Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.8% → 81.6%

Time: 33.8s

Precision: binary64

Cost: 46344

• 27.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \tan k \cdot \left(1 + \left(1 + t_1\right)\right)\\ t_3 := 2 + t_1\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+180}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{t_3 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot 0.5\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot t_2}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-8}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{\tan k}}{t_3 \cdot \left(\sin k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\ell \cdot \cos k\right)\right)}{t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

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 (pow (/ k t) 2.0))
        (t_2 (* (tan k) (+ 1.0 (+ 1.0 t_1))))
        (t_3 (+ 2.0 t_1)))
   (if (<= t -1.4e+180)
     (pow
      (*
       (/ 1.0 (cbrt (* t_3 (* (* (sin k) (tan k)) 0.5))))
       (/ (pow (cbrt l) 2.0) t))
      3.0)
     (if (<= t -5.8e+62)
       (/ 2.0 (* (pow (* (cbrt (sin k)) (* t (cbrt (pow l -2.0)))) 3.0) t_2))
       (if (<= t -6e-8)
         (* l (/ (* l (/ 2.0 (tan k))) (* t_3 (* (sin k) (pow t 3.0)))))
         (if (<= t -5e-99)
           (/ 2.0 (* t_2 (* (/ (pow t 3.0) l) (/ k l))))
           (if (<= t -4e-172)
             (/ (* l (* (/ 2.0 (pow (* k (sin k)) 2.0)) (* l (cos k)))) t)
             (if (<= t 7.5e+30)
               (/
                2.0
                (/ k (* (/ (cos k) k) (/ l (/ (pow (sin k) 2.0) (/ l t))))))
               (/ (/ l t) (/ (pow (* t k) 2.0) l))))))))))

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 = pow((k / t), 2.0);
	double t_2 = tan(k) * (1.0 + (1.0 + t_1));
	double t_3 = 2.0 + t_1;
	double tmp;
	if (t <= -1.4e+180) {
		tmp = pow(((1.0 / cbrt((t_3 * ((sin(k) * tan(k)) * 0.5)))) * (pow(cbrt(l), 2.0) / t)), 3.0);
	} else if (t <= -5.8e+62) {
		tmp = 2.0 / (pow((cbrt(sin(k)) * (t * cbrt(pow(l, -2.0)))), 3.0) * t_2);
	} else if (t <= -6e-8) {
		tmp = l * ((l * (2.0 / tan(k))) / (t_3 * (sin(k) * pow(t, 3.0))));
	} else if (t <= -5e-99) {
		tmp = 2.0 / (t_2 * ((pow(t, 3.0) / l) * (k / l)));
	} else if (t <= -4e-172) {
		tmp = (l * ((2.0 / pow((k * sin(k)), 2.0)) * (l * cos(k)))) / t;
	} else if (t <= 7.5e+30) {
		tmp = 2.0 / (k / ((cos(k) / k) * (l / (pow(sin(k), 2.0) / (l / t)))));
	} else {
		tmp = (l / t) / (pow((t * k), 2.0) / l);
	}
	return tmp;
}

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));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = Math.tan(k) * (1.0 + (1.0 + t_1));
	double t_3 = 2.0 + t_1;
	double tmp;
	if (t <= -1.4e+180) {
		tmp = Math.pow(((1.0 / Math.cbrt((t_3 * ((Math.sin(k) * Math.tan(k)) * 0.5)))) * (Math.pow(Math.cbrt(l), 2.0) / t)), 3.0);
	} else if (t <= -5.8e+62) {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) * (t * Math.cbrt(Math.pow(l, -2.0)))), 3.0) * t_2);
	} else if (t <= -6e-8) {
		tmp = l * ((l * (2.0 / Math.tan(k))) / (t_3 * (Math.sin(k) * Math.pow(t, 3.0))));
	} else if (t <= -5e-99) {
		tmp = 2.0 / (t_2 * ((Math.pow(t, 3.0) / l) * (k / l)));
	} else if (t <= -4e-172) {
		tmp = (l * ((2.0 / Math.pow((k * Math.sin(k)), 2.0)) * (l * Math.cos(k)))) / t;
	} else if (t <= 7.5e+30) {
		tmp = 2.0 / (k / ((Math.cos(k) / k) * (l / (Math.pow(Math.sin(k), 2.0) / (l / t)))));
	} else {
		tmp = (l / t) / (Math.pow((t * k), 2.0) / l);
	}
	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(k / t) ^ 2.0
	t_2 = Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_1)))
	t_3 = Float64(2.0 + t_1)
	tmp = 0.0
	if (t <= -1.4e+180)
		tmp = Float64(Float64(1.0 / cbrt(Float64(t_3 * Float64(Float64(sin(k) * tan(k)) * 0.5)))) * Float64((cbrt(l) ^ 2.0) / t)) ^ 3.0;
	elseif (t <= -5.8e+62)
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) * Float64(t * cbrt((l ^ -2.0)))) ^ 3.0) * t_2));
	elseif (t <= -6e-8)
		tmp = Float64(l * Float64(Float64(l * Float64(2.0 / tan(k))) / Float64(t_3 * Float64(sin(k) * (t ^ 3.0)))));
	elseif (t <= -5e-99)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64((t ^ 3.0) / l) * Float64(k / l))));
	elseif (t <= -4e-172)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / (Float64(k * sin(k)) ^ 2.0)) * Float64(l * cos(k)))) / t);
	elseif (t <= 7.5e+30)
		tmp = Float64(2.0 / Float64(k / Float64(Float64(cos(k) / k) * Float64(l / Float64((sin(k) ^ 2.0) / Float64(l / t))))));
	else
		tmp = Float64(Float64(l / t) / Float64((Float64(t * k) ^ 2.0) / l));
	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[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[t, -1.4e+180], N[Power[N[(N[(1.0 / N[Power[N[(t$95$3 * N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], If[LessEqual[t, -5.8e+62], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Power[l, -2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -6e-8], N[(l * N[(N[(l * N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$3 * N[(N[Sin[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -5e-99], N[(2.0 / N[(t$95$2 * N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4e-172], N[(N[(l * N[(N[(2.0 / N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 7.5e+30], N[(2.0 / N[(k / N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(l / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l / t), $MachinePrecision] / N[(N[Power[N[(t * k), $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if t < -1.40000000000000006e180

   1. Initial program 67.7%

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

   2. Simplified 56.5%

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

      [Start] 67.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 67.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/l/ [<=] 67.7	\[ \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]
      *-commutative [=>] 67.7	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]
      associate-*r/ [=>] 67.7	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]
      associate-/l* [=>] 67.7	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
      associate-/r/ [=>] 51.6	\[ \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Applied egg-rr 83.0%

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

      [Start] 56.5	\[ \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right) \]
      add-cube-cbrt [=>] 56.5	\[ \color{blue}{\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \cdot \sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)}} \]
      pow3 [=>] 56.5	\[ \color{blue}{{\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)}\right)}^{3}} \]
      cbrt-prod [=>] 56.5	\[ {\color{blue}{\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}} \cdot \ell}\right)}}^{3} \]
      associate-*l/ [=>] 51.6	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt[3]{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}\right)}^{3} \]
      cbrt-div [=>] 51.6	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \color{blue}{\frac{\sqrt[3]{\ell \cdot \ell}}{\sqrt[3]{{t}^{3}}}}\right)}^{3} \]
      cbrt-unprod [<=] 56.5	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]
      pow2 [=>] 56.5	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]
      rem-cbrt-cube [=>] 83.0	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\color{blue}{t}}\right)}^{3} \]

   4. Applied egg-rr 86.2%

      \[\leadsto {\left(\color{blue}{\frac{1}{\sqrt[3]{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot 0.5}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      Step-by-step derivation

      [Start] 83.0	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      clear-num [=>] 83.0	\[ {\left(\sqrt[3]{\color{blue}{\frac{1}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}{2}}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      cbrt-div [=>] 86.2	\[ {\left(\color{blue}{\frac{\sqrt[3]{1}}{\sqrt[3]{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}{2}}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      metadata-eval [=>] 86.2	\[ {\left(\frac{\color{blue}{1}}{\sqrt[3]{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}{2}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      div-inv [=>] 86.2	\[ {\left(\frac{1}{\sqrt[3]{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{1}{2}}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      metadata-eval [=>] 86.2	\[ {\left(\frac{1}{\sqrt[3]{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \color{blue}{0.5}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]

   5. Simplified 86.2%

      \[\leadsto {\left(\color{blue}{\frac{1}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot 0.5\right)}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      Step-by-step derivation

      [Start] 86.2	\[ {\left(\frac{1}{\sqrt[3]{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot 0.5}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      associate-*l* [=>] 86.2	\[ {\left(\frac{1}{\sqrt[3]{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot 0.5\right)}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]

   if -1.40000000000000006e180 < t < -5.79999999999999968e62

   1. Initial program 54.0%

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

   2. Simplified 54.0%

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

      [Start] 54.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 54.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 54.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 87.1%

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

      [Start] 54.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      add-cube-cbrt [=>] 53.7	\[ \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right) \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      pow3 [=>] 53.7	\[ \frac{2}{\color{blue}{{\left(\sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}\right)}^{3}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      *-commutative [=>] 53.7	\[ \frac{2}{{\left(\sqrt[3]{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      cbrt-prod [=>] 53.8	\[ \frac{2}{{\color{blue}{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      div-inv [=>] 53.7	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\color{blue}{{t}^{3} \cdot \frac{1}{\ell \cdot \ell}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      cbrt-prod [=>] 58.1	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \color{blue}{\left(\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      rem-cbrt-cube [=>] 84.8	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(\color{blue}{t} \cdot \sqrt[3]{\frac{1}{\ell \cdot \ell}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      pow2 [=>] 84.8	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{\frac{1}{\color{blue}{{\ell}^{2}}}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      pow-flip [=>] 87.1	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{\color{blue}{{\ell}^{\left(-2\right)}}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      metadata-eval [=>] 87.1	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{\color{blue}{-2}}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if -5.79999999999999968e62 < t < -5.99999999999999946e-8

   1. Initial program 68.7%

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

   2. Simplified 68.4%

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

      [Start] 68.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 68.7	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 68.6	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 68.2	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 68.4	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 68.4	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 68.4	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 68.2	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 68.2	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 68.4	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 68.4	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Applied egg-rr 34.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right)\right)} - 1} \]
      Step-by-step derivation

      [Start] 68.4	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \]
      expm1-log1p-u [=>] 51.7	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)\right)} \]
      expm1-udef [=>] 28.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)} - 1} \]
      associate-*l* [=>] 34.7	\[ e^{\mathsf{log1p}\left(\color{blue}{\ell \cdot \left(\ell \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right)}\right)} - 1 \]
      associate-/r* [=>] 34.7	\[ e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \color{blue}{\frac{\frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}}\right)\right)} - 1 \]

   4. Simplified 91.2%

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

      [Start] 34.7	\[ e^{\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right)\right)} - 1 \]
      expm1-def [=>] 65.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right)\right)\right)} \]
      expm1-log1p [=>] 83.0	\[ \color{blue}{\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}\right)} \]
      associate-*r/ [=>] 91.2	\[ \ell \cdot \color{blue}{\frac{\ell \cdot \frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)}} \]
      *-commutative [<=] 91.2	\[ \ell \cdot \frac{\ell \cdot \frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sin k \cdot {t}^{3}\right)}} \]

   if -5.99999999999999946e-8 < t < -4.99999999999999969e-99

   1. Initial program 60.3%

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

   2. Simplified 60.3%

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

      [Start] 60.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 60.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 60.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in k around 0 53.9%

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

   4. Simplified 87.0%

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

      [Start] 53.9	\[ \frac{2}{\frac{k \cdot {t}^{3}}{{\ell}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      *-commutative [=>] 53.9	\[ \frac{2}{\frac{\color{blue}{{t}^{3} \cdot k}}{{\ell}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow2 [=>] 53.9	\[ \frac{2}{\frac{{t}^{3} \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 87.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if -4.99999999999999969e-99 < t < -4.0000000000000002e-172

   1. Initial program 34.2%

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

   2. Simplified 34.2%

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

      [Start] 34.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 34.2	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l* [=>] 34.2	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-*r* [=>] 34.2	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      +-commutative [=>] 34.2	\[ \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-+r+ [=>] 34.2	\[ \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      metadata-eval [=>] 34.2	\[ \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Taylor expanded in k around inf 86.4%

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

   4. Simplified 81.9%

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

      [Start] 86.4	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 82.0	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 82.0	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 82.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]
      unpow2 [=>] 82.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{t}}} \]
      associate-/l* [=>] 81.9	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}} \]

   5. Applied egg-rr 81.7%

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

      [Start] 81.9	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{\frac{t}{\ell}}}} \]
      associate-*r/ [=>] 81.9	\[ \frac{2}{\color{blue}{\frac{\frac{k \cdot k}{\cos k} \cdot {\sin k}^{2}}{\frac{\ell}{\frac{t}{\ell}}}}} \]
      associate-/r/ [=>] 81.7	\[ \color{blue}{\frac{2}{\frac{k \cdot k}{\cos k} \cdot {\sin k}^{2}} \cdot \frac{\ell}{\frac{t}{\ell}}} \]
      associate-*l/ [=>] 81.7	\[ \frac{2}{\color{blue}{\frac{\left(k \cdot k\right) \cdot {\sin k}^{2}}{\cos k}}} \cdot \frac{\ell}{\frac{t}{\ell}} \]
      pow2 [=>] 81.7	\[ \frac{2}{\frac{\color{blue}{{k}^{2}} \cdot {\sin k}^{2}}{\cos k}} \cdot \frac{\ell}{\frac{t}{\ell}} \]
      pow-prod-down [=>] 81.7	\[ \frac{2}{\frac{\color{blue}{{\left(k \cdot \sin k\right)}^{2}}}{\cos k}} \cdot \frac{\ell}{\frac{t}{\ell}} \]
      associate-/r/ [=>] 81.7	\[ \frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k}} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \ell\right)} \]
      *-commutative [<=] 81.7	\[ \frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k}} \cdot \color{blue}{\left(\ell \cdot \frac{\ell}{t}\right)} \]

   6. Applied egg-rr 95.1%

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

      [Start] 81.7	\[ \frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k}} \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]
      associate-*r* [=>] 90.9	\[ \color{blue}{\left(\frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k}} \cdot \ell\right) \cdot \frac{\ell}{t}} \]
      associate-*r/ [=>] 95.1	\[ \color{blue}{\frac{\left(\frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k}} \cdot \ell\right) \cdot \ell}{t}} \]
      associate-/r/ [=>] 95.1	\[ \frac{\left(\color{blue}{\left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \cos k\right)} \cdot \ell\right) \cdot \ell}{t} \]
      associate-*l* [=>] 95.1	\[ \frac{\color{blue}{\left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\cos k \cdot \ell\right)\right)} \cdot \ell}{t} \]

   if -4.0000000000000002e-172 < t < 7.49999999999999973e30

   1. Initial program 42.0%

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

   2. Simplified 41.0%

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

      [Start] 42.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 42.0	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l* [=>] 41.0	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-*r* [=>] 41.0	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      +-commutative [=>] 41.0	\[ \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-+r+ [=>] 41.0	\[ \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      metadata-eval [=>] 41.0	\[ \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Taylor expanded in k around inf 67.1%

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

   4. Simplified 78.4%

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

      [Start] 67.1	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 65.6	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 65.6	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 66.5	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]
      unpow2 [=>] 66.5	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{t}}} \]
      associate-/l* [=>] 78.4	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}} \]

   5. Applied egg-rr 84.6%

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

      [Start] 78.4	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{\frac{t}{\ell}}}} \]
      associate-/l* [=>] 78.4	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{\frac{t}{\ell}}}} \]
      clear-num [=>] 78.4	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\frac{1}{\frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}}}} \]
      frac-times [=>] 84.4	\[ \frac{2}{\color{blue}{\frac{k \cdot 1}{\frac{\cos k}{k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}}}} \]
      *-commutative [<=] 84.4	\[ \frac{2}{\frac{\color{blue}{1 \cdot k}}{\frac{\cos k}{k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}}} \]
      *-un-lft-identity [<=] 84.4	\[ \frac{2}{\frac{\color{blue}{k}}{\frac{\cos k}{k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}}} \]
      associate-/r/ [=>] 84.4	\[ \frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\color{blue}{\frac{\ell}{t} \cdot \ell}}{{\sin k}^{2}}}} \]
      *-commutative [<=] 84.4	\[ \frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\color{blue}{\ell \cdot \frac{\ell}{t}}}{{\sin k}^{2}}}} \]
      associate-/l* [=>] 84.6	\[ \frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \color{blue}{\frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}} \]

   if 7.49999999999999973e30 < t

   1. Initial program 62.3%

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

   2. Simplified 50.8%

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

      [Start] 62.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 62.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l* [=>] 50.8	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-*r* [=>] 50.8	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      +-commutative [=>] 50.8	\[ \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-+r+ [=>] 50.8	\[ \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      metadata-eval [=>] 50.8	\[ \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Taylor expanded in k around 0 52.8%

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

   4. Simplified 53.3%

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

      [Start] 52.8	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 52.8	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      *-commutative [=>] 52.8	\[ \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]
      times-frac [=>] 53.3	\[ \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]
      unpow2 [=>] 53.3	\[ \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{\left(\frac{1}{t} \cdot \frac{\ell}{t \cdot t}\right)} \cdot \frac{\ell}{k \cdot k} \]
      Step-by-step derivation

      [Start] 53.3	\[ \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{k \cdot k} \]
      *-un-lft-identity [=>] 53.3	\[ \frac{\color{blue}{1 \cdot \ell}}{{t}^{3}} \cdot \frac{\ell}{k \cdot k} \]
      cube-mult [=>] 53.3	\[ \frac{1 \cdot \ell}{\color{blue}{t \cdot \left(t \cdot t\right)}} \cdot \frac{\ell}{k \cdot k} \]
      times-frac [=>] 55.3	\[ \color{blue}{\left(\frac{1}{t} \cdot \frac{\ell}{t \cdot t}\right)} \cdot \frac{\ell}{k \cdot k} \]

   6. Applied egg-rr 55.3%

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

      [Start] 55.3	\[ \left(\frac{1}{t} \cdot \frac{\ell}{t \cdot t}\right) \cdot \frac{\ell}{k \cdot k} \]
      associate-/r* [=>] 55.3	\[ \left(\frac{1}{t} \cdot \color{blue}{\frac{\frac{\ell}{t}}{t}}\right) \cdot \frac{\ell}{k \cdot k} \]
      frac-times [=>] 55.3	\[ \color{blue}{\frac{1 \cdot \frac{\ell}{t}}{t \cdot t}} \cdot \frac{\ell}{k \cdot k} \]
      *-un-lft-identity [<=] 55.3	\[ \frac{\color{blue}{\frac{\ell}{t}}}{t \cdot t} \cdot \frac{\ell}{k \cdot k} \]

   7. Applied egg-rr 92.0%

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

      [Start] 55.3	\[ \frac{\frac{\ell}{t}}{t \cdot t} \cdot \frac{\ell}{k \cdot k} \]
      frac-times [=>] 62.5	\[ \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}} \]
      associate-/l* [=>] 62.6	\[ \color{blue}{\frac{\frac{\ell}{t}}{\frac{\left(t \cdot t\right) \cdot \left(k \cdot k\right)}{\ell}}} \]
      pow2 [=>] 62.6	\[ \frac{\frac{\ell}{t}}{\frac{\color{blue}{{t}^{2}} \cdot \left(k \cdot k\right)}{\ell}} \]
      pow2 [=>] 62.6	\[ \frac{\frac{\ell}{t}}{\frac{{t}^{2} \cdot \color{blue}{{k}^{2}}}{\ell}} \]
      pow-prod-down [=>] 92.0	\[ \frac{\frac{\ell}{t}}{\frac{\color{blue}{{\left(t \cdot k\right)}^{2}}}{\ell}} \]

3. Recombined 7 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+180}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\sin k \cdot \tan k\right) \cdot 0.5\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{+62}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-8}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-172}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\ell \cdot \cos k\right)\right)}{t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	82.3%
Cost	72712

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{t_1 \cdot \tan k} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+180}:\\ \;\;\;\;{\left(\frac{1}{\sqrt[3]{t_1 \cdot \left(\left(\sin k \cdot \tan k\right) \cdot 0.5\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{+59}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot \ell\right)}{{t_2}^{2}}}{t_2}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(t_1 \cdot \frac{\sin k}{\frac{\ell}{\tan k}}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 2
Accuracy	81.5%
Cost	46344

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ t_3 := \tan k \cdot \left(1 + \left(1 + t_1\right)\right)\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+180}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{\frac{\frac{2}{\sin k \cdot \tan k}}{t_2}}\right)}^{3}\\ \mathbf{elif}\;t \leq -6.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \left(t \cdot \sqrt[3]{{\ell}^{-2}}\right)\right)}^{3} \cdot t_3}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-8}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot \frac{2}{\tan k}}{t_2 \cdot \left(\sin k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-170}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\ell \cdot \cos k\right)\right)}{t}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+29}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 3
Accuracy	81.5%
Cost	46084

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-5}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}\right)}^{3}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-172}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\ell \cdot \cos k\right)\right)}{t}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	81.3%
Cost	46084

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-7}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{\frac{\frac{2}{\sin k \cdot \tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-172}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\ell \cdot \cos k\right)\right)}{t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+36}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	83.5%
Cost	27080

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\frac{\ell}{\tan k}}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	80.1%
Cost	20808

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-8}:\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t \cdot k}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-99}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\ell \cdot \cos k\right)\right)}{t}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+33}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 7
Accuracy	81.3%
Cost	20752

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-170}:\\ \;\;\;\;\frac{\ell \cdot \left(\frac{2}{{\left(k \cdot \sin k\right)}^{2}} \cdot \left(\ell \cdot \cos k\right)\right)}{t}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\frac{\ell}{t}}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	75.7%
Cost	14156

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-100}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-242}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot \sin k\right)\right)}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-136}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+52}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	80.2%
Cost	14156

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	75.0%
Cost	7884

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 0.0013:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k \cdot k}{\ell \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	75.0%
Cost	7884

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 0.0008:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	72.8%
Cost	7436

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -1.75 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-50}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Accuracy	68.8%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 14
Accuracy	68.8%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -1.95 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\left(t \cdot t\right) \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{k}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 15
Accuracy	64.8%
Cost	832

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

Alternative 16
Accuracy	64.9%
Cost	832

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

Alternative 17
Accuracy	68.2%
Cost	832

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

Reproduce

herbie shell --seed 2023161 
(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))))