Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.0% → 73.5%

Time: 18.2s

Alternatives: 15

Speedup: 3.8×

• 27.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 5.985091557265312e+111
  2. l = 1.584699178793835e+115
  3. k = 4.091818798690425e-42

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: 54.0% 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: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-214}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \ell}{{\left(t \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{k}\right)\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.55e-214)
   (/
    (* l (/ (* 2.0 l) (pow (* t (* (cbrt (tan k)) (cbrt k))) 3.0)))
    (+ 2.0 (pow (/ k t) 2.0)))
   (if (<= k 1.55e+22)
     (/ (/ l k) (/ (pow t 3.0) (/ l k)))
     (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t (pow (sin k) 2.0))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.55e-214) {
		tmp = (l * ((2.0 * l) / pow((t * (cbrt(tan(k)) * cbrt(k))), 3.0))) / (2.0 + pow((k / t), 2.0));
	} else if (k <= 1.55e+22) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.55e-214) {
		tmp = (l * ((2.0 * l) / Math.pow((t * (Math.cbrt(Math.tan(k)) * Math.cbrt(k))), 3.0))) / (2.0 + Math.pow((k / t), 2.0));
	} else if (k <= 1.55e+22) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.55e-214)
		tmp = Float64(Float64(l * Float64(Float64(2.0 * l) / (Float64(t * Float64(cbrt(tan(k)) * cbrt(k))) ^ 3.0))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	elseif (k <= 1.55e+22)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.55e-214], N[(N[(l * N[(N[(2.0 * l), $MachinePrecision] / N[Power[N[(t * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.55e+22], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $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 3 regimes

2. if k < 1.55000000000000002e-214

   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. Step-by-step derivation

      1. associate-/r* 50.3%

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

      2. associate-*l* 44.4%

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

      3. sqr-neg 44.4%

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

      4. associate-*l* 50.3%

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

      5. *-commutative 50.3%

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

      6. sqr-neg 50.3%

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

      7. associate-*l/ 50.2%

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

      8. associate-*r/ 50.8%

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

      9. associate-/r/ 50.8%

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

   3. Simplified 50.8%

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

      1. add-cube-cbrt 50.7%

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

      2. pow3 50.7%

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

      3. *-commutative 50.7%

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

      4. cbrt-prod 50.7%

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

      5. cbrt-prod 50.6%

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

      6. rem-cbrt-cube 58.9%

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

   5. Applied egg-rr 58.9%

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

      1. associate-*l/ 58.9%

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

      2. associate-*l* 58.9%

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

   7. Applied egg-rr 58.9%

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

      1. unpow2 58.9%

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

      2. *-commutative 58.9%

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

      3. associate-*r/ 58.9%

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

      4. unpow2 58.9%

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

      5. associate-*l* 67.5%

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

      6. associate-*r/ 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in k around 0 66.3%

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

   if 1.55000000000000002e-214 < k < 1.5500000000000001e22

   1. Initial program 65.1%

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

      1. associate-/r* 65.1%

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

      2. associate-*l* 65.0%

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

      3. sqr-neg 65.0%

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

      4. associate-*l* 65.1%

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

      5. *-commutative 65.1%

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

      6. sqr-neg 65.1%

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

      7. associate-*l/ 65.0%

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

      8. associate-*r/ 65.0%

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

      9. associate-/r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in k around 0 70.0%

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

      1. associate-/r* 68.3%

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

      2. unpow2 68.3%

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

      3. unpow2 68.3%

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

      4. times-frac 73.4%

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

   6. Simplified 73.4%

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

      1. add-exp-log 51.7%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 51.7%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 51.7%

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

      1. add-exp-log 73.4%

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

      2. pow2 73.4%

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

      3. associate-/l* 78.4%

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

   10. Applied egg-rr 78.4%

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

   if 1.5500000000000001e22 < k

   1. Initial program 48.0%

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

      1. associate-/r* 48.0%

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

      2. associate-*l* 48.0%

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

      3. sqr-neg 48.0%

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

      4. associate-*l* 48.0%

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

      5. *-commutative 48.0%

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

      6. sqr-neg 48.0%

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

      7. associate-*l/ 48.0%

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

      8. associate-*r/ 48.0%

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

      9. associate-/r/ 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in k around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 69.8%

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

      4. unpow2 69.8%

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

      5. associate-*l* 69.8%

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

      6. unpow2 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in k around inf 69.9%

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

      1. times-frac 65.8%

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

      2. unpow2 65.8%

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

      3. unpow2 65.8%

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

      4. times-frac 74.7%

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

      5. unpow2 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-214}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \ell}{{\left(t \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{k}\right)\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+288}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \ell}{{\left(t \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{3}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        1e+288)
     (/
      (* l (/ (* 2.0 l) (pow (* t (* (cbrt (sin k)) (cbrt (tan k)))) 3.0)))
      (+ 2.0 t_1))
     (/ 2.0 (/ (/ (* t (pow (sin k) 2.0)) (pow (/ l k) 2.0)) (cos k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+288) {
		tmp = (l * ((2.0 * l) / pow((t * (cbrt(sin(k)) * cbrt(tan(k)))), 3.0))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((t * pow(sin(k), 2.0)) / pow((l / k), 2.0)) / cos(k));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+288) {
		tmp = (l * ((2.0 * l) / Math.pow((t * (Math.cbrt(Math.sin(k)) * Math.cbrt(Math.tan(k)))), 3.0))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((t * Math.pow(Math.sin(k), 2.0)) / Math.pow((l / k), 2.0)) / Math.cos(k));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 1e+288)
		tmp = Float64(Float64(l * Float64(Float64(2.0 * l) / (Float64(t * Float64(cbrt(sin(k)) * cbrt(tan(k)))) ^ 3.0))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * (sin(k) ^ 2.0)) / (Float64(l / k) ^ 2.0)) / cos(k)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+288], N[(N[(l * N[(N[(2.0 * l), $MachinePrecision] / N[Power[N[(t * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $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))) < 1e288

   1. Initial program 76.5%

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

      1. associate-/r* 76.5%

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

      2. associate-*l* 70.8%

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

      3. sqr-neg 70.8%

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

      4. associate-*l* 76.5%

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

      5. *-commutative 76.5%

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

      6. sqr-neg 76.5%

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

      7. associate-*l/ 76.4%

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

      8. associate-*r/ 76.9%

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

      9. associate-/r/ 76.9%

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

   3. Simplified 76.9%

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

      1. add-cube-cbrt 76.8%

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

      2. pow3 76.8%

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

      3. *-commutative 76.8%

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

      4. cbrt-prod 76.8%

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

      5. cbrt-prod 76.7%

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

      6. rem-cbrt-cube 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. associate-*l/ 85.2%

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

      2. associate-*l* 85.2%

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

   7. Applied egg-rr 85.2%

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

      1. unpow2 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-*r/ 85.1%

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

      4. unpow2 85.1%

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

      5. associate-*l* 87.7%

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

      6. associate-*r/ 88.0%

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

   9. Simplified 88.0%

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

   if 1e288 < (/.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 18.7%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

      3. associate-*l* 18.7%

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

      4. sqrt-prod 18.6%

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

      5. sqrt-div 18.6%

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

      6. sqrt-pow1 21.7%

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

      7. metadata-eval 21.7%

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

      8. sqrt-prod 10.9%

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

      9. add-sqr-sqrt 26.6%

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

   3. Applied egg-rr 26.6%

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

   4. Taylor expanded in k around 0 16.8%

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

      1. unpow2 16.8%

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

      2. unpow2 16.8%

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

   6. Simplified 16.8%

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

   7. Taylor expanded in t around 0 44.2%

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

      1. associate-/r* 44.2%

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

      2. *-commutative 44.2%

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

      3. associate-/l* 44.3%

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

      4. unpow2 44.3%

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

      5. unpow2 44.3%

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

      6. times-frac 60.1%

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

      7. unpow2 60.1%

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

   9. Simplified 60.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+288}:\\ \;\;\;\;\frac{\ell \cdot \frac{2 \cdot \ell}{{\left(t \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k}\right)\right)}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \]

Alternative 3: 76.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+288}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        1e+288)
     (/
      (* l (* l (/ 2.0 (pow (* (cbrt (tan k)) (* t (cbrt (sin k)))) 3.0))))
      (+ 2.0 t_1))
     (/ 2.0 (/ (/ (* t (pow (sin k) 2.0)) (pow (/ l k) 2.0)) (cos k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+288) {
		tmp = (l * (l * (2.0 / pow((cbrt(tan(k)) * (t * cbrt(sin(k)))), 3.0)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((t * pow(sin(k), 2.0)) / pow((l / k), 2.0)) / cos(k));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+288) {
		tmp = (l * (l * (2.0 / Math.pow((Math.cbrt(Math.tan(k)) * (t * Math.cbrt(Math.sin(k)))), 3.0)))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((t * Math.pow(Math.sin(k), 2.0)) / Math.pow((l / k), 2.0)) / Math.cos(k));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 1e+288)
		tmp = Float64(Float64(l * Float64(l * Float64(2.0 / (Float64(cbrt(tan(k)) * Float64(t * cbrt(sin(k)))) ^ 3.0)))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * (sin(k) ^ 2.0)) / (Float64(l / k) ^ 2.0)) / cos(k)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+288], N[(N[(l * N[(l * N[(2.0 / N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $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))) < 1e288

   1. Initial program 76.5%

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

      1. associate-/r* 76.5%

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

      2. associate-*l* 70.8%

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

      3. sqr-neg 70.8%

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

      4. associate-*l* 76.5%

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

      5. *-commutative 76.5%

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

      6. sqr-neg 76.5%

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

      7. associate-*l/ 76.4%

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

      8. associate-*r/ 76.9%

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

      9. associate-/r/ 76.9%

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

   3. Simplified 76.9%

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

      1. add-cube-cbrt 76.8%

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

      2. pow3 76.8%

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

      3. *-commutative 76.8%

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

      4. cbrt-prod 76.8%

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

      5. cbrt-prod 76.7%

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

      6. rem-cbrt-cube 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. associate-*l/ 85.2%

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

      2. associate-*l* 85.2%

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

   7. Applied egg-rr 85.2%

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

      1. unpow2 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-*r/ 85.1%

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

      4. unpow2 85.1%

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

      5. associate-*l* 87.7%

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

      6. associate-*r* 87.7%

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

   9. Simplified 87.7%

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

   if 1e288 < (/.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 18.7%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

      3. associate-*l* 18.7%

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

      4. sqrt-prod 18.6%

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

      5. sqrt-div 18.6%

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

      6. sqrt-pow1 21.7%

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

      7. metadata-eval 21.7%

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

      8. sqrt-prod 10.9%

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

      9. add-sqr-sqrt 26.6%

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

   3. Applied egg-rr 26.6%

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

   4. Taylor expanded in k around 0 16.8%

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

      1. unpow2 16.8%

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

      2. unpow2 16.8%

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

   6. Simplified 16.8%

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

   7. Taylor expanded in t around 0 44.2%

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

      1. associate-/r* 44.2%

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

      2. *-commutative 44.2%

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

      3. associate-/l* 44.3%

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

      4. unpow2 44.3%

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

      5. unpow2 44.3%

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

      6. times-frac 60.1%

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

      7. unpow2 60.1%

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

   9. Simplified 60.1%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+288}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \]

Alternative 4: 76.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 10^{+288}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\frac{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}{\ell}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        1e+288)
     (/
      (* l (/ 2.0 (/ (pow (* (cbrt (tan k)) (* t (cbrt (sin k)))) 3.0) l)))
      (+ 2.0 t_1))
     (/ 2.0 (/ (/ (* t (pow (sin k) 2.0)) (pow (/ l k) 2.0)) (cos k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+288) {
		tmp = (l * (2.0 / (pow((cbrt(tan(k)) * (t * cbrt(sin(k)))), 3.0) / l))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((t * pow(sin(k), 2.0)) / pow((l / k), 2.0)) / cos(k));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 1e+288) {
		tmp = (l * (2.0 / (Math.pow((Math.cbrt(Math.tan(k)) * (t * Math.cbrt(Math.sin(k)))), 3.0) / l))) / (2.0 + t_1);
	} else {
		tmp = 2.0 / (((t * Math.pow(Math.sin(k), 2.0)) / Math.pow((l / k), 2.0)) / Math.cos(k));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 1e+288)
		tmp = Float64(Float64(l * Float64(2.0 / Float64((Float64(cbrt(tan(k)) * Float64(t * cbrt(sin(k)))) ^ 3.0) / l))) / Float64(2.0 + t_1));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * (sin(k) ^ 2.0)) / (Float64(l / k) ^ 2.0)) / cos(k)));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+288], N[(N[(l * N[(2.0 / N[(N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Cos[k], $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))) < 1e288

   1. Initial program 76.5%

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

      1. associate-/r* 76.5%

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

      2. associate-*l* 70.8%

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

      3. sqr-neg 70.8%

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

      4. associate-*l* 76.5%

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

      5. *-commutative 76.5%

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

      6. sqr-neg 76.5%

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

      7. associate-*l/ 76.4%

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

      8. associate-*r/ 76.9%

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

      9. associate-/r/ 76.9%

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

   3. Simplified 76.9%

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

      1. add-cube-cbrt 76.8%

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

      2. pow3 76.8%

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

      3. *-commutative 76.8%

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

      4. cbrt-prod 76.8%

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

      5. cbrt-prod 76.7%

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

      6. rem-cbrt-cube 85.2%

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

   5. Applied egg-rr 85.2%

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

      1. associate-*l/ 85.2%

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

      2. associate-*l* 85.2%

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

   7. Applied egg-rr 85.2%

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

      1. unpow2 85.2%

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

      2. *-commutative 85.2%

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

      3. associate-*r/ 85.1%

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

      4. unpow2 85.1%

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

      5. associate-*l* 87.7%

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

      6. associate-*r* 87.7%

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

   9. Simplified 87.7%

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

       1. associate-*r/ 87.9%

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

       2. associate-*l* 88.0%

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

   11. Applied egg-rr 88.0%

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

       1. *-commutative 88.0%

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

       2. associate-/l* 88.0%

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

       3. associate-*r* 87.9%

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

       4. *-commutative 87.9%

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

   13. Simplified 87.9%

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

   if 1e288 < (/.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 18.7%

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

      1. add-sqr-sqrt 18.7%

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

      2. pow2 18.7%

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

      3. associate-*l* 18.7%

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

      4. sqrt-prod 18.6%

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

      5. sqrt-div 18.6%

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

      6. sqrt-pow1 21.7%

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

      7. metadata-eval 21.7%

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

      8. sqrt-prod 10.9%

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

      9. add-sqr-sqrt 26.6%

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

   3. Applied egg-rr 26.6%

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

   4. Taylor expanded in k around 0 16.8%

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

      1. unpow2 16.8%

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

      2. unpow2 16.8%

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

   6. Simplified 16.8%

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

   7. Taylor expanded in t around 0 44.2%

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

      1. associate-/r* 44.2%

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

      2. *-commutative 44.2%

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

      3. associate-/l* 44.3%

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

      4. unpow2 44.3%

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

      5. unpow2 44.3%

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

      6. times-frac 60.1%

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

      7. unpow2 60.1%

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

   9. Simplified 60.1%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 10^{+288}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\frac{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}{\cos k}}\\ \end{array} \]

Alternative 5: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 5.8e-214)
   (/
    (* l (* l (/ 2.0 (pow (* (cbrt (tan k)) (* t (cbrt k))) 3.0))))
    (+ 2.0 (pow (/ k t) 2.0)))
   (if (<= k 1.75e+22)
     (/ (/ l k) (/ (pow t 3.0) (/ l k)))
     (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t (pow (sin k) 2.0))))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-214) {
		tmp = (l * (l * (2.0 / pow((cbrt(tan(k)) * (t * cbrt(k))), 3.0)))) / (2.0 + pow((k / t), 2.0));
	} else if (k <= 1.75e+22) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 5.8e-214) {
		tmp = (l * (l * (2.0 / Math.pow((Math.cbrt(Math.tan(k)) * (t * Math.cbrt(k))), 3.0)))) / (2.0 + Math.pow((k / t), 2.0));
	} else if (k <= 1.75e+22) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 5.8e-214)
		tmp = Float64(Float64(l * Float64(l * Float64(2.0 / (Float64(cbrt(tan(k)) * Float64(t * cbrt(k))) ^ 3.0)))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	elseif (k <= 1.75e+22)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 5.8e-214], N[(N[(l * N[(l * N[(2.0 / N[Power[N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.75e+22], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $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 3 regimes

2. if k < 5.7999999999999997e-214

   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. Step-by-step derivation

      1. associate-/r* 50.3%

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

      2. associate-*l* 44.4%

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

      3. sqr-neg 44.4%

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

      4. associate-*l* 50.3%

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

      5. *-commutative 50.3%

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

      6. sqr-neg 50.3%

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

      7. associate-*l/ 50.2%

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

      8. associate-*r/ 50.8%

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

      9. associate-/r/ 50.8%

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

   3. Simplified 50.8%

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

      1. add-cube-cbrt 50.7%

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

      2. pow3 50.7%

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

      3. *-commutative 50.7%

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

      4. cbrt-prod 50.7%

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

      5. cbrt-prod 50.6%

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

      6. rem-cbrt-cube 58.9%

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

   5. Applied egg-rr 58.9%

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

      1. associate-*l/ 58.9%

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

      2. associate-*l* 58.9%

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

   7. Applied egg-rr 58.9%

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

      1. unpow2 58.9%

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

      2. *-commutative 58.9%

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

      3. associate-*r/ 58.9%

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

      4. unpow2 58.9%

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

      5. associate-*l* 67.5%

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

      6. associate-*r* 67.5%

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

   9. Simplified 67.5%

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

   10. Taylor expanded in k around 0 65.6%

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

   if 5.7999999999999997e-214 < k < 1.75e22

   1. Initial program 65.1%

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

      1. associate-/r* 65.1%

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

      2. associate-*l* 65.0%

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

      3. sqr-neg 65.0%

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

      4. associate-*l* 65.1%

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

      5. *-commutative 65.1%

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

      6. sqr-neg 65.1%

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

      7. associate-*l/ 65.0%

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

      8. associate-*r/ 65.0%

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

      9. associate-/r/ 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in k around 0 70.0%

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

      1. associate-/r* 68.3%

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

      2. unpow2 68.3%

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

      3. unpow2 68.3%

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

      4. times-frac 73.4%

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

   6. Simplified 73.4%

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

      1. add-exp-log 51.7%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 51.7%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 51.7%

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

      1. add-exp-log 73.4%

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

      2. pow2 73.4%

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

      3. associate-/l* 78.4%

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

   10. Applied egg-rr 78.4%

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

   if 1.75e22 < k

   1. Initial program 48.0%

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

      1. associate-/r* 48.0%

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

      2. associate-*l* 48.0%

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

      3. sqr-neg 48.0%

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

      4. associate-*l* 48.0%

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

      5. *-commutative 48.0%

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

      6. sqr-neg 48.0%

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

      7. associate-*l/ 48.0%

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

      8. associate-*r/ 48.0%

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

      9. associate-/r/ 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in k around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 69.8%

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

      4. unpow2 69.8%

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

      5. associate-*l* 69.8%

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

      6. unpow2 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in k around inf 69.9%

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

      1. times-frac 65.8%

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

      2. unpow2 65.8%

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

      3. unpow2 65.8%

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

      4. times-frac 74.7%

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

      5. unpow2 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.8 \cdot 10^{-214}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{2}{{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6: 72.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 6.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 6.3e+22)
   (/ (/ l k) (/ (pow t 3.0) (/ l k)))
   (* 2.0 (* (pow (/ l k) 2.0) (/ (cos k) (* t (pow (sin k) 2.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.3e+22) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * (pow((l / k), 2.0) * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 6.3d+22) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = 2.0d0 * (((l / k) ** 2.0d0) * (cos(k) / (t * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 6.3e+22) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * (Math.pow((l / k), 2.0) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 6.3e+22:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = 2.0 * (math.pow((l / k), 2.0) * (math.cos(k) / (t * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 6.3e+22)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64((Float64(l / k) ^ 2.0) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 6.3e+22)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = 2.0 * (((l / k) ^ 2.0) * (cos(k) / (t * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 6.3e+22], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $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 2 regimes

2. if k < 6.30000000000000021e22

   1. Initial program 54.6%

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

      1. associate-/r* 54.6%

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

      2. associate-*l* 50.4%

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

      3. sqr-neg 50.4%

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

      4. associate-*l* 54.6%

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

      5. *-commutative 54.6%

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

      6. sqr-neg 54.6%

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

      7. associate-*l/ 54.6%

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

      8. associate-*r/ 55.0%

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

      9. associate-/r/ 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in k around 0 53.9%

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

      1. associate-/r* 53.4%

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

      2. unpow2 53.4%

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

      3. unpow2 53.4%

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

      4. times-frac 65.6%

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

   6. Simplified 65.6%

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

      1. add-exp-log 42.1%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 42.1%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 42.1%

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

      1. add-exp-log 65.6%

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

      2. pow2 65.6%

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

      3. associate-/l* 68.5%

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

   10. Applied egg-rr 68.5%

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

   if 6.30000000000000021e22 < k

   1. Initial program 48.0%

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

      1. associate-/r* 48.0%

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

      2. associate-*l* 48.0%

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

      3. sqr-neg 48.0%

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

      4. associate-*l* 48.0%

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

      5. *-commutative 48.0%

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

      6. sqr-neg 48.0%

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

      7. associate-*l/ 48.0%

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

      8. associate-*r/ 48.0%

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

      9. associate-/r/ 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in k around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 69.8%

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

      4. unpow2 69.8%

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

      5. associate-*l* 69.8%

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

      6. unpow2 69.8%

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

   6. Simplified 69.8%

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

   7. Taylor expanded in k around inf 69.9%

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

      1. times-frac 65.8%

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

      2. unpow2 65.8%

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

      3. unpow2 65.8%

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

      4. times-frac 74.7%

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

      5. unpow2 74.7%

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

   9. Simplified 74.7%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.3 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 7: 69.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.75e+22)
   (/ (/ l k) (/ (pow t 3.0) (/ l k)))
   (* (/ 2.0 (* k (* t k))) (/ (* (* l l) (cos k)) (pow (sin k) 2.0)))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e+22) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = (2.0 / (k * (t * k))) * (((l * l) * cos(k)) / pow(sin(k), 2.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d+22) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = (2.0d0 / (k * (t * k))) * (((l * l) * cos(k)) / (sin(k) ** 2.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e+22) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = (2.0 / (k * (t * k))) * (((l * l) * Math.cos(k)) / Math.pow(Math.sin(k), 2.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.75e+22:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = (2.0 / (k * (t * k))) * (((l * l) * math.cos(k)) / math.pow(math.sin(k), 2.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.75e+22)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(Float64(l * l) * cos(k)) / (sin(k) ^ 2.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.75e+22)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = (2.0 / (k * (t * k))) * (((l * l) * cos(k)) / (sin(k) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.75e+22], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l * l), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e22

   1. Initial program 54.6%

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

      1. associate-/r* 54.6%

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

      2. associate-*l* 50.4%

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

      3. sqr-neg 50.4%

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

      4. associate-*l* 54.6%

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

      5. *-commutative 54.6%

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

      6. sqr-neg 54.6%

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

      7. associate-*l/ 54.6%

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

      8. associate-*r/ 55.0%

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

      9. associate-/r/ 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in k around 0 53.9%

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

      1. associate-/r* 53.4%

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

      2. unpow2 53.4%

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

      3. unpow2 53.4%

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

      4. times-frac 65.6%

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

   6. Simplified 65.6%

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

      1. add-exp-log 42.1%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 42.1%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 42.1%

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

      1. add-exp-log 65.6%

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

      2. pow2 65.6%

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

      3. associate-/l* 68.5%

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

   10. Applied egg-rr 68.5%

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

   if 1.75e22 < k

   1. Initial program 48.0%

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

      1. associate-/r* 48.0%

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

      2. associate-*l* 48.0%

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

      3. sqr-neg 48.0%

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

      4. associate-*l* 48.0%

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

      5. *-commutative 48.0%

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

      6. sqr-neg 48.0%

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

      7. associate-*l/ 48.0%

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

      8. associate-*r/ 48.0%

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

      9. associate-/r/ 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in k around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 69.8%

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

      4. unpow2 69.8%

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

      5. associate-*l* 69.8%

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

      6. unpow2 69.8%

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

   6. Simplified 69.8%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2}}\\ \end{array} \]

Alternative 8: 69.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{{\sin k}^{2} \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= k 1.75e+22)
   (/ (/ l k) (/ (pow t 3.0) (/ l k)))
   (/ (* 2.0 (* l (* l (cos k)))) (* (pow (sin k) 2.0) (* k (* t k))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e+22) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = (2.0 * (l * (l * cos(k)))) / (pow(sin(k), 2.0) * (k * (t * k)));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= 1.75d+22) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = (2.0d0 * (l * (l * cos(k)))) / ((sin(k) ** 2.0d0) * (k * (t * k)))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (k <= 1.75e+22) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = (2.0 * (l * (l * Math.cos(k)))) / (Math.pow(Math.sin(k), 2.0) * (k * (t * k)));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if k <= 1.75e+22:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = (2.0 * (l * (l * math.cos(k)))) / (math.pow(math.sin(k), 2.0) * (k * (t * k)))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (k <= 1.75e+22)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(Float64(2.0 * Float64(l * Float64(l * cos(k)))) / Float64((sin(k) ^ 2.0) * Float64(k * Float64(t * k))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= 1.75e+22)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = (2.0 * (l * (l * cos(k)))) / ((sin(k) ^ 2.0) * (k * (t * k)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[k, 1.75e+22], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.75e22

   1. Initial program 54.6%

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

      1. associate-/r* 54.6%

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

      2. associate-*l* 50.4%

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

      3. sqr-neg 50.4%

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

      4. associate-*l* 54.6%

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

      5. *-commutative 54.6%

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

      6. sqr-neg 54.6%

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

      7. associate-*l/ 54.6%

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

      8. associate-*r/ 55.0%

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

      9. associate-/r/ 55.0%

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

   3. Simplified 55.0%

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

   4. Taylor expanded in k around 0 53.9%

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

      1. associate-/r* 53.4%

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

      2. unpow2 53.4%

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

      3. unpow2 53.4%

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

      4. times-frac 65.6%

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

   6. Simplified 65.6%

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

      1. add-exp-log 42.1%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 42.1%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 42.1%

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

      1. add-exp-log 65.6%

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

      2. pow2 65.6%

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

      3. associate-/l* 68.5%

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

   10. Applied egg-rr 68.5%

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

   if 1.75e22 < k

   1. Initial program 48.0%

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

      1. associate-/r* 48.0%

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

      2. associate-*l* 48.0%

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

      3. sqr-neg 48.0%

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

      4. associate-*l* 48.0%

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

      5. *-commutative 48.0%

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

      6. sqr-neg 48.0%

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

      7. associate-*l/ 48.0%

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

      8. associate-*r/ 48.0%

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

      9. associate-/r/ 48.0%

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

   3. Simplified 48.0%

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

   4. Taylor expanded in k around inf 69.9%

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

      1. associate-*r/ 69.9%

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

      2. associate-*r* 69.9%

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

      3. times-frac 69.8%

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

      4. unpow2 69.8%

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

      5. associate-*l* 69.8%

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

      6. unpow2 69.8%

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

   6. Simplified 69.8%

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

      1. frac-times 69.8%

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

      2. associate-*l* 69.9%

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

   8. Applied egg-rr 69.9%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \left(\ell \cdot \cos k\right)\right)}{{\sin k}^{2} \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]

Alternative 9: 60.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;e^{2 \cdot \log \left(\frac{\ell}{k}\right) - 3 \cdot \log t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -1.35e-108)
   (/ (/ l k) (/ (pow t 3.0) (/ l k)))
   (if (<= t 1.65e-34)
     (* 2.0 (/ (* l (/ l (pow k 4.0))) t))
     (exp (- (* 2.0 (log (/ l k))) (* 3.0 (log t)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.35e-108) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else if (t <= 1.65e-34) {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	} else {
		tmp = exp(((2.0 * log((l / k))) - (3.0 * log(t))));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.35d-108)) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else if (t <= 1.65d-34) then
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    else
        tmp = exp(((2.0d0 * log((l / k))) - (3.0d0 * log(t))))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.35e-108) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else if (t <= 1.65e-34) {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	} else {
		tmp = Math.exp(((2.0 * Math.log((l / k))) - (3.0 * Math.log(t))));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -1.35e-108:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	elif t <= 1.65e-34:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	else:
		tmp = math.exp(((2.0 * math.log((l / k))) - (3.0 * math.log(t))))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -1.35e-108)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	elseif (t <= 1.65e-34)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	else
		tmp = exp(Float64(Float64(2.0 * log(Float64(l / k))) - Float64(3.0 * log(t))));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.35e-108)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	elseif (t <= 1.65e-34)
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	else
		tmp = exp(((2.0 * log((l / k))) - (3.0 * log(t))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -1.35e-108], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.65e-34], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(2.0 * N[Log[N[(l / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(3.0 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.35000000000000002e-108

   1. Initial program 68.6%

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

      1. associate-/r* 68.5%

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

      2. associate-*l* 66.0%

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

      3. sqr-neg 66.0%

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

      4. associate-*l* 68.5%

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

      5. *-commutative 68.5%

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

      6. sqr-neg 68.5%

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

      7. associate-*l/ 68.5%

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

      8. associate-*r/ 69.4%

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

      9. associate-/r/ 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in k around 0 63.1%

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

      1. associate-/r* 63.0%

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

      2. unpow2 63.0%

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

      3. unpow2 63.0%

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

      4. times-frac 71.8%

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

   6. Simplified 71.8%

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

      1. add-exp-log 44.2%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 44.2%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 44.2%

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

      1. add-exp-log 71.8%

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

      2. pow2 71.8%

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

      3. associate-/l* 74.0%

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

   10. Applied egg-rr 74.0%

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

   if -1.35000000000000002e-108 < t < 1.64999999999999991e-34

   1. Initial program 34.9%

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

      1. associate-/r* 34.9%

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

      2. associate-*l* 34.8%

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

      3. sqr-neg 34.8%

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

      4. associate-*l* 34.9%

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

      5. *-commutative 34.9%

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

      6. sqr-neg 34.9%

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

      7. associate-*l/ 34.8%

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

      8. associate-*r/ 34.8%

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

      9. associate-/r/ 34.8%

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

   3. Simplified 34.8%

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

   4. Taylor expanded in k around inf 65.0%

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

      1. associate-*r/ 65.0%

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

      2. associate-*r* 65.1%

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

      3. times-frac 66.2%

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

      4. unpow2 66.2%

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

      5. associate-*l* 66.2%

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

      6. unpow2 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in k around 0 57.2%

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

      1. unpow2 57.2%

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

      2. *-commutative 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in l around 0 57.2%

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

       1. unpow2 57.2%

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

       2. *-commutative 57.2%

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

       3. times-frac 70.5%

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

   12. Simplified 70.5%

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

       1. associate-*l/ 70.6%

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

   14. Applied egg-rr 70.6%

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

   if 1.64999999999999991e-34 < t

   1. Initial program 56.6%

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

      1. associate-/r* 56.6%

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

      2. associate-*l* 47.6%

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

      3. sqr-neg 47.6%

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

      4. associate-*l* 56.6%

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

      5. *-commutative 56.6%

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

      6. sqr-neg 56.6%

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

      7. associate-*l/ 56.7%

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

      8. associate-*r/ 56.5%

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

      9. associate-/r/ 56.5%

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

   3. Simplified 56.5%

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

   4. Taylor expanded in k around 0 44.9%

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

      1. associate-/r* 43.4%

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

      2. unpow2 43.4%

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

      3. unpow2 43.4%

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

      4. times-frac 59.3%

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

   6. Simplified 59.3%

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

      1. add-exp-log 58.6%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 58.6%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 58.6%

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

      1. log-div 58.3%

         \[\leadsto e^{\color{blue}{\log \left({\left(\frac{\ell}{k}\right)}^{2}\right) - \log \left({t}^{3}\right)}} \]

      2. pow-to-exp 29.5%

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

      3. add-log-exp 32.0%

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

      4. pow-to-exp 32.0%

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

      5. add-log-exp 34.6%

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

   10. Applied egg-rr 34.6%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;e^{2 \cdot \log \left(\frac{\ell}{k}\right) - 3 \cdot \log t}\\ \end{array} \]

Alternative 10: 69.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-88}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 10^{+208}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (<= t -1.35e-108)
   (/ (/ l k) (/ (pow t 3.0) (/ l k)))
   (if (<= t 6e-88)
     (* 2.0 (/ (* l (/ l (pow k 4.0))) t))
     (if (<= t 1e+208)
       (/
        2.0
        (*
         (pow (* k (/ (pow t 1.5) l)) 2.0)
         (+ 1.0 (+ 1.0 (/ (* k k) (* t t))))))
       (/
        (* l (* 2.0 (/ l (* k (* (pow t 3.0) k)))))
        (+ 2.0 (pow (/ k t) 2.0)))))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.35e-108) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else if (t <= 6e-88) {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	} else if (t <= 1e+208) {
		tmp = 2.0 / (pow((k * (pow(t, 1.5) / l)), 2.0) * (1.0 + (1.0 + ((k * k) / (t * t)))));
	} else {
		tmp = (l * (2.0 * (l / (k * (pow(t, 3.0) * k))))) / (2.0 + pow((k / t), 2.0));
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= (-1.35d-108)) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else if (t <= 6d-88) then
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    else if (t <= 1d+208) then
        tmp = 2.0d0 / (((k * ((t ** 1.5d0) / l)) ** 2.0d0) * (1.0d0 + (1.0d0 + ((k * k) / (t * t)))))
    else
        tmp = (l * (2.0d0 * (l / (k * ((t ** 3.0d0) * k))))) / (2.0d0 + ((k / t) ** 2.0d0))
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if (t <= -1.35e-108) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else if (t <= 6e-88) {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	} else if (t <= 1e+208) {
		tmp = 2.0 / (Math.pow((k * (Math.pow(t, 1.5) / l)), 2.0) * (1.0 + (1.0 + ((k * k) / (t * t)))));
	} else {
		tmp = (l * (2.0 * (l / (k * (Math.pow(t, 3.0) * k))))) / (2.0 + Math.pow((k / t), 2.0));
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if t <= -1.35e-108:
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	elif t <= 6e-88:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	elif t <= 1e+208:
		tmp = 2.0 / (math.pow((k * (math.pow(t, 1.5) / l)), 2.0) * (1.0 + (1.0 + ((k * k) / (t * t)))))
	else:
		tmp = (l * (2.0 * (l / (k * (math.pow(t, 3.0) * k))))) / (2.0 + math.pow((k / t), 2.0))
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if (t <= -1.35e-108)
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	elseif (t <= 6e-88)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	elseif (t <= 1e+208)
		tmp = Float64(2.0 / Float64((Float64(k * Float64((t ^ 1.5) / l)) ^ 2.0) * Float64(1.0 + Float64(1.0 + Float64(Float64(k * k) / Float64(t * t))))));
	else
		tmp = Float64(Float64(l * Float64(2.0 * Float64(l / Float64(k * Float64((t ^ 3.0) * k))))) / Float64(2.0 + (Float64(k / t) ^ 2.0)));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= -1.35e-108)
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	elseif (t <= 6e-88)
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	elseif (t <= 1e+208)
		tmp = 2.0 / (((k * ((t ^ 1.5) / l)) ^ 2.0) * (1.0 + (1.0 + ((k * k) / (t * t)))));
	else
		tmp = (l * (2.0 * (l / (k * ((t ^ 3.0) * k))))) / (2.0 + ((k / t) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[LessEqual[t, -1.35e-108], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6e-88], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e+208], N[(2.0 / N[(N[Power[N[(k * N[(N[Power[t, 1.5], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 + N[(1.0 + N[(N[(k * k), $MachinePrecision] / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l * N[(2.0 * N[(l / N[(k * N[(N[Power[t, 3.0], $MachinePrecision] * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.35000000000000002e-108

   1. Initial program 68.6%

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

      1. associate-/r* 68.5%

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

      2. associate-*l* 66.0%

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

      3. sqr-neg 66.0%

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

      4. associate-*l* 68.5%

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

      5. *-commutative 68.5%

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

      6. sqr-neg 68.5%

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

      7. associate-*l/ 68.5%

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

      8. associate-*r/ 69.4%

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

      9. associate-/r/ 69.4%

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

   3. Simplified 69.4%

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

   4. Taylor expanded in k around 0 63.1%

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

      1. associate-/r* 63.0%

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

      2. unpow2 63.0%

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

      3. unpow2 63.0%

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

      4. times-frac 71.8%

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

   6. Simplified 71.8%

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

      1. add-exp-log 44.2%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 44.2%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 44.2%

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

      1. add-exp-log 71.8%

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

      2. pow2 71.8%

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

      3. associate-/l* 74.0%

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

   10. Applied egg-rr 74.0%

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

   if -1.35000000000000002e-108 < t < 5.9999999999999999e-88

   1. Initial program 33.4%

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

      1. associate-/r* 33.4%

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

      2. associate-*l* 33.4%

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

      3. sqr-neg 33.4%

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

      4. associate-*l* 33.4%

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

      5. *-commutative 33.4%

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

      6. sqr-neg 33.4%

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

      7. associate-*l/ 33.4%

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

      8. associate-*r/ 33.3%

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

      9. associate-/r/ 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in k around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. associate-*r* 64.4%

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

      3. times-frac 65.5%

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

      4. unpow2 65.5%

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

      5. associate-*l* 65.6%

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

      6. unpow2 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in k around 0 57.1%

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

      1. unpow2 57.1%

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

      2. *-commutative 57.1%

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

   9. Simplified 57.1%

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

   10. Taylor expanded in l around 0 57.1%

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

       1. unpow2 57.1%

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

       2. *-commutative 57.1%

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

       3. times-frac 71.2%

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

   12. Simplified 71.2%

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

       1. associate-*l/ 71.3%

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

   14. Applied egg-rr 71.3%

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

   if 5.9999999999999999e-88 < t < 9.9999999999999998e207

   1. Initial program 57.0%

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

      1. add-sqr-sqrt 47.0%

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

      2. pow2 47.0%

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

      3. associate-*l* 40.4%

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

      4. sqrt-prod 40.4%

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

      5. sqrt-div 40.4%

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

      6. sqrt-pow1 44.3%

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

      7. metadata-eval 44.3%

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

      8. sqrt-prod 26.3%

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

      9. add-sqr-sqrt 50.5%

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

   3. Applied egg-rr 50.5%

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

   4. Taylor expanded in k around 0 48.5%

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

      1. unpow2 48.5%

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

      2. unpow2 48.5%

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

   6. Simplified 48.5%

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

   7. Taylor expanded in k around 0 72.8%

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

   if 9.9999999999999998e207 < t

   1. Initial program 56.6%

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

      1. associate-/r* 56.6%

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

      2. associate-*l* 44.3%

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

      3. sqr-neg 44.3%

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

      4. associate-*l* 56.6%

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

      5. *-commutative 56.6%

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

      6. sqr-neg 56.6%

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

      7. associate-*l/ 56.6%

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

      8. associate-*r/ 56.6%

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

      9. associate-/r/ 56.6%

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

   3. Simplified 56.6%

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

      1. add-cube-cbrt 56.6%

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

      2. pow3 56.6%

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

      3. *-commutative 56.6%

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

      4. cbrt-prod 56.6%

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

      5. cbrt-prod 56.6%

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

      6. rem-cbrt-cube 63.9%

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

   5. Applied egg-rr 63.9%

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

      1. associate-*l/ 63.9%

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

      2. associate-*l* 63.9%

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

   7. Applied egg-rr 63.9%

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

      1. unpow2 63.9%

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

      2. *-commutative 63.9%

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

      3. associate-*r/ 63.9%

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

      4. unpow2 63.9%

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

      5. associate-*l* 75.0%

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

      6. associate-*r* 75.0%

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

   9. Simplified 75.0%

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

   10. Taylor expanded in k around 0 55.2%

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

       1. unpow2 55.2%

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

       2. associate-*l* 67.7%

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

   12. Simplified 67.7%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-88}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 10^{+208}:\\ \;\;\;\;\frac{2}{{\left(k \cdot \frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(1 + \left(1 + \frac{k \cdot k}{t \cdot t}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternative 11: 62.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108} \lor \neg \left(t \leq 3 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.35e-108) (not (<= t 3e-32)))
   (* (/ l (pow t 3.0)) (/ l (* k k)))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-108) || !(t <= 3e-32)) {
		tmp = (l / pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.35d-108)) .or. (.not. (t <= 3d-32))) then
        tmp = (l / (t ** 3.0d0)) * (l / (k * k))
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-108) || !(t <= 3e-32)) {
		tmp = (l / Math.pow(t, 3.0)) * (l / (k * k));
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -1.35e-108) or not (t <= 3e-32):
		tmp = (l / math.pow(t, 3.0)) * (l / (k * k))
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.35e-108) || !(t <= 3e-32))
		tmp = Float64(Float64(l / (t ^ 3.0)) * Float64(l / Float64(k * k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.35e-108) || ~((t <= 3e-32)))
		tmp = (l / (t ^ 3.0)) * (l / (k * k));
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.35e-108], N[Not[LessEqual[t, 3e-32]], $MachinePrecision]], N[(N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35000000000000002e-108 or 3e-32 < t

   1. Initial program 63.5%

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

      1. associate-/r* 63.5%

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

      2. associate-*l* 58.2%

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

      3. sqr-neg 58.2%

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

      4. associate-*l* 63.5%

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

      5. *-commutative 63.5%

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

      6. sqr-neg 63.5%

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

      7. associate-*l/ 63.5%

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

      8. associate-*r/ 64.0%

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

      9. associate-/r/ 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in k around 0 55.4%

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

      1. unpow2 55.4%

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

      2. *-commutative 55.4%

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

      3. unpow2 55.4%

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

   6. Simplified 55.4%

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

      1. times-frac 59.8%

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

   8. Applied egg-rr 59.8%

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

   if -1.35000000000000002e-108 < t < 3e-32

   1. Initial program 34.9%

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

      1. associate-/r* 34.9%

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

      2. associate-*l* 34.8%

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

      3. sqr-neg 34.8%

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

      4. associate-*l* 34.9%

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

      5. *-commutative 34.9%

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

      6. sqr-neg 34.9%

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

      7. associate-*l/ 34.8%

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

      8. associate-*r/ 34.8%

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

      9. associate-/r/ 34.8%

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

   3. Simplified 34.8%

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

   4. Taylor expanded in k around inf 65.0%

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

      1. associate-*r/ 65.0%

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

      2. associate-*r* 65.1%

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

      3. times-frac 66.2%

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

      4. unpow2 66.2%

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

      5. associate-*l* 66.2%

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

      6. unpow2 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in k around 0 57.2%

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

      1. unpow2 57.2%

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

      2. *-commutative 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in l around 0 57.2%

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

       1. unpow2 57.2%

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

       2. *-commutative 57.2%

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

       3. times-frac 70.5%

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

   12. Simplified 70.5%

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

       1. associate-*l/ 70.6%

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

   14. Applied egg-rr 70.6%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108} \lor \neg \left(t \leq 3 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 12: 66.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108} \lor \neg \left(t \leq 7.6 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.35e-108) (not (<= t 7.6e-81)))
   (/ (* (/ l k) (/ l k)) (pow t 3.0))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-108) || !(t <= 7.6e-81)) {
		tmp = ((l / k) * (l / k)) / pow(t, 3.0);
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.35d-108)) .or. (.not. (t <= 7.6d-81))) then
        tmp = ((l / k) * (l / k)) / (t ** 3.0d0)
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-108) || !(t <= 7.6e-81)) {
		tmp = ((l / k) * (l / k)) / Math.pow(t, 3.0);
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -1.35e-108) or not (t <= 7.6e-81):
		tmp = ((l / k) * (l / k)) / math.pow(t, 3.0)
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.35e-108) || !(t <= 7.6e-81))
		tmp = Float64(Float64(Float64(l / k) * Float64(l / k)) / (t ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.35e-108) || ~((t <= 7.6e-81)))
		tmp = ((l / k) * (l / k)) / (t ^ 3.0);
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.35e-108], N[Not[LessEqual[t, 7.6e-81]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35000000000000002e-108 or 7.5999999999999997e-81 < t

   1. Initial program 63.4%

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

      1. associate-/r* 63.4%

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

      2. associate-*l* 58.3%

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

      3. sqr-neg 58.3%

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

      4. associate-*l* 63.4%

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

      5. *-commutative 63.4%

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

      6. sqr-neg 63.4%

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

      7. associate-*l/ 63.4%

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

      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 54.5%

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

      1. associate-/r* 54.4%

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

      2. unpow2 54.4%

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

      3. unpow2 54.4%

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

      4. times-frac 66.4%

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

   6. Simplified 66.4%

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

   if -1.35000000000000002e-108 < t < 7.5999999999999997e-81

   1. Initial program 33.4%

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

      1. associate-/r* 33.4%

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

      2. associate-*l* 33.4%

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

      3. sqr-neg 33.4%

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

      4. associate-*l* 33.4%

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

      5. *-commutative 33.4%

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

      6. sqr-neg 33.4%

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

      7. associate-*l/ 33.4%

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

      8. associate-*r/ 33.3%

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

      9. associate-/r/ 33.3%

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

   3. Simplified 33.3%

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

   4. Taylor expanded in k around inf 64.3%

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

      1. associate-*r/ 64.3%

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

      2. associate-*r* 64.4%

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

      3. times-frac 65.5%

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

      4. unpow2 65.5%

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

      5. associate-*l* 65.6%

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

      6. unpow2 65.6%

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

   6. Simplified 65.6%

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

   7. Taylor expanded in k around 0 57.1%

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

      1. unpow2 57.1%

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

      2. *-commutative 57.1%

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

   9. Simplified 57.1%

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

   10. Taylor expanded in l around 0 57.1%

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

       1. unpow2 57.1%

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

       2. *-commutative 57.1%

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

       3. times-frac 71.2%

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

   12. Simplified 71.2%

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

       1. associate-*l/ 71.3%

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

   14. Applied egg-rr 71.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108} \lor \neg \left(t \leq 7.6 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 13: 67.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108} \lor \neg \left(t \leq 1.65 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k)
 :precision binary64
 (if (or (<= t -1.35e-108) (not (<= t 1.65e-34)))
   (/ (/ l k) (/ (pow t 3.0) (/ l k)))
   (* 2.0 (/ (* l (/ l (pow k 4.0))) t))))

C

k = abs(k);
double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-108) || !(t <= 1.65e-34)) {
		tmp = (l / k) / (pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * ((l * (l / pow(k, 4.0))) / t);
	}
	return tmp;
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-1.35d-108)) .or. (.not. (t <= 1.65d-34))) then
        tmp = (l / k) / ((t ** 3.0d0) / (l / k))
    else
        tmp = 2.0d0 * ((l * (l / (k ** 4.0d0))) / t)
    end if
    code = tmp
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -1.35e-108) || !(t <= 1.65e-34)) {
		tmp = (l / k) / (Math.pow(t, 3.0) / (l / k));
	} else {
		tmp = 2.0 * ((l * (l / Math.pow(k, 4.0))) / t);
	}
	return tmp;
}

Python

k = abs(k)
def code(t, l, k):
	tmp = 0
	if (t <= -1.35e-108) or not (t <= 1.65e-34):
		tmp = (l / k) / (math.pow(t, 3.0) / (l / k))
	else:
		tmp = 2.0 * ((l * (l / math.pow(k, 4.0))) / t)
	return tmp

Julia

k = abs(k)
function code(t, l, k)
	tmp = 0.0
	if ((t <= -1.35e-108) || !(t <= 1.65e-34))
		tmp = Float64(Float64(l / k) / Float64((t ^ 3.0) / Float64(l / k)));
	else
		tmp = Float64(2.0 * Float64(Float64(l * Float64(l / (k ^ 4.0))) / t));
	end
	return tmp
end

MATLAB

k = abs(k)
function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -1.35e-108) || ~((t <= 1.65e-34)))
		tmp = (l / k) / ((t ^ 3.0) / (l / k));
	else
		tmp = 2.0 * ((l * (l / (k ^ 4.0))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := If[Or[LessEqual[t, -1.35e-108], N[Not[LessEqual[t, 1.65e-34]], $MachinePrecision]], N[(N[(l / k), $MachinePrecision] / N[(N[Power[t, 3.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(l * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35000000000000002e-108 or 1.64999999999999991e-34 < t

   1. Initial program 63.5%

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

      1. associate-/r* 63.5%

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

      2. associate-*l* 58.2%

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

      3. sqr-neg 58.2%

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

      4. associate-*l* 63.5%

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

      5. *-commutative 63.5%

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

      6. sqr-neg 63.5%

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

      7. associate-*l/ 63.5%

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

      8. associate-*r/ 64.0%

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

      9. associate-/r/ 64.0%

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

   3. Simplified 64.0%

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

   4. Taylor expanded in k around 0 55.4%

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

      1. associate-/r* 54.7%

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

      2. unpow2 54.7%

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

      3. unpow2 54.7%

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

      4. times-frac 66.5%

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

   6. Simplified 66.5%

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

      1. add-exp-log 50.3%

         \[\leadsto \color{blue}{e^{\log \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\right)}} \]

      2. pow2 50.3%

         \[\leadsto e^{\log \left(\frac{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{t}^{3}}\right)} \]

   8. Applied egg-rr 50.3%

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

      1. add-exp-log 66.5%

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

      2. pow2 66.5%

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

      3. associate-/l* 70.1%

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

   10. Applied egg-rr 70.1%

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

   if -1.35000000000000002e-108 < t < 1.64999999999999991e-34

   1. Initial program 34.9%

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

      1. associate-/r* 34.9%

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

      2. associate-*l* 34.8%

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

      3. sqr-neg 34.8%

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

      4. associate-*l* 34.9%

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

      5. *-commutative 34.9%

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

      6. sqr-neg 34.9%

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

      7. associate-*l/ 34.8%

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

      8. associate-*r/ 34.8%

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

      9. associate-/r/ 34.8%

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

   3. Simplified 34.8%

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

   4. Taylor expanded in k around inf 65.0%

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

      1. associate-*r/ 65.0%

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

      2. associate-*r* 65.1%

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

      3. times-frac 66.2%

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

      4. unpow2 66.2%

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

      5. associate-*l* 66.2%

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

      6. unpow2 66.2%

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

   6. Simplified 66.2%

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

   7. Taylor expanded in k around 0 57.2%

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

      1. unpow2 57.2%

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

      2. *-commutative 57.2%

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

   9. Simplified 57.2%

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

   10. Taylor expanded in l around 0 57.2%

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

       1. unpow2 57.2%

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

       2. *-commutative 57.2%

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

       3. times-frac 70.5%

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

   12. Simplified 70.5%

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

       1. associate-*l/ 70.6%

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

   14. Applied egg-rr 70.6%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-108} \lor \neg \left(t \leq 1.65 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{{t}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 14: 57.2% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \left(\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{t}\right) \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (* (/ l (pow k 4.0)) (/ l t))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l / pow(k, 4.0)) * (l / t));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l / (k ** 4.0d0)) * (l / t))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l / Math.pow(k, 4.0)) * (l / t));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((l / math.pow(k, 4.0)) * (l / t))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l / (k ^ 4.0)) * Float64(l / t)))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l / (k ^ 4.0)) * (l / t));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. associate-/r* 53.4%

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

   2. associate-*l* 50.0%

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

   3. sqr-neg 50.0%

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

   4. associate-*l* 53.4%

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

   5. *-commutative 53.4%

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

   6. sqr-neg 53.4%

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

   7. associate-*l/ 53.4%

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

   8. associate-*r/ 53.7%

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

   9. associate-/r/ 53.7%

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

3. Simplified 53.7%

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

4. Taylor expanded in k around inf 52.8%

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

   1. associate-*r/ 52.8%

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

   2. associate-*r* 52.8%

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

   3. times-frac 54.0%

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

   4. unpow2 54.0%

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

   5. associate-*l* 54.0%

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

   6. unpow2 54.0%

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

6. Simplified 54.0%

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

7. Taylor expanded in k around 0 46.6%

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

   1. unpow2 46.6%

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

   2. *-commutative 46.6%

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

9. Simplified 46.6%

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

10. Taylor expanded in l around 0 46.6%

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

    1. unpow2 46.6%

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

    2. *-commutative 46.6%

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

    3. times-frac 52.1%

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

12. Simplified 52.1%

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

13. Final simplification 52.1%

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

Alternative 15: 56.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} k = |k|\\ \\ 2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}} \end{array} \]

FPCore

NOTE: k should be positive before calling this function
(FPCore (t l k) :precision binary64 (* 2.0 (/ (* l (/ l t)) (pow k 4.0))))

C

k = abs(k);
double code(double t, double l, double k) {
	return 2.0 * ((l * (l / t)) / pow(k, 4.0));
}

Fortran

NOTE: k should be positive before calling this function
real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 * ((l * (l / t)) / (k ** 4.0d0))
end function

Java

k = Math.abs(k);
public static double code(double t, double l, double k) {
	return 2.0 * ((l * (l / t)) / Math.pow(k, 4.0));
}

Python

k = abs(k)
def code(t, l, k):
	return 2.0 * ((l * (l / t)) / math.pow(k, 4.0))

Julia

k = abs(k)
function code(t, l, k)
	return Float64(2.0 * Float64(Float64(l * Float64(l / t)) / (k ^ 4.0)))
end

MATLAB

k = abs(k)
function tmp = code(t, l, k)
	tmp = 2.0 * ((l * (l / t)) / (k ^ 4.0));
end

Wolfram

NOTE: k should be positive before calling this function
code[t_, l_, k_] := N[(2.0 * N[(N[(l * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. associate-/r* 53.4%

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

   2. associate-*l* 50.0%

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

   3. sqr-neg 50.0%

      \[\leadsto \frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\left(-\ell\right) \cdot \left(-\ell\right)}} \cdot \left(\sin k \cdot \tan k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   4. associate-*l* 53.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right) \cdot \tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   5. *-commutative 53.4%

      \[\leadsto \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\left(-\ell\right) \cdot \left(-\ell\right)} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   6. sqr-neg 53.4%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \left(\frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   7. associate-*l/ 53.4%

      \[\leadsto \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   8. associate-*r/ 53.7%

      \[\leadsto \frac{\frac{2}{\color{blue}{\frac{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

   9. associate-/r/ 53.7%

      \[\leadsto \frac{\color{blue}{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]

3. Simplified 53.7%

   \[\leadsto \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)} \cdot \left(\ell \cdot \ell\right)}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

4. Taylor expanded in k around inf 52.8%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
5. Step-by-step derivation

   1. associate-*r/ 52.8%

      \[\leadsto \color{blue}{\frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]

   2. associate-*r* 52.8%

      \[\leadsto \frac{2 \cdot \left({\ell}^{2} \cdot \cos k\right)}{\color{blue}{\left({k}^{2} \cdot t\right) \cdot {\sin k}^{2}}} \]

   3. times-frac 54.0%

      \[\leadsto \color{blue}{\frac{2}{{k}^{2} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}}} \]

   4. unpow2 54.0%

      \[\leadsto \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

   5. associate-*l* 54.0%

      \[\leadsto \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{{\ell}^{2} \cdot \cos k}{{\sin k}^{2}} \]

   6. unpow2 54.0%

      \[\leadsto \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{\sin k}^{2}} \]

6. Simplified 54.0%

   \[\leadsto \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\sin k}^{2}}} \]

7. Taylor expanded in k around 0 46.6%

   \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
8. Step-by-step derivation

   1. unpow2 46.6%

      \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

   2. *-commutative 46.6%

      \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

9. Simplified 46.6%

   \[\leadsto \color{blue}{2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}} \]

10. Taylor expanded in l around 0 46.6%

    \[\leadsto 2 \cdot \color{blue}{\frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]
11. Step-by-step derivation

    1. unpow2 46.6%

       \[\leadsto 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]

    2. *-commutative 46.6%

       \[\leadsto 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]

    3. times-frac 52.1%

       \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

12. Simplified 52.1%

    \[\leadsto 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]
13. Step-by-step derivation

    1. associate-*r/ 52.1%

       \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

14. Applied egg-rr 52.1%

    \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]

15. Final simplification 52.1%

    \[\leadsto 2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}} \]

Reproduce

herbie shell --seed 2023292 
(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))))