Toniolo and Linder, Equation (10-)

Percentage Accurate: 24.0% → 73.9%

Time: 31.8s

Precision: binary64

Cost: 27724

• 75.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := \left(k \cdot k\right) \cdot 0.041666666666666664\\ t_2 := 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\\ t_3 := 0.008333333333333333 \cdot \frac{\ell}{t} + \frac{\ell}{t} \cdot -0.027777777777777776\\ t_4 := \frac{\ell}{\tan k}\\ t_5 := \frac{1}{k \cdot k}\\ \mathbf{if}\;k \leq -8.6 \cdot 10^{+86}:\\ \;\;\;\;\left(-2 \cdot \left(k \cdot t_3\right) + \left(-2 \cdot \left({k}^{3} \cdot \left(\frac{\ell}{t} \cdot 0.001388888888888889 + \left(t_3 \cdot 0.16666666666666666 + \frac{\ell}{t} \cdot -0.0001984126984126984\right)\right)\right) + \left(0.3333333333333333 \cdot \frac{\ell}{k \cdot t} + 2 \cdot \frac{\ell}{t \cdot {k}^{3}}\right)\right)\right) \cdot t_4\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;t_4 \cdot \frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+100}:\\ \;\;\;\;\left(t_5 + \left(\mathsf{fma}\left(-0.001388888888888889, {k}^{4}, t_1\right) - 0.5\right)\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_5 + \left(t_1 - 0.5\right)\right)\\ \end{array} \]

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))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (* k k) 0.041666666666666664))
        (t_2 (* 2.0 (/ (/ l (/ t l)) (pow (sin k) 2.0))))
        (t_3
         (+
          (* 0.008333333333333333 (/ l t))
          (* (/ l t) -0.027777777777777776)))
        (t_4 (/ l (tan k)))
        (t_5 (/ 1.0 (* k k))))
   (if (<= k -8.6e+86)
     (*
      (+
       (* -2.0 (* k t_3))
       (+
        (*
         -2.0
         (*
          (pow k 3.0)
          (+
           (* (/ l t) 0.001388888888888889)
           (+
            (* t_3 0.16666666666666666)
            (* (/ l t) -0.0001984126984126984)))))
        (+
         (* 0.3333333333333333 (/ l (* k t)))
         (* 2.0 (/ l (* t (pow k 3.0)))))))
      t_4)
     (if (<= k 2.1e+79)
       (* t_4 (/ (/ (/ (* l 2.0) (* k t)) k) (sin k)))
       (if (<= k 2.6e+100)
         (* (+ t_5 (- (fma -0.001388888888888889 (pow k 4.0) t_1) 0.5)) t_2)
         (* t_2 (+ t_5 (- t_1 0.5))))))))

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));
}

↓

double code(double t, double l, double k) {
	double t_1 = (k * k) * 0.041666666666666664;
	double t_2 = 2.0 * ((l / (t / l)) / pow(sin(k), 2.0));
	double t_3 = (0.008333333333333333 * (l / t)) + ((l / t) * -0.027777777777777776);
	double t_4 = l / tan(k);
	double t_5 = 1.0 / (k * k);
	double tmp;
	if (k <= -8.6e+86) {
		tmp = ((-2.0 * (k * t_3)) + ((-2.0 * (pow(k, 3.0) * (((l / t) * 0.001388888888888889) + ((t_3 * 0.16666666666666666) + ((l / t) * -0.0001984126984126984))))) + ((0.3333333333333333 * (l / (k * t))) + (2.0 * (l / (t * pow(k, 3.0))))))) * t_4;
	} else if (k <= 2.1e+79) {
		tmp = t_4 * ((((l * 2.0) / (k * t)) / k) / sin(k));
	} else if (k <= 2.6e+100) {
		tmp = (t_5 + (fma(-0.001388888888888889, pow(k, 4.0), t_1) - 0.5)) * t_2;
	} else {
		tmp = t_2 * (t_5 + (t_1 - 0.5));
	}
	return tmp;
}

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

↓

function code(t, l, k)
	t_1 = Float64(Float64(k * k) * 0.041666666666666664)
	t_2 = Float64(2.0 * Float64(Float64(l / Float64(t / l)) / (sin(k) ^ 2.0)))
	t_3 = Float64(Float64(0.008333333333333333 * Float64(l / t)) + Float64(Float64(l / t) * -0.027777777777777776))
	t_4 = Float64(l / tan(k))
	t_5 = Float64(1.0 / Float64(k * k))
	tmp = 0.0
	if (k <= -8.6e+86)
		tmp = Float64(Float64(Float64(-2.0 * Float64(k * t_3)) + Float64(Float64(-2.0 * Float64((k ^ 3.0) * Float64(Float64(Float64(l / t) * 0.001388888888888889) + Float64(Float64(t_3 * 0.16666666666666666) + Float64(Float64(l / t) * -0.0001984126984126984))))) + Float64(Float64(0.3333333333333333 * Float64(l / Float64(k * t))) + Float64(2.0 * Float64(l / Float64(t * (k ^ 3.0))))))) * t_4);
	elseif (k <= 2.1e+79)
		tmp = Float64(t_4 * Float64(Float64(Float64(Float64(l * 2.0) / Float64(k * t)) / k) / sin(k)));
	elseif (k <= 2.6e+100)
		tmp = Float64(Float64(t_5 + Float64(fma(-0.001388888888888889, (k ^ 4.0), t_1) - 0.5)) * t_2);
	else
		tmp = Float64(t_2 * Float64(t_5 + Float64(t_1 - 0.5)));
	end
	return tmp
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]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(k * k), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.008333333333333333 * N[(l / t), $MachinePrecision]), $MachinePrecision] + N[(N[(l / t), $MachinePrecision] * -0.027777777777777776), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(1.0 / N[(k * k), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -8.6e+86], N[(N[(N[(-2.0 * N[(k * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[Power[k, 3.0], $MachinePrecision] * N[(N[(N[(l / t), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] + N[(N[(t$95$3 * 0.16666666666666666), $MachinePrecision] + N[(N[(l / t), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(l / N[(t * N[Power[k, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[k, 2.1e+79], N[(t$95$4 * N[(N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.6e+100], N[(N[(t$95$5 + N[(N[(-0.001388888888888889 * N[Power[k, 4.0], $MachinePrecision] + t$95$1), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$2 * N[(t$95$5 + N[(t$95$1 - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -8.6000000000000004e86

   1. Initial program 2.6%

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

   2. Simplified 2.9%

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

      [Start] 2.6%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 2.6%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 2.6%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 2.6%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 2.6%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 2.6%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 2.7%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 2.7%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 2.9%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 2.9%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 2.9%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 2.9%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 5.2%

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

   4. Simplified 5.2%

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

      [Start] 5.2%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 5.2%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 9.2%

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

      [Start] 5.2%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l/ [=>] 5.2%	\[ \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]
      associate-*l* [=>] 9.2%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   6. Simplified 3.7%

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

      [Start] 9.2%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)} \]
      associate-*l/ [<=] 9.2%	\[ \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]
      associate-*r* [=>] 3.7%	\[ \color{blue}{\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   7. Taylor expanded in k around 0 48.2%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(k \cdot \left(0.008333333333333333 \cdot \frac{\ell}{t} + -0.027777777777777776 \cdot \frac{\ell}{t}\right)\right) + \left(-2 \cdot \left({k}^{3} \cdot \left(0.001388888888888889 \cdot \frac{\ell}{t} + \left(0.16666666666666666 \cdot \left(0.008333333333333333 \cdot \frac{\ell}{t} + -0.027777777777777776 \cdot \frac{\ell}{t}\right) + -0.0001984126984126984 \cdot \frac{\ell}{t}\right)\right)\right) + \left(0.3333333333333333 \cdot \frac{\ell}{k \cdot t} + 2 \cdot \frac{\ell}{{k}^{3} \cdot t}\right)\right)\right)} \cdot \frac{\ell}{\tan k} \]

   if -8.6000000000000004e86 < k < 2.10000000000000008e79

   1. Initial program 38.6%

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

   2. Simplified 50.7%

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

      [Start] 38.6%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 38.6%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 38.6%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 38.6%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 38.0%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 38.0%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 38.1%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 38.1%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 43.1%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 43.1%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 43.1%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 50.7%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 85.8%

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

   4. Simplified 85.8%

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

      [Start] 85.8%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 85.8%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 88.8%

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

      [Start] 85.8%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l/ [=>] 85.8%	\[ \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]
      associate-*l* [=>] 88.8%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   6. Simplified 93.0%

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

      [Start] 88.8%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)} \]
      associate-*l/ [<=] 88.8%	\[ \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]
      associate-*r* [=>] 93.0%	\[ \color{blue}{\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]

   7. Taylor expanded in k around inf 88.5%

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

   8. Simplified 94.3%

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

      [Start] 88.5%	\[ \left(2 \cdot \frac{\ell}{{k}^{2} \cdot \left(\sin k \cdot t\right)}\right) \cdot \frac{\ell}{\tan k} \]
      associate-*r/ [=>] 88.5%	\[ \color{blue}{\frac{2 \cdot \ell}{{k}^{2} \cdot \left(\sin k \cdot t\right)}} \cdot \frac{\ell}{\tan k} \]
      *-commutative [=>] 88.5%	\[ \frac{2 \cdot \ell}{\color{blue}{\left(\sin k \cdot t\right) \cdot {k}^{2}}} \cdot \frac{\ell}{\tan k} \]
      associate-*r* [<=] 88.5%	\[ \frac{2 \cdot \ell}{\color{blue}{\sin k \cdot \left(t \cdot {k}^{2}\right)}} \cdot \frac{\ell}{\tan k} \]
      *-commutative [<=] 88.5%	\[ \frac{2 \cdot \ell}{\sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}} \cdot \frac{\ell}{\tan k} \]
      unpow2 [=>] 88.5%	\[ \frac{2 \cdot \ell}{\sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \cdot \frac{\ell}{\tan k} \]
      associate-*r* [<=] 89.1%	\[ \frac{2 \cdot \ell}{\sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}} \cdot \frac{\ell}{\tan k} \]
      associate-/l/ [<=] 93.1%	\[ \color{blue}{\frac{\frac{2 \cdot \ell}{k \cdot \left(k \cdot t\right)}}{\sin k}} \cdot \frac{\ell}{\tan k} \]
      *-commutative [=>] 93.1%	\[ \frac{\frac{2 \cdot \ell}{\color{blue}{\left(k \cdot t\right) \cdot k}}}{\sin k} \cdot \frac{\ell}{\tan k} \]
      associate-/r* [=>] 94.3%	\[ \frac{\color{blue}{\frac{\frac{2 \cdot \ell}{k \cdot t}}{k}}}{\sin k} \cdot \frac{\ell}{\tan k} \]

   if 2.10000000000000008e79 < k < 2.6000000000000002e100

   1. Initial program 0.2%

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

   2. Simplified 0.2%

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

      [Start] 0.2%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.2%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.2%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 0.2%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 0.2%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 0.2%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 0.2%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 0.2%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 0.2%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 0.2%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 0.2%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 0.2%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 16.1%

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

   4. Simplified 16.1%

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

      [Start] 16.1%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 16.1%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Taylor expanded in k around inf 16.1%

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

   6. Simplified 29.5%

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

      [Start] 16.1%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 16.1%	\[ \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot 2} \]
      times-frac [=>] 29.4%	\[ \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \cdot 2 \]
      associate-*l* [=>] 29.4%	\[ \color{blue}{\frac{\cos k}{{k}^{2}} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2} \cdot t} \cdot 2\right)} \]
      unpow2 [=>] 29.4%	\[ \frac{\cos k}{\color{blue}{k \cdot k}} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2} \cdot t} \cdot 2\right) \]
      *-commutative [=>] 29.4%	\[ \frac{\cos k}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{\color{blue}{t \cdot {\sin k}^{2}}} \cdot 2\right) \]
      associate-/r* [=>] 29.4%	\[ \frac{\cos k}{k \cdot k} \cdot \left(\color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{\sin k}^{2}}} \cdot 2\right) \]
      unpow2 [=>] 29.4%	\[ \frac{\cos k}{k \cdot k} \cdot \left(\frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{\sin k}^{2}} \cdot 2\right) \]
      associate-/l* [=>] 29.5%	\[ \frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{\sin k}^{2}} \cdot 2\right) \]

   7. Taylor expanded in k around 0 85.7%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{{k}^{2}} + \left(-0.001388888888888889 \cdot {k}^{4} + 0.041666666666666664 \cdot {k}^{2}\right)\right) - 0.5\right)} \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]

   8. Simplified 85.7%

      \[\leadsto \color{blue}{\left(\frac{1}{k \cdot k} + \left(\mathsf{fma}\left(-0.001388888888888889, {k}^{4}, \left(k \cdot k\right) \cdot 0.041666666666666664\right) - 0.5\right)\right)} \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      Step-by-step derivation

      [Start] 85.7%	\[ \left(\left(\frac{1}{{k}^{2}} + \left(-0.001388888888888889 \cdot {k}^{4} + 0.041666666666666664 \cdot {k}^{2}\right)\right) - 0.5\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      associate--l+ [=>] 85.7%	\[ \color{blue}{\left(\frac{1}{{k}^{2}} + \left(\left(-0.001388888888888889 \cdot {k}^{4} + 0.041666666666666664 \cdot {k}^{2}\right) - 0.5\right)\right)} \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      unpow2 [=>] 85.7%	\[ \left(\frac{1}{\color{blue}{k \cdot k}} + \left(\left(-0.001388888888888889 \cdot {k}^{4} + 0.041666666666666664 \cdot {k}^{2}\right) - 0.5\right)\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      fma-def [=>] 85.7%	\[ \left(\frac{1}{k \cdot k} + \left(\color{blue}{\mathsf{fma}\left(-0.001388888888888889, {k}^{4}, 0.041666666666666664 \cdot {k}^{2}\right)} - 0.5\right)\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      *-commutative [=>] 85.7%	\[ \left(\frac{1}{k \cdot k} + \left(\mathsf{fma}\left(-0.001388888888888889, {k}^{4}, \color{blue}{{k}^{2} \cdot 0.041666666666666664}\right) - 0.5\right)\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      unpow2 [=>] 85.7%	\[ \left(\frac{1}{k \cdot k} + \left(\mathsf{fma}\left(-0.001388888888888889, {k}^{4}, \color{blue}{\left(k \cdot k\right)} \cdot 0.041666666666666664\right) - 0.5\right)\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]

   if 2.6000000000000002e100 < k

   1. Initial program 0.8%

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

   2. Simplified 1.0%

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

      [Start] 0.8%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.8%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.8%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 0.8%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 0.8%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 0.8%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 0.8%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 0.8%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 1.0%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 1.0%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 1.0%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 1.0%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 1.1%

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

   4. Simplified 1.1%

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

      [Start] 1.1%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 1.1%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Taylor expanded in k around inf 1.1%

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

   6. Simplified 3.6%

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

      [Start] 1.1%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 1.1%	\[ \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot 2} \]
      times-frac [=>] 3.5%	\[ \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \cdot 2 \]
      associate-*l* [=>] 3.5%	\[ \color{blue}{\frac{\cos k}{{k}^{2}} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2} \cdot t} \cdot 2\right)} \]
      unpow2 [=>] 3.5%	\[ \frac{\cos k}{\color{blue}{k \cdot k}} \cdot \left(\frac{{\ell}^{2}}{{\sin k}^{2} \cdot t} \cdot 2\right) \]
      *-commutative [=>] 3.5%	\[ \frac{\cos k}{k \cdot k} \cdot \left(\frac{{\ell}^{2}}{\color{blue}{t \cdot {\sin k}^{2}}} \cdot 2\right) \]
      associate-/r* [=>] 3.5%	\[ \frac{\cos k}{k \cdot k} \cdot \left(\color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{\sin k}^{2}}} \cdot 2\right) \]
      unpow2 [=>] 3.5%	\[ \frac{\cos k}{k \cdot k} \cdot \left(\frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{\sin k}^{2}} \cdot 2\right) \]
      associate-/l* [=>] 3.6%	\[ \frac{\cos k}{k \cdot k} \cdot \left(\frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{\sin k}^{2}} \cdot 2\right) \]

   7. Taylor expanded in k around 0 40.7%

      \[\leadsto \color{blue}{\left(\left(\frac{1}{{k}^{2}} + 0.041666666666666664 \cdot {k}^{2}\right) - 0.5\right)} \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]

   8. Simplified 40.7%

      \[\leadsto \color{blue}{\left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right)} \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      Step-by-step derivation

      [Start] 40.7%	\[ \left(\left(\frac{1}{{k}^{2}} + 0.041666666666666664 \cdot {k}^{2}\right) - 0.5\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      associate--l+ [=>] 40.7%	\[ \color{blue}{\left(\frac{1}{{k}^{2}} + \left(0.041666666666666664 \cdot {k}^{2} - 0.5\right)\right)} \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      unpow2 [=>] 40.7%	\[ \left(\frac{1}{\color{blue}{k \cdot k}} + \left(0.041666666666666664 \cdot {k}^{2} - 0.5\right)\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      *-commutative [=>] 40.7%	\[ \left(\frac{1}{k \cdot k} + \left(\color{blue}{{k}^{2} \cdot 0.041666666666666664} - 0.5\right)\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]
      unpow2 [=>] 40.7%	\[ \left(\frac{1}{k \cdot k} + \left(\color{blue}{\left(k \cdot k\right)} \cdot 0.041666666666666664 - 0.5\right)\right) \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}} \cdot 2\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{+86}:\\ \;\;\;\;\left(-2 \cdot \left(k \cdot \left(0.008333333333333333 \cdot \frac{\ell}{t} + \frac{\ell}{t} \cdot -0.027777777777777776\right)\right) + \left(-2 \cdot \left({k}^{3} \cdot \left(\frac{\ell}{t} \cdot 0.001388888888888889 + \left(\left(0.008333333333333333 \cdot \frac{\ell}{t} + \frac{\ell}{t} \cdot -0.027777777777777776\right) \cdot 0.16666666666666666 + \frac{\ell}{t} \cdot -0.0001984126984126984\right)\right)\right) + \left(0.3333333333333333 \cdot \frac{\ell}{k \cdot t} + 2 \cdot \frac{\ell}{t \cdot {k}^{3}}\right)\right)\right) \cdot \frac{\ell}{\tan k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+100}:\\ \;\;\;\;\left(\frac{1}{k \cdot k} + \left(\mathsf{fma}\left(-0.001388888888888889, {k}^{4}, \left(k \cdot k\right) \cdot 0.041666666666666664\right) - 0.5\right)\right) \cdot \left(2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \cdot \left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	73.9%
Cost	27724

\[\begin{array}{l} t_1 := \left(k \cdot k\right) \cdot 0.041666666666666664\\ t_2 := 2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\\ t_3 := 0.008333333333333333 \cdot \frac{\ell}{t} + \frac{\ell}{t} \cdot -0.027777777777777776\\ t_4 := \frac{\ell}{\tan k}\\ t_5 := \frac{1}{k \cdot k}\\ \mathbf{if}\;k \leq -8.6 \cdot 10^{+86}:\\ \;\;\;\;\left(-2 \cdot \left(k \cdot t_3\right) + \left(-2 \cdot \left({k}^{3} \cdot \left(\frac{\ell}{t} \cdot 0.001388888888888889 + \left(t_3 \cdot 0.16666666666666666 + \frac{\ell}{t} \cdot -0.0001984126984126984\right)\right)\right) + \left(0.3333333333333333 \cdot \frac{\ell}{k \cdot t} + 2 \cdot \frac{\ell}{t \cdot {k}^{3}}\right)\right)\right) \cdot t_4\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+79}:\\ \;\;\;\;t_4 \cdot \frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+100}:\\ \;\;\;\;\left(t_5 + \left(\mathsf{fma}\left(-0.001388888888888889, {k}^{4}, t_1\right) - 0.5\right)\right) \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(t_5 + \left(t_1 - 0.5\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	73.4%
Cost	23620

\[\begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ t_2 := 0.008333333333333333 \cdot \frac{\ell}{t} + \frac{\ell}{t} \cdot -0.027777777777777776\\ \mathbf{if}\;k \leq -5.4 \cdot 10^{+88}:\\ \;\;\;\;\left(-2 \cdot \left(k \cdot t_2\right) + \left(-2 \cdot \left({k}^{3} \cdot \left(\frac{\ell}{t} \cdot 0.001388888888888889 + \left(t_2 \cdot 0.16666666666666666 + \frac{\ell}{t} \cdot -0.0001984126984126984\right)\right)\right) + \left(0.3333333333333333 \cdot \frac{\ell}{k \cdot t} + 2 \cdot \frac{\ell}{t \cdot {k}^{3}}\right)\right)\right) \cdot t_1\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+97}:\\ \;\;\;\;t_1 \cdot \frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \cdot \left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	72.8%
Cost	14601

\[\begin{array}{l} \mathbf{if}\;k \leq -1 \cdot 10^{+79} \lor \neg \left(k \leq 3.7 \cdot 10^{+96}\right):\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \cdot \left(\frac{1}{k \cdot k} + \left(\left(k \cdot k\right) \cdot 0.041666666666666664 - 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k}\\ \end{array} \]

Alternative 4
Accuracy	62.2%
Cost	14084

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2.75 \cdot 10^{+187}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right) \cdot \left(\frac{1}{k \cdot k} + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k}\\ \end{array} \]

Alternative 5
Accuracy	63.9%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+27}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{\frac{\sin k}{\frac{\ell}{t}}} \cdot \frac{\ell}{k}}{\tan k}\\ \end{array} \]

Alternative 6
Accuracy	61.4%
Cost	13760

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

Alternative 7
Accuracy	61.4%
Cost	13760

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

Alternative 8
Accuracy	61.5%
Cost	13760

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

Alternative 9
Accuracy	62.3%
Cost	13760

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

Alternative 10
Accuracy	57.4%
Cost	7748

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k} \cdot \frac{\ell}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{k \cdot k}\right)\\ \end{array} \]

Alternative 11
Accuracy	58.0%
Cost	7744

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

Alternative 12
Accuracy	55.2%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{\frac{k \cdot t}{\ell}}}{k}\\ \end{array} \]

Alternative 13
Accuracy	55.5%
Cost	7492

\[\begin{array}{l} \mathbf{if}\;k \leq -5500000000:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell \cdot 2}{k \cdot t}}{k}}{\sin k} \cdot \frac{\ell}{k}\\ \end{array} \]

Alternative 14
Accuracy	53.0%
Cost	7168

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

Alternative 15
Accuracy	46.1%
Cost	960

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

Alternative 16
Accuracy	51.8%
Cost	960

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

Alternative 17
Accuracy	10.4%
Cost	704

\[-0.3333333333333333 \cdot \left(\frac{\ell}{k \cdot t} \cdot \frac{\ell}{k}\right) \]

Alternative 18
Accuracy	10.8%
Cost	704

\[-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \]

Alternative 19
Accuracy	11.5%
Cost	704

\[-0.3333333333333333 \cdot \frac{\frac{\ell}{\frac{k \cdot t}{\ell}}}{k} \]

Reproduce

herbie shell --seed 2023255 
(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))))