(k) / l)) * ((2.0d0 / (t ** 3.0d0)) / k)) / (2.0d0 + ((k / t) ** 2.0d0))
    else
        tmp = 2.0d0 / ((k * (t * (k / l))) * (((sin(k) ** 2.0d0) / l) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.35e-224) {
		tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
	} else if (k <= 2.05e-135) {
		tmp = ((l / (Math.sin(k) / l)) * ((2.0 / Math.pow(t, 3.0)) / k)) / (2.0 + Math.pow((k / t), 2.0));
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 1.35e-224:
		tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l))
	elif k <= 2.05e-135:
		tmp = ((l / (math.sin(k) / l)) * ((2.0 / math.pow(t, 3.0)) / k)) / (2.0 + math.pow((k / t), 2.0))
	else:
		tmp = 2.0 / ((k * (t * (k / l))) * ((math.pow(math.sin(k), 2.0) / l) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 1.35e-224)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k * Float64(k / l))) * Float64(Float64(k * k) / l)));
	elseif (k <= 2.05e-135)
		tmp = Float64(Float64(Float64(l / Float64(sin(k) / l)) * Float64(Float64(2.0 / (t ^ 3.0)) / k)) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.35e-224)
		tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
	elseif (k <= 2.05e-135)
		tmp = ((l / (sin(k) / l)) * ((2.0 / (t ^ 3.0)) / k)) / (2.0 + ((k / t) ^ 2.0));
	else
		tmp = 2.0 / ((k * (t * (k / l))) * (((sin(k) ^ 2.0) / l) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 1.35e-224], N[(2.0 / N[(N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.05e-135], N[(N[(N[(l / N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 1.34999999999999999e-224

   1. Initial program 50.3%

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

   2. Taylor expanded in t around 0 59.4%

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

      1. associate-*r* 59.4%

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

      2. unpow2 59.4%

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

      3. associate-*l* 59.4%

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

      4. times-frac 66.5%

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

      5. unpow2 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. unpow2 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in k around 0 60.4%

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

      1. associate-/l* 61.8%

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

      2. unpow2 61.8%

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

   10. Simplified 61.8%

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

   11. Taylor expanded in k around 0 60.4%

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

       1. associate-*l/ 66.9%

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

       2. unpow2 66.9%

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

       3. associate-*l/ 69.9%

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

       4. associate-/r/ 69.9%

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

       5. *-commutative 69.9%

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

       6. associate-/r/ 69.9%

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

   13. Simplified 60.7%

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

   if 1.34999999999999999e-224 < k < 2.05000000000000005e-135

   1. Initial program 73.8%

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

      1. associate-/r* 73.8%

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

      2. associate-*l* 51.7%

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

      3. sqr-neg 51.7%

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

      4. associate-*l* 73.8%

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

      5. *-commutative 73.8%

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

      6. sqr-neg 73.8%

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

      7. associate-/r* 73.8%

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

   3. Simplified 82.3%

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

      1. div-inv 82.3%

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

      2. associate-/r/ 82.4%

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

   5. Applied egg-rr 82.4%

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

      1. associate-*r/ 82.4%

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

      2. *-rgt-identity 82.4%

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

      3. *-commutative 82.4%

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

      4. associate-/l* 82.2%

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

      5. associate-/l/ 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in k around 0 82.5%

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

      1. *-commutative 82.5%

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

      2. associate-/r* 82.4%

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

   10. Simplified 82.4%

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

   if 2.05000000000000005e-135 < k

   1. Initial program 44.3%

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

   2. Taylor expanded in t around 0 67.5%

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

      1. associate-*r* 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*l* 67.5%

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

      4. times-frac 71.5%

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

      5. unpow2 71.5%

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

   4. Simplified 71.5%

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

   5. Taylor expanded in k around inf 67.5%

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

      1. associate-*r* 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*r* 69.8%

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

      4. unpow2 69.8%

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

      5. associate-*r* 69.8%

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

      6. times-frac 73.8%

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

      7. associate-*r* 71.5%

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

      8. unpow2 71.5%

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

      9. associate-/l* 76.7%

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

      10. unpow2 76.7%

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

      11. associate-/r* 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in k around 0 71.5%

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

      1. associate-*l/ 76.7%

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

      2. unpow2 76.7%

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

      3. associate-*l/ 81.3%

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

      4. associate-/r/ 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-/r/ 81.3%

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

   10. Simplified 81.3%

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

   11. Taylor expanded in t around 0 71.5%

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

       1. associate-*l/ 76.7%

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

       2. unpow2 76.7%

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

       3. associate-*r/ 81.3%

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

       4. associate-*l* 83.6%

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

   13. Simplified 83.6%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.35 \cdot 10^{-224}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{\frac{2}{{t}^{3}}}{k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 11: 67.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 7.5e-134)
   (/
    2.0
    (*
     (+ 1.0 (+ (pow (/ k t) 2.0) 1.0))
     (* (tan k) (* (/ k l) (/ (pow t 3.0) l)))))
   (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.5e-134) {
		tmp = 2.0 / ((1.0 + (pow((k / t), 2.0) + 1.0)) * (tan(k) * ((k / l) * (pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 7.5d-134) then
        tmp = 2.0d0 / ((1.0d0 + (((k / t) ** 2.0d0) + 1.0d0)) * (tan(k) * ((k / l) * ((t ** 3.0d0) / l))))
    else
        tmp = 2.0d0 / ((k * (t * (k / l))) * (((sin(k) ** 2.0d0) / l) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 7.5e-134) {
		tmp = 2.0 / ((1.0 + (Math.pow((k / t), 2.0) + 1.0)) * (Math.tan(k) * ((k / l) * (Math.pow(t, 3.0) / l))));
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 7.5e-134:
		tmp = 2.0 / ((1.0 + (math.pow((k / t), 2.0) + 1.0)) * (math.tan(k) * ((k / l) * (math.pow(t, 3.0) / l))))
	else:
		tmp = 2.0 / ((k * (t * (k / l))) * ((math.pow(math.sin(k), 2.0) / l) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 7.5e-134)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)) * Float64(tan(k) * Float64(Float64(k / l) * Float64((t ^ 3.0) / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 7.5e-134)
		tmp = 2.0 / ((1.0 + (((k / t) ^ 2.0) + 1.0)) * (tan(k) * ((k / l) * ((t ^ 3.0) / l))));
	else
		tmp = 2.0 / ((k * (t * (k / l))) * (((sin(k) ^ 2.0) / l) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 7.5e-134], N[(2.0 / N[(N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.50000000000000048e-134

   1. Initial program 51.8%

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

   2. Taylor expanded in k around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

      3. times-frac 55.6%

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

   4. Simplified 55.6%

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

   if 7.50000000000000048e-134 < k

   1. Initial program 44.3%

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

   2. Taylor expanded in t around 0 67.5%

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

      1. associate-*r* 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*l* 67.5%

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

      4. times-frac 71.5%

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

      5. unpow2 71.5%

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

   4. Simplified 71.5%

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

   5. Taylor expanded in k around inf 67.5%

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

      1. associate-*r* 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*r* 69.8%

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

      4. unpow2 69.8%

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

      5. associate-*r* 69.8%

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

      6. times-frac 73.8%

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

      7. associate-*r* 71.5%

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

      8. unpow2 71.5%

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

      9. associate-/l* 76.7%

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

      10. unpow2 76.7%

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

      11. associate-/r* 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in k around 0 71.5%

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

      1. associate-*l/ 76.7%

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

      2. unpow2 76.7%

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

      3. associate-*l/ 81.3%

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

      4. associate-/r/ 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-/r/ 81.3%

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

   10. Simplified 81.3%

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

   11. Taylor expanded in t around 0 71.5%

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

       1. associate-*l/ 76.7%

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

       2. unpow2 76.7%

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

       3. associate-*r/ 81.3%

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

       4. associate-*l* 83.6%

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

   13. Simplified 83.6%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 12: 67.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \frac{k \cdot \frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 6.6e-136)
   (/
    2.0
    (*
     (+ 1.0 (+ (pow (/ k t) 2.0) 1.0))
     (* (tan k) (/ (* k (/ (pow t 3.0) l)) l))))
   (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-136) {
		tmp = 2.0 / ((1.0 + (pow((k / t), 2.0) + 1.0)) * (tan(k) * ((k * (pow(t, 3.0) / l)) / l)));
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.6d-136) then
        tmp = 2.0d0 / ((1.0d0 + (((k / t) ** 2.0d0) + 1.0d0)) * (tan(k) * ((k * ((t ** 3.0d0) / l)) / l)))
    else
        tmp = 2.0d0 / ((k * (t * (k / l))) * (((sin(k) ** 2.0d0) / l) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.6e-136) {
		tmp = 2.0 / ((1.0 + (Math.pow((k / t), 2.0) + 1.0)) * (Math.tan(k) * ((k * (Math.pow(t, 3.0) / l)) / l)));
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 6.6e-136:
		tmp = 2.0 / ((1.0 + (math.pow((k / t), 2.0) + 1.0)) * (math.tan(k) * ((k * (math.pow(t, 3.0) / l)) / l)))
	else:
		tmp = 2.0 / ((k * (t * (k / l))) * ((math.pow(math.sin(k), 2.0) / l) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 6.6e-136)
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64((Float64(k / t) ^ 2.0) + 1.0)) * Float64(tan(k) * Float64(Float64(k * Float64((t ^ 3.0) / l)) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.6e-136)
		tmp = 2.0 / ((1.0 + (((k / t) ^ 2.0) + 1.0)) * (tan(k) * ((k * ((t ^ 3.0) / l)) / l)));
	else
		tmp = 2.0 / ((k * (t * (k / l))) * (((sin(k) ^ 2.0) / l) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 6.6e-136], N[(2.0 / N[(N[(1.0 + N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[(k * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.60000000000000035e-136

   1. Initial program 51.8%

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

   2. Taylor expanded in k around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

      3. times-frac 55.6%

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

   4. Simplified 55.6%

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

      1. associate-*r/ 55.7%

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

   6. Applied egg-rr 55.7%

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

   if 6.60000000000000035e-136 < k

   1. Initial program 44.3%

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

   2. Taylor expanded in t around 0 67.5%

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

      1. associate-*r* 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*l* 67.5%

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

      4. times-frac 71.5%

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

      5. unpow2 71.5%

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

   4. Simplified 71.5%

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

   5. Taylor expanded in k around inf 67.5%

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

      1. associate-*r* 67.5%

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

      2. unpow2 67.5%

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

      3. associate-*r* 69.8%

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

      4. unpow2 69.8%

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

      5. associate-*r* 69.8%

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

      6. times-frac 73.8%

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

      7. associate-*r* 71.5%

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

      8. unpow2 71.5%

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

      9. associate-/l* 76.7%

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

      10. unpow2 76.7%

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

      11. associate-/r* 76.7%

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

   7. Simplified 76.7%

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

   8. Taylor expanded in k around 0 71.5%

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

      1. associate-*l/ 76.7%

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

      2. unpow2 76.7%

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

      3. associate-*l/ 81.3%

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

      4. associate-/r/ 81.3%

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

      5. *-commutative 81.3%

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

      6. associate-/r/ 81.3%

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

   10. Simplified 81.3%

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

   11. Taylor expanded in t around 0 71.5%

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

       1. associate-*l/ 76.7%

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

       2. unpow2 76.7%

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

       3. associate-*r/ 81.3%

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

       4. associate-*l* 83.6%

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

   13. Simplified 83.6%

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

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.6 \cdot 10^{-136}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \frac{k \cdot \frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 13: 62.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{k \cdot k}{\ell}\\ \mathbf{if}\;k \leq 2.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot t_1}\\ \mathbf{elif}\;k \leq 1.36 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{k}}{{t}^{3}} \cdot \frac{\cos k}{\sin k}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{t_1}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (/ (* k k) l)))
   (if (<= k 2.8e-224)
     (/ 2.0 (* (* t (* k (/ k l))) t_1))
     (if (<= k 1.36e-133)
       (* (/ (/ (* l l) k) (pow t 3.0)) (/ (cos k) (sin k)))
       (if (<= k 1.8e+16)
         (/ 2.0 (* (/ (* k k) (/ l t)) (/ t_1 (cos k))))
         (*
          2.0
          (* (* l (/ l (* k k))) (/ (cos k) (* t (pow (sin k) 2.0))))))))))

C

double code(double t, double l, double k) {
	double t_1 = (k * k) / l;
	double tmp;
	if (k <= 2.8e-224) {
		tmp = 2.0 / ((t * (k * (k / l))) * t_1);
	} else if (k <= 1.36e-133) {
		tmp = (((l * l) / k) / pow(t, 3.0)) * (cos(k) / sin(k));
	} else if (k <= 1.8e+16) {
		tmp = 2.0 / (((k * k) / (l / t)) * (t_1 / cos(k)));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k * k) / l
    if (k <= 2.8d-224) then
        tmp = 2.0d0 / ((t * (k * (k / l))) * t_1)
    else if (k <= 1.36d-133) then
        tmp = (((l * l) / k) / (t ** 3.0d0)) * (cos(k) / sin(k))
    else if (k <= 1.8d+16) then
        tmp = 2.0d0 / (((k * k) / (l / t)) * (t_1 / cos(k)))
    else
        tmp = 2.0d0 * ((l * (l / (k * k))) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double t_1 = (k * k) / l;
	double tmp;
	if (k <= 2.8e-224) {
		tmp = 2.0 / ((t * (k * (k / l))) * t_1);
	} else if (k <= 1.36e-133) {
		tmp = (((l * l) / k) / Math.pow(t, 3.0)) * (Math.cos(k) / Math.sin(k));
	} else if (k <= 1.8e+16) {
		tmp = 2.0 / (((k * k) / (l / t)) * (t_1 / Math.cos(k)));
	} else {
		tmp = 2.0 * ((l * (l / (k * k))) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	t_1 = (k * k) / l
	tmp = 0
	if k <= 2.8e-224:
		tmp = 2.0 / ((t * (k * (k / l))) * t_1)
	elif k <= 1.36e-133:
		tmp = (((l * l) / k) / math.pow(t, 3.0)) * (math.cos(k) / math.sin(k))
	elif k <= 1.8e+16:
		tmp = 2.0 / (((k * k) / (l / t)) * (t_1 / math.cos(k)))
	else:
		tmp = 2.0 * ((l * (l / (k * k))) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	t_1 = Float64(Float64(k * k) / l)
	tmp = 0.0
	if (k <= 2.8e-224)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k * Float64(k / l))) * t_1));
	elseif (k <= 1.36e-133)
		tmp = Float64(Float64(Float64(Float64(l * l) / k) / (t ^ 3.0)) * Float64(cos(k) / sin(k)));
	elseif (k <= 1.8e+16)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l / t)) * Float64(t_1 / cos(k))));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / Float64(k * k))) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	t_1 = (k * k) / l;
	tmp = 0.0;
	if (k <= 2.8e-224)
		tmp = 2.0 / ((t * (k * (k / l))) * t_1);
	elseif (k <= 1.36e-133)
		tmp = (((l * l) / k) / (t ^ 3.0)) * (cos(k) / sin(k));
	elseif (k <= 1.8e+16)
		tmp = 2.0 / (((k * k) / (l / t)) * (t_1 / cos(k)));
	else
		tmp = 2.0 * ((l * (l / (k * k))) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]}, If[LessEqual[k, 2.8e-224], N[(2.0 / N[(N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.36e-133], N[(N[(N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.8e+16], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if k < 2.7999999999999998e-224

   1. Initial program 50.3%

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

   2. Taylor expanded in t around 0 59.4%

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

      1. associate-*r* 59.4%

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

      2. unpow2 59.4%

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

      3. associate-*l* 59.4%

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

      4. times-frac 66.5%

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

      5. unpow2 66.5%

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

   4. Simplified 66.5%

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

   5. Taylor expanded in k around 0 60.4%

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

      1. unpow2 60.4%

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

   7. Simplified 60.4%

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

   8. Taylor expanded in k around 0 60.4%

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

      1. associate-/l* 61.8%

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

      2. unpow2 61.8%

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

   10. Simplified 61.8%

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

   11. Taylor expanded in k around 0 60.4%

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

       1. associate-*l/ 66.9%

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

       2. unpow2 66.9%

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

       3. associate-*l/ 69.9%

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       4. associate-/r/ 69.9%

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       5. *-commutative 69.9%

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       6. associate-/r/ 69.9%

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

   13. Simplified 60.7%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right)} \cdot \frac{k \cdot k}{\ell}} \]

   if 2.7999999999999998e-224 < k < 1.35999999999999991e-133

   1. Initial program 73.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 73.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 51.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 51.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 73.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 73.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 73.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 82.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 82.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 82.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 82.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 82.3%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 82.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\left({t}^{3} \cdot k\right)}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Simplified 82.3%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\left({t}^{3} \cdot k\right)}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Taylor expanded in t around inf 82.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{k \cdot \left({t}^{3} \cdot \sin k\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 82.5%

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \sin k}} \]

      2. times-frac 82.4%

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k \cdot {t}^{3}} \cdot \frac{\cos k}{\sin k}} \]

      3. associate-/r* 82.2%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{k}}{{t}^{3}}} \cdot \frac{\cos k}{\sin k} \]

      4. unpow2 82.2%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{k}}{{t}^{3}} \cdot \frac{\cos k}{\sin k} \]

   9. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{k}}{{t}^{3}} \cdot \frac{\cos k}{\sin k}} \]

   if 1.35999999999999991e-133 < k < 1.8e16

   1. Initial program 50.5%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 58.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* 58.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 58.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      3. associate-*l* 58.2%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      4. times-frac 64.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      5. unpow2 64.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

   4. Simplified 64.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. Taylor expanded in k around inf 58.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   6. Step-by-step derivation

      1. associate-*r* 58.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 58.2%

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      3. associate-*r* 58.2%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      4. unpow2 58.2%

         \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      5. associate-*r* 58.2%

         \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      6. times-frac 64.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      7. associate-*r* 64.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      8. unpow2 64.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2}} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      9. associate-/l* 77.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      10. unpow2 77.3%

          \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      11. associate-/r* 77.3%

          \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}} \]

   7. Simplified 77.3%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}} \]

   8. Taylor expanded in k around 0 72.9%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\color{blue}{\frac{{k}^{2}}{\ell}}}{\cos k}} \]
   9. Step-by-step derivation

      1. unpow2 72.9%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\frac{\color{blue}{k \cdot k}}{\ell}}{\cos k}} \]

   10. Simplified 72.9%

       \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\color{blue}{\frac{k \cdot k}{\ell}}}{\cos k}} \]

   if 1.8e16 < k

   1. Initial program 40.1%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 40.1%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 40.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 40.1%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 40.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 40.1%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 40.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 40.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. div-inv 40.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. associate-/r/ 40.1%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}\right)} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 40.1%

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 40.0%

         \[\leadsto \color{blue}{\frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}\right) \cdot 1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. *-rgt-identity 40.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. *-commutative 40.0%

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \ell}{\sin k} \cdot \frac{\frac{2}{\tan k}}{{t}^{3}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/l* 40.1%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \frac{\frac{2}{\tan k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. associate-/l/ 40.0%

         \[\leadsto \frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \color{blue}{\frac{2}{{t}^{3} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 40.0%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{2}{{t}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   8. Taylor expanded in k around inf 74.3%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   9. Step-by-step derivation

      1. times-frac 70.2%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]

      2. unpow2 70.2%

         \[\leadsto 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

      3. associate-*r/ 74.4%

         \[\leadsto 2 \cdot \left(\color{blue}{\left(\ell \cdot \frac{\ell}{{k}^{2}}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

      4. unpow2 74.4%

         \[\leadsto 2 \cdot \left(\left(\ell \cdot \frac{\ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]

   10. Simplified 74.4%

       \[\leadsto \color{blue}{2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.8 \cdot 10^{-224}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{elif}\;k \leq 1.36 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{k}}{{t}^{3}} \cdot \frac{\cos k}{\sin k}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 14: 65.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-226}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{k}}{{t}^{3}} \cdot \frac{\cos k}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= k 8e-226)
   (/ 2.0 (* (* t (* k (/ k l))) (/ (* k k) l)))
   (if (<= k 1.25e-134)
     (* (/ (/ (* l l) k) (pow t 3.0)) (/ (cos k) (sin k)))
     (/ 2.0 (* (* k (* t (/ k l))) (/ (/ (pow (sin k) 2.0) l) (cos k)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (k <= 8e-226) {
		tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
	} else if (k <= 1.25e-134) {
		tmp = (((l * l) / k) / pow(t, 3.0)) * (cos(k) / sin(k));
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((pow(sin(k), 2.0) / l) / cos(k)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 8d-226) then
        tmp = 2.0d0 / ((t * (k * (k / l))) * ((k * k) / l))
    else if (k <= 1.25d-134) then
        tmp = (((l * l) / k) / (t ** 3.0d0)) * (cos(k) / sin(k))
    else
        tmp = 2.0d0 / ((k * (t * (k / l))) * (((sin(k) ** 2.0d0) / l) / cos(k)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 8e-226) {
		tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
	} else if (k <= 1.25e-134) {
		tmp = (((l * l) / k) / Math.pow(t, 3.0)) * (Math.cos(k) / Math.sin(k));
	} else {
		tmp = 2.0 / ((k * (t * (k / l))) * ((Math.pow(Math.sin(k), 2.0) / l) / Math.cos(k)));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if k <= 8e-226:
		tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l))
	elif k <= 1.25e-134:
		tmp = (((l * l) / k) / math.pow(t, 3.0)) * (math.cos(k) / math.sin(k))
	else:
		tmp = 2.0 / ((k * (t * (k / l))) * ((math.pow(math.sin(k), 2.0) / l) / math.cos(k)))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (k <= 8e-226)
		tmp = Float64(2.0 / Float64(Float64(t * Float64(k * Float64(k / l))) * Float64(Float64(k * k) / l)));
	elseif (k <= 1.25e-134)
		tmp = Float64(Float64(Float64(Float64(l * l) / k) / (t ^ 3.0)) * Float64(cos(k) / sin(k)));
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(k / l))) * Float64(Float64((sin(k) ^ 2.0) / l) / cos(k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 8e-226)
		tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
	elseif (k <= 1.25e-134)
		tmp = (((l * l) / k) / (t ^ 3.0)) * (cos(k) / sin(k));
	else
		tmp = 2.0 / ((k * (t * (k / l))) * (((sin(k) ^ 2.0) / l) / cos(k)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[k, 8e-226], N[(2.0 / N[(N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.25e-134], N[(N[(N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(k * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if k < 7.99999999999999937e-226

   1. Initial program 50.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 59.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* 59.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 59.4%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      3. associate-*l* 59.4%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      4. times-frac 66.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      5. unpow2 66.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

   4. Simplified 66.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. Taylor expanded in k around 0 60.4%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}} \]
   6. Step-by-step derivation

      1. unpow2 60.4%

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}} \]

   7. Simplified 60.4%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot k}{\ell}}} \]

   8. Taylor expanded in k around 0 60.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
   9. Step-by-step derivation

      1. associate-/l* 61.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

      2. unpow2 61.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{k \cdot k}{\ell}} \]

   10. Simplified 61.8%

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

   11. Taylor expanded in k around 0 60.4%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
   12. Step-by-step derivation

       1. associate-*l/ 66.9%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot t\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       2. unpow2 66.9%

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       3. associate-*l/ 69.9%

          \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       4. associate-/r/ 69.9%

          \[\leadsto \frac{2}{\left(\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       5. *-commutative 69.9%

          \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       6. associate-/r/ 69.9%

          \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

   13. Simplified 60.7%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right)} \cdot \frac{k \cdot k}{\ell}} \]

   if 7.99999999999999937e-226 < k < 1.2500000000000001e-134

   1. Initial program 73.8%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 73.8%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 51.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 51.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 73.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 73.8%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 73.8%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 82.5%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 82.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 82.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 82.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 82.3%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\left(k \cdot {t}^{3}\right)}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
   5. Step-by-step derivation

      1. *-commutative 82.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\left({t}^{3} \cdot k\right)}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   6. Simplified 82.3%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\left({t}^{3} \cdot k\right)}} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Taylor expanded in t around inf 82.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2} \cdot \cos k}{k \cdot \left({t}^{3} \cdot \sin k\right)}} \]
   8. Step-by-step derivation

      1. associate-*r* 82.5%

         \[\leadsto \frac{{\ell}^{2} \cdot \cos k}{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot \sin k}} \]

      2. times-frac 82.4%

         \[\leadsto \color{blue}{\frac{{\ell}^{2}}{k \cdot {t}^{3}} \cdot \frac{\cos k}{\sin k}} \]

      3. associate-/r* 82.2%

         \[\leadsto \color{blue}{\frac{\frac{{\ell}^{2}}{k}}{{t}^{3}}} \cdot \frac{\cos k}{\sin k} \]

      4. unpow2 82.2%

         \[\leadsto \frac{\frac{\color{blue}{\ell \cdot \ell}}{k}}{{t}^{3}} \cdot \frac{\cos k}{\sin k} \]

   9. Simplified 82.2%

      \[\leadsto \color{blue}{\frac{\frac{\ell \cdot \ell}{k}}{{t}^{3}} \cdot \frac{\cos k}{\sin k}} \]

   if 1.2500000000000001e-134 < k

   1. Initial program 44.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 67.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* 67.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 67.5%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      3. associate-*l* 67.5%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      4. times-frac 71.5%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      5. unpow2 71.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

   4. Simplified 71.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. Taylor expanded in k around inf 67.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   6. Step-by-step derivation

      1. associate-*r* 67.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 67.5%

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      3. associate-*r* 69.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      4. unpow2 69.8%

         \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      5. associate-*r* 69.8%

         \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      6. times-frac 73.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      7. associate-*r* 71.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      8. unpow2 71.5%

         \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2}} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      9. associate-/l* 76.7%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      10. unpow2 76.7%

          \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      11. associate-/r* 76.7%

          \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}} \]

   7. Simplified 76.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}} \]

   8. Taylor expanded in k around 0 71.5%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]
   9. Step-by-step derivation

      1. associate-*l/ 76.7%

         \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot t\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

      2. unpow2 76.7%

         \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

      3. associate-*l/ 81.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

      4. associate-/r/ 81.3%

         \[\leadsto \frac{2}{\left(\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

      5. *-commutative 81.3%

         \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

      6. associate-/r/ 81.3%

         \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

   10. Simplified 81.3%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

   11. Taylor expanded in t around 0 71.5%

       \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]
   12. Step-by-step derivation

       1. associate-*l/ 76.7%

          \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot t\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       2. unpow2 76.7%

          \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       3. associate-*r/ 81.3%

          \[\leadsto \frac{2}{\left(\color{blue}{\left(k \cdot \frac{k}{\ell}\right)} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

       4. associate-*l* 83.6%

          \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(\frac{k}{\ell} \cdot t\right)\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

   13. Simplified 83.6%

       \[\leadsto \frac{2}{\color{blue}{\left(k \cdot \left(\frac{k}{\ell} \cdot t\right)\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-226}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k \cdot k}{\ell}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-134}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{k}}{{t}^{3}} \cdot \frac{\cos k}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}\\ \end{array} \]

Alternative 15: 68.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3800000000000:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 3800000000000.0)
   (* 2.0 (* (/ l (* t (* k k))) (/ (* l (cos k)) (pow (sin k) 2.0))))
   (* (/ (/ l k) k) (/ l (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 3800000000000.0) {
		tmp = 2.0 * ((l / (t * (k * k))) * ((l * cos(k)) / pow(sin(k), 2.0)));
	} else {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 3800000000000.0d0) then
        tmp = 2.0d0 * ((l / (t * (k * k))) * ((l * cos(k)) / (sin(k) ** 2.0d0)))
    else
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 3800000000000.0) {
		tmp = 2.0 * ((l / (t * (k * k))) * ((l * Math.cos(k)) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 3800000000000.0:
		tmp = 2.0 * ((l / (t * (k * k))) * ((l * math.cos(k)) / math.pow(math.sin(k), 2.0)))
	else:
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 3800000000000.0)
		tmp = Float64(2.0 * Float64(Float64(l / Float64(t * Float64(k * k))) * Float64(Float64(l * cos(k)) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 3800000000000.0)
		tmp = 2.0 * ((l / (t * (k * k))) * ((l * cos(k)) / (sin(k) ^ 2.0)));
	else
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 3800000000000.0], N[(2.0 * N[(N[(l / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.8e12

   1. Initial program 46.3%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 46.3%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 44.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 44.9%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 46.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 46.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 46.3%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 46.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 46.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 45.7%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 45.7%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around inf 67.6%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   5. Step-by-step derivation

      1. unpow2 67.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*l* 67.6%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. associate-*r* 67.6%

         \[\leadsto 2 \cdot \frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

      4. times-frac 74.1%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2} \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)} \]

      5. unpow2 74.1%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right) \]

   6. Simplified 74.1%

      \[\leadsto \color{blue}{2 \cdot \left(\frac{\ell}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)} \]

   if 3.8e12 < t

   1. Initial program 58.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 59.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 52.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 52.2%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 59.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 59.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 63.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 61.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 61.5%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 48.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 48.9%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 53.9%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 53.9%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 53.9%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 53.9%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{{t}^{3}} \]
   8. Step-by-step derivation

      1. unpow2 53.9%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

      2. associate-/r* 57.1%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

   9. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3800000000000:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 16: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= l 4.7e-181)
   (/ 2.0 (* (/ (* k k) l) (* (* k k) (/ 1.0 (/ l t)))))
   (* 2.0 (* (/ (/ l k) k) (/ (* l (cos k)) (* t (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.7e-181) {
		tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
	} else {
		tmp = 2.0 * (((l / k) / k) * ((l * cos(k)) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (l <= 4.7d-181) then
        tmp = 2.0d0 / (((k * k) / l) * ((k * k) * (1.0d0 / (l / t))))
    else
        tmp = 2.0d0 * (((l / k) / k) * ((l * cos(k)) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (l <= 4.7e-181) {
		tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
	} else {
		tmp = 2.0 * (((l / k) / k) * ((l * Math.cos(k)) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if l <= 4.7e-181:
		tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))))
	else:
		tmp = 2.0 * (((l / k) / k) * ((l * math.cos(k)) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (l <= 4.7e-181)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * k) * Float64(1.0 / Float64(l / t)))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / k) * Float64(Float64(l * cos(k)) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (l <= 4.7e-181)
		tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
	else
		tmp = 2.0 * (((l / k) / k) * ((l * cos(k)) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[l, 4.7e-181], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(1.0 / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 4.6999999999999998e-181

   1. Initial program 49.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 58.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* 58.7%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 58.7%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      3. associate-*l* 58.7%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      4. times-frac 65.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      5. unpow2 65.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

   4. Simplified 65.0%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. Taylor expanded in k around 0 56.7%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}} \]
   6. Step-by-step derivation

      1. unpow2 56.7%

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}} \]

   7. Simplified 56.7%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot k}{\ell}}} \]

   8. Taylor expanded in k around 0 56.7%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
   9. Step-by-step derivation

      1. associate-/l* 61.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

      2. unpow2 61.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{k \cdot k}{\ell}} \]

   10. Simplified 61.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]
   11. Step-by-step derivation

       1. div-inv 62.5%

          \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot \frac{k \cdot k}{\ell}} \]

   12. Applied egg-rr 62.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot \frac{k \cdot k}{\ell}} \]

   if 4.6999999999999998e-181 < l

   1. Initial program 49.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 49.7%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 47.5%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 47.5%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 49.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 49.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 49.7%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-/r* 49.7%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 51.0%

      \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   4. Step-by-step derivation

      1. div-inv 51.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. associate-/r/ 51.0%

         \[\leadsto \color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}\right)} \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 51.0%

      \[\leadsto \color{blue}{\left(\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}\right) \cdot \frac{1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
   6. Step-by-step derivation

      1. associate-*r/ 51.0%

         \[\leadsto \color{blue}{\frac{\left(\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}\right) \cdot 1}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

      2. *-rgt-identity 51.0%

         \[\leadsto \frac{\color{blue}{\frac{\frac{2}{\tan k}}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      3. *-commutative 51.0%

         \[\leadsto \frac{\color{blue}{\frac{\ell \cdot \ell}{\sin k} \cdot \frac{\frac{2}{\tan k}}{{t}^{3}}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      4. associate-/l* 51.4%

         \[\leadsto \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \frac{\frac{2}{\tan k}}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

      5. associate-/l/ 51.4%

         \[\leadsto \frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \color{blue}{\frac{2}{{t}^{3} \cdot \tan k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   7. Simplified 51.4%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{2}{{t}^{3} \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   8. Taylor expanded in k around inf 63.2%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
   9. Step-by-step derivation

      1. unpow2 63.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      2. associate-*r* 63.2%

         \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)} \]

      3. times-frac 67.4%

         \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{2}} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}\right)} \]

      4. unpow2 67.4%

         \[\leadsto 2 \cdot \left(\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}\right) \]

      5. associate-/r* 69.3%

         \[\leadsto 2 \cdot \left(\color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}\right) \]

   10. Simplified 69.3%

       \[\leadsto \color{blue}{2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 4.7 \cdot 10^{-181}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell \cdot \cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 17: 64.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 7.2e-76)
   (/ 2.0 (* (/ (* k k) (/ l t)) (/ (/ (* k k) l) (cos k))))
   (* (/ (/ l k) k) (/ l (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.2e-76) {
		tmp = 2.0 / (((k * k) / (l / t)) * (((k * k) / l) / cos(k)));
	} else {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 7.2d-76) then
        tmp = 2.0d0 / (((k * k) / (l / t)) * (((k * k) / l) / cos(k)))
    else
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.2e-76) {
		tmp = 2.0 / (((k * k) / (l / t)) * (((k * k) / l) / Math.cos(k)));
	} else {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 7.2e-76:
		tmp = 2.0 / (((k * k) / (l / t)) * (((k * k) / l) / math.cos(k)))
	else:
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 7.2e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / Float64(l / t)) * Float64(Float64(Float64(k * k) / l) / cos(k))));
	else
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 7.2e-76)
		tmp = 2.0 / (((k * k) / (l / t)) * (((k * k) / l) / cos(k)));
	else
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 7.2e-76], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.2000000000000001e-76

   1. Initial program 43.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 66.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* 66.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 66.8%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      3. associate-*l* 66.7%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      4. times-frac 73.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      5. unpow2 73.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

   4. Simplified 73.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. Taylor expanded in k around inf 66.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   6. Step-by-step derivation

      1. associate-*r* 66.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 66.8%

         \[\leadsto \frac{2}{\frac{\left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      3. associate-*r* 71.1%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot {\sin k}^{2}}{{\ell}^{2} \cdot \cos k}} \]

      4. unpow2 71.1%

         \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      5. associate-*r* 71.1%

         \[\leadsto \frac{2}{\frac{\left(k \cdot \left(k \cdot t\right)\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      6. times-frac 78.3%

         \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      7. associate-*r* 73.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      8. unpow2 73.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{{k}^{2}} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      9. associate-/l* 76.2%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      10. unpow2 76.2%

          \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

      11. associate-/r* 76.2%

          \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \color{blue}{\frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}} \]

   7. Simplified 76.2%

      \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}}} \]

   8. Taylor expanded in k around 0 67.6%

      \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\color{blue}{\frac{{k}^{2}}{\ell}}}{\cos k}} \]
   9. Step-by-step derivation

      1. unpow2 67.6%

         \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\frac{\color{blue}{k \cdot k}}{\ell}}{\cos k}} \]

   10. Simplified 67.6%

       \[\leadsto \frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\color{blue}{\frac{k \cdot k}{\ell}}}{\cos k}} \]

   if 7.2000000000000001e-76 < t

   1. Initial program 61.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 62.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 56.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 56.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 62.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 62.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 62.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 65.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 63.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 63.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 51.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 55.8%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 55.8%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 55.8%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{{t}^{3}} \]
   8. Step-by-step derivation

      1. unpow2 55.8%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

      2. associate-/r* 59.5%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

   9. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\frac{\ell}{t}} \cdot \frac{\frac{k \cdot k}{\ell}}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 18: 60.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 1.95e-28)
   (/ 2.0 (* (/ (* k k) l) (/ (* k k) (/ l t))))
   (* (/ l (pow t 3.0)) (/ l (* k k)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.95e-28) {
		tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
	} else {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.95d-28) then
        tmp = 2.0d0 / (((k * k) / l) * ((k * k) / (l / t)))
    else
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.95e-28) {
		tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
	} else {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 1.95e-28:
		tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)))
	else:
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.95e-28)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * k) / Float64(l / t))));
	else
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.95e-28)
		tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
	else
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 1.95e-28], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.94999999999999999e-28

   1. Initial program 44.7%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 67.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* 67.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 67.4%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      3. associate-*l* 67.3%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      4. times-frac 73.8%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      5. unpow2 73.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

   4. Simplified 73.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. Taylor expanded in k around 0 63.4%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}} \]
   6. Step-by-step derivation

      1. unpow2 63.4%

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}} \]

   7. Simplified 63.4%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot k}{\ell}}} \]

   8. Taylor expanded in k around 0 63.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
   9. Step-by-step derivation

      1. associate-/l* 67.0%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

      2. unpow2 67.0%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{k \cdot k}{\ell}} \]

   10. Simplified 67.0%

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

   if 1.94999999999999999e-28 < t

   1. Initial program 61.0%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 61.2%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 55.4%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 55.4%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 61.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 61.2%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 61.2%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 64.6%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 63.3%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 63.3%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 63.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 50.1%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 50.1%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 54.4%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 54.4%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 54.4%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.95 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \end{array} \]

Alternative 19: 62.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (if (<= t 7.5e-76)
   (/ 2.0 (* (/ (* k k) l) (/ (* k k) (/ l t))))
   (* (/ (/ l k) k) (/ l (pow t 3.0)))))

C

double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.5e-76) {
		tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
	} else {
		tmp = ((l / k) / k) * (l / pow(t, 3.0));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 7.5d-76) then
        tmp = 2.0d0 / (((k * k) / l) * ((k * k) / (l / t)))
    else
        tmp = ((l / k) / k) * (l / (t ** 3.0d0))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 7.5e-76) {
		tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
	} else {
		tmp = ((l / k) / k) * (l / Math.pow(t, 3.0));
	}
	return tmp;
}

Python

def code(t, l, k):
	tmp = 0
	if t <= 7.5e-76:
		tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)))
	else:
		tmp = ((l / k) / k) * (l / math.pow(t, 3.0))
	return tmp

Julia

function code(t, l, k)
	tmp = 0.0
	if (t <= 7.5e-76)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * k) / Float64(l / t))));
	else
		tmp = Float64(Float64(Float64(l / k) / k) * Float64(l / (t ^ 3.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 7.5e-76)
		tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
	else
		tmp = ((l / k) / k) * (l / (t ^ 3.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := If[LessEqual[t, 7.5e-76], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.4999999999999997e-76

   1. Initial program 43.6%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

   2. Taylor expanded in t around 0 66.8%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
   3. Step-by-step derivation

      1. associate-*r* 66.8%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

      2. unpow2 66.8%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

      3. associate-*l* 66.7%

         \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

      4. times-frac 73.4%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

      5. unpow2 73.4%

         \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

   4. Simplified 73.4%

      \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. Taylor expanded in k around 0 63.1%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}} \]
   6. Step-by-step derivation

      1. unpow2 63.1%

         \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}} \]

   7. Simplified 63.1%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot k}{\ell}}} \]

   8. Taylor expanded in k around 0 63.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
   9. Step-by-step derivation

      1. associate-/l* 66.9%

         \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

      2. unpow2 66.9%

         \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{k \cdot k}{\ell}} \]

   10. Simplified 66.9%

       \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

   if 7.4999999999999997e-76 < t

   1. Initial program 61.9%

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/r* 62.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

      2. associate-*l* 56.7%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      3. sqr-neg 56.7%

         \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      4. associate-*l* 62.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      5. *-commutative 62.0%

         \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      6. sqr-neg 62.0%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      7. associate-*l/ 65.1%

         \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      8. associate-*r/ 63.9%

         \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

      9. associate-/r/ 63.9%

         \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   3. Simplified 63.9%

      \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   4. Taylor expanded in k around 0 51.8%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]
   5. Step-by-step derivation

      1. unpow2 51.8%

         \[\leadsto \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]

      2. times-frac 55.8%

         \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}} \cdot \frac{\ell}{{t}^{3}}} \]

      3. unpow2 55.8%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

   6. Simplified 55.8%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}} \]

   7. Taylor expanded in l around 0 55.8%

      \[\leadsto \color{blue}{\frac{\ell}{{k}^{2}}} \cdot \frac{\ell}{{t}^{3}} \]
   8. Step-by-step derivation

      1. unpow2 55.8%

         \[\leadsto \frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\ell}{{t}^{3}} \]

      2. associate-/r* 59.5%

         \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]

   9. Simplified 59.5%

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k}}{k}} \cdot \frac{\ell}{{t}^{3}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{t}^{3}}\\ \end{array} \]

Alternative 20: 59.2% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (/ (* k k) l) (* (* k k) (/ 1.0 (/ l t))))))

C

double code(double t, double l, double k) {
	return 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((k * k) / l) * ((k * k) * (1.0d0 / (l / t))))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
}

Python

def code(t, l, k):
	return 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * k) * Float64(1.0 / Float64(l / t)))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((k * k) / l) * ((k * k) * (1.0 / (l / t))));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] * N[(1.0 / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 60.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* 60.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   2. unpow2 60.8%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

   3. associate-*l* 60.8%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

   4. times-frac 66.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. unpow2 66.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

4. Simplified 66.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

5. Taylor expanded in k around 0 57.7%

   \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}} \]
6. Step-by-step derivation

   1. unpow2 57.7%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}} \]

7. Simplified 57.7%

   \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot k}{\ell}}} \]

8. Taylor expanded in k around 0 57.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
9. Step-by-step derivation

   1. associate-/l* 61.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

   2. unpow2 61.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{k \cdot k}{\ell}} \]

10. Simplified 61.1%

    \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]
11. Step-by-step derivation

    1. div-inv 61.4%

       \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot \frac{k \cdot k}{\ell}} \]

12. Applied egg-rr 61.4%

    \[\leadsto \frac{2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)} \cdot \frac{k \cdot k}{\ell}} \]

13. Final simplification 61.4%

    \[\leadsto \frac{2}{\frac{k \cdot k}{\ell} \cdot \left(\left(k \cdot k\right) \cdot \frac{1}{\frac{\ell}{t}}\right)} \]

Alternative 21: 57.1% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k \cdot k}{\ell}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (* t (* k (/ k l))) (/ (* k k) l))))

C

double code(double t, double l, double k) {
	return 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / ((t * (k * (k / l))) * ((k * k) / l))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
}

Python

def code(t, l, k):
	return 2.0 / ((t * (k * (k / l))) * ((k * k) / l))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(t * Float64(k * Float64(k / l))) * Float64(Float64(k * k) / l)))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / ((t * (k * (k / l))) * ((k * k) / l));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 60.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* 60.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   2. unpow2 60.8%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

   3. associate-*l* 60.8%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

   4. times-frac 66.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. unpow2 66.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

4. Simplified 66.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

5. Taylor expanded in k around 0 57.7%

   \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}} \]
6. Step-by-step derivation

   1. unpow2 57.7%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}} \]

7. Simplified 57.7%

   \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot k}{\ell}}} \]

8. Taylor expanded in k around 0 57.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
9. Step-by-step derivation

   1. associate-/l* 61.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

   2. unpow2 61.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{k \cdot k}{\ell}} \]

10. Simplified 61.1%

    \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

11. Taylor expanded in k around 0 57.7%

    \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
12. Step-by-step derivation

    1. associate-*l/ 68.5%

       \[\leadsto \frac{2}{\color{blue}{\left(\frac{{k}^{2}}{\ell} \cdot t\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

    2. unpow2 68.5%

       \[\leadsto \frac{2}{\left(\frac{\color{blue}{k \cdot k}}{\ell} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

    3. associate-*l/ 71.9%

       \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

    4. associate-/r/ 71.9%

       \[\leadsto \frac{2}{\left(\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot t\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

    5. *-commutative 71.9%

       \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right)} \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

    6. associate-/r/ 71.9%

       \[\leadsto \frac{2}{\left(t \cdot \color{blue}{\left(\frac{k}{\ell} \cdot k\right)}\right) \cdot \frac{\frac{{\sin k}^{2}}{\ell}}{\cos k}} \]

13. Simplified 59.7%

    \[\leadsto \frac{2}{\color{blue}{\left(t \cdot \left(\frac{k}{\ell} \cdot k\right)\right)} \cdot \frac{k \cdot k}{\ell}} \]

14. Final simplification 59.7%

    \[\leadsto \frac{2}{\left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right) \cdot \frac{k \cdot k}{\ell}} \]

Alternative 22: 58.9% accurate, 28.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}} \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/ 2.0 (* (/ (* k k) l) (/ (* k k) (/ l t)))))

C

double code(double t, double l, double k) {
	return 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((k * k) / l) * ((k * k) / (l / t)))
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
}

Python

def code(t, l, k):
	return 2.0 / (((k * k) / l) * ((k * k) / (l / t)))

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(k * k) / l) * Float64(Float64(k * k) / Float64(l / t))))
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((k * k) / l) * ((k * k) / (l / t)));
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.3%

   \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]

2. Taylor expanded in t around 0 60.8%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}} \]
3. Step-by-step derivation

   1. associate-*r* 60.8%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}}{{\ell}^{2} \cdot \cos k}} \]

   2. unpow2 60.8%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}} \]

   3. associate-*l* 60.8%

      \[\leadsto \frac{2}{\frac{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}} \]

   4. times-frac 66.6%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

   5. unpow2 66.6%

      \[\leadsto \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}} \]

4. Simplified 66.6%

   \[\leadsto \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell \cdot \cos k}}} \]

5. Taylor expanded in k around 0 57.7%

   \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{{k}^{2}}{\ell}}} \]
6. Step-by-step derivation

   1. unpow2 57.7%

      \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}} \]

7. Simplified 57.7%

   \[\leadsto \frac{2}{\frac{\left(k \cdot k\right) \cdot t}{\ell} \cdot \color{blue}{\frac{k \cdot k}{\ell}}} \]

8. Taylor expanded in k around 0 57.7%

   \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{\ell}} \cdot \frac{k \cdot k}{\ell}} \]
9. Step-by-step derivation

   1. associate-/l* 61.1%

      \[\leadsto \frac{2}{\color{blue}{\frac{{k}^{2}}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

   2. unpow2 61.1%

      \[\leadsto \frac{2}{\frac{\color{blue}{k \cdot k}}{\frac{\ell}{t}} \cdot \frac{k \cdot k}{\ell}} \]

10. Simplified 61.1%

    \[\leadsto \frac{2}{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}} \cdot \frac{k \cdot k}{\ell}} \]

11. Final simplification 61.1%

    \[\leadsto \frac{2}{\frac{k \cdot k}{\ell} \cdot \frac{k \cdot k}{\frac{\ell}{t}}} \]

Reproduce

herbie shell --seed 2023285 
(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))))