Toniolo and Linder, Equation (10+)

Average Accuracy: 50.1% → 87.1%

Time: 45.9s

Precision: binary64

Cost: 46544

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot t_2\right)\right)\right)\\ \mathbf{if}\;k \leq -5.4 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{t_3}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+108}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0))))))
        (t_2 (+ 2.0 (pow (/ k t) 2.0)))
        (t_3 (* t (* (/ t l) (* (tan k) (* (sin k) t_2))))))
   (if (<= k -5.4e+39)
     t_1
     (if (<= k -6.8e-76)
       (/ 2.0 (/ t_3 (/ l t)))
       (if (<= k 3.8e-260)
         (/ (/ l (* k t)) (* (/ t l) (* k t)))
         (if (<= k 7.8e-161)
           (/
            2.0
            (*
             t_2
             (pow
              (/ (* t (cbrt (tan k))) (* (cbrt l) (cbrt (/ l (sin k)))))
              3.0)))
           (if (<= k 2.35e+108) (/ (* 2.0 (/ l t)) t_3) t_1)))))))

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 = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	double t_2 = 2.0 + pow((k / t), 2.0);
	double t_3 = t * ((t / l) * (tan(k) * (sin(k) * t_2)));
	double tmp;
	if (k <= -5.4e+39) {
		tmp = t_1;
	} else if (k <= -6.8e-76) {
		tmp = 2.0 / (t_3 / (l / t));
	} else if (k <= 3.8e-260) {
		tmp = (l / (k * t)) / ((t / l) * (k * t));
	} else if (k <= 7.8e-161) {
		tmp = 2.0 / (t_2 * pow(((t * cbrt(tan(k))) / (cbrt(l) * cbrt((l / sin(k))))), 3.0));
	} else if (k <= 2.35e+108) {
		tmp = (2.0 * (l / t)) / t_3;
	} else {
		tmp = t_1;
	}
	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 = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	double t_2 = 2.0 + Math.pow((k / t), 2.0);
	double t_3 = t * ((t / l) * (Math.tan(k) * (Math.sin(k) * t_2)));
	double tmp;
	if (k <= -5.4e+39) {
		tmp = t_1;
	} else if (k <= -6.8e-76) {
		tmp = 2.0 / (t_3 / (l / t));
	} else if (k <= 3.8e-260) {
		tmp = (l / (k * t)) / ((t / l) * (k * t));
	} else if (k <= 7.8e-161) {
		tmp = 2.0 / (t_2 * Math.pow(((t * Math.cbrt(Math.tan(k))) / (Math.cbrt(l) * Math.cbrt((l / Math.sin(k))))), 3.0));
	} else if (k <= 2.35e+108) {
		tmp = (2.0 * (l / t)) / t_3;
	} else {
		tmp = t_1;
	}
	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(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))))
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_3 = Float64(t * Float64(Float64(t / l) * Float64(tan(k) * Float64(sin(k) * t_2))))
	tmp = 0.0
	if (k <= -5.4e+39)
		tmp = t_1;
	elseif (k <= -6.8e-76)
		tmp = Float64(2.0 / Float64(t_3 / Float64(l / t)));
	elseif (k <= 3.8e-260)
		tmp = Float64(Float64(l / Float64(k * t)) / Float64(Float64(t / l) * Float64(k * t)));
	elseif (k <= 7.8e-161)
		tmp = Float64(2.0 / Float64(t_2 * (Float64(Float64(t * cbrt(tan(k))) / Float64(cbrt(l) * cbrt(Float64(l / sin(k))))) ^ 3.0)));
	elseif (k <= 2.35e+108)
		tmp = Float64(Float64(2.0 * Float64(l / t)) / t_3);
	else
		tmp = t_1;
	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[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -5.4e+39], t$95$1, If[LessEqual[k, -6.8e-76], N[(2.0 / N[(t$95$3 / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.8e-260], N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(N[(t / l), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.8e-161], N[(2.0 / N[(t$95$2 * N[Power[N[(N[(t * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 1/3], $MachinePrecision] * N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.35e+108], N[(N[(2.0 * N[(l / t), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -5.40000000000000007e39 or 2.3499999999999998e108 < k

   1. Initial program 49.0%

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

   2. Simplified 49.0%

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

      [Start] 49.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 49.0	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 49.0	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 49.0	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 49.0	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 49.0	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 49.0	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 49.0	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 67.6%

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

   4. Simplified 86.8%

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

      [Start] 67.6	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 67.6	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 65.0	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 65.0	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 65.0	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 86.8	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 86.8	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if -5.40000000000000007e39 < k < -6.7999999999999998e-76

   1. Initial program 55.1%

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

   2. Simplified 55.1%

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

      [Start] 55.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 55.1	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 55.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 55.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 55.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 55.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 55.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 55.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 55.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 70.3%

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

      [Start] 55.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow3 [=>] 55.1	\[ \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 70.3	\[ \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. Applied egg-rr 85.6%

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

      [Start] 70.3	\[ \frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l* [=>] 74.1	\[ \frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]
      associate-/l* [=>] 85.1	\[ \frac{2}{\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \]
      associate-*l/ [=>] 85.6	\[ \frac{2}{\color{blue}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}{\frac{\ell}{t}}}} \]
      *-commutative [=>] 85.6	\[ \frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k\right)}\right)}{\frac{\ell}{t}}} \]
      associate-*l* [=>] 85.6	\[ \frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\right)}\right)}{\frac{\ell}{t}}} \]

   if -6.7999999999999998e-76 < k < 3.8000000000000003e-260

   1. Initial program 43.9%

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

   2. Simplified 21.2%

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

      [Start] 43.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 43.9	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 43.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 43.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 43.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 21.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 21.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 21.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 21.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 24.4%

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

      [Start] 21.2	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow3 [=>] 21.2	\[ \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 24.4	\[ \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. Applied egg-rr 30.6%

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

      [Start] 24.4	\[ \frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      *-commutative [=>] 24.4	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      associate-/r* [=>] 22.4	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}}} \]
      clear-num [=>] 22.4	\[ \frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}} \]
      un-div-inv [=>] 22.4	\[ \frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\color{blue}{\frac{\frac{t \cdot t}{\ell}}{\frac{\ell}{t}}}} \]
      associate-/r/ [=>] 26.2	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{t \cdot t}{\ell}} \cdot \frac{\ell}{t}} \]
      *-commutative [=>] 26.2	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k}}}{\frac{t \cdot t}{\ell}} \cdot \frac{\ell}{t} \]
      associate-*l* [=>] 26.2	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}}{\frac{t \cdot t}{\ell}} \cdot \frac{\ell}{t} \]
      div-inv [=>] 26.2	\[ \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{\color{blue}{\left(t \cdot t\right) \cdot \frac{1}{\ell}}} \cdot \frac{\ell}{t} \]
      associate-*l* [=>] 30.6	\[ \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{\color{blue}{t \cdot \left(t \cdot \frac{1}{\ell}\right)}} \cdot \frac{\ell}{t} \]
      div-inv [<=] 30.6	\[ \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t \cdot \color{blue}{\frac{t}{\ell}}} \cdot \frac{\ell}{t} \]

   5. Taylor expanded in k around 0 25.7%

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

   6. Simplified 52.7%

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

      [Start] 25.7	\[ \frac{\ell}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\ell}{t} \]
      unpow2 [=>] 25.7	\[ \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}} \cdot \frac{\ell}{t} \]
      associate-*l* [=>] 52.7	\[ \frac{\ell}{\color{blue}{k \cdot \left(k \cdot {t}^{2}\right)}} \cdot \frac{\ell}{t} \]
      unpow2 [=>] 52.7	\[ \frac{\ell}{k \cdot \left(k \cdot \color{blue}{\left(t \cdot t\right)}\right)} \cdot \frac{\ell}{t} \]

   7. Applied egg-rr 91.4%

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

      [Start] 52.7	\[ \frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \frac{\ell}{t} \]
      associate-*r/ [=>] 51.5	\[ \color{blue}{\frac{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \ell}{t}} \]
      associate-/l* [=>] 52.4	\[ \color{blue}{\frac{\frac{\ell}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}}{\frac{t}{\ell}}} \]
      add-sqr-sqrt [=>] 52.3	\[ \frac{\frac{\ell}{\color{blue}{\sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)} \cdot \sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}}}}{\frac{t}{\ell}} \]
      associate-/r* [=>] 52.3	\[ \frac{\color{blue}{\frac{\frac{\ell}{\sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}}}{\sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}}}}{\frac{t}{\ell}} \]
      associate-/l/ [=>] 52.4	\[ \color{blue}{\frac{\frac{\ell}{\sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}}}{\frac{t}{\ell} \cdot \sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}}} \]
      associate-*r* [=>] 25.7	\[ \frac{\frac{\ell}{\sqrt{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}}}}{\frac{t}{\ell} \cdot \sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]
      sqrt-prod [=>] 25.7	\[ \frac{\frac{\ell}{\color{blue}{\sqrt{k \cdot k} \cdot \sqrt{t \cdot t}}}}{\frac{t}{\ell} \cdot \sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]
      sqrt-unprod [<=] 5.4	\[ \frac{\frac{\ell}{\color{blue}{\left(\sqrt{k} \cdot \sqrt{k}\right)} \cdot \sqrt{t \cdot t}}}{\frac{t}{\ell} \cdot \sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]
      add-sqr-sqrt [<=] 35.7	\[ \frac{\frac{\ell}{\color{blue}{k} \cdot \sqrt{t \cdot t}}}{\frac{t}{\ell} \cdot \sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]
      sqrt-prod [=>] 16.2	\[ \frac{\frac{\ell}{k \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}}}{\frac{t}{\ell} \cdot \sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]
      add-sqr-sqrt [<=] 43.3	\[ \frac{\frac{\ell}{k \cdot \color{blue}{t}}}{\frac{t}{\ell} \cdot \sqrt{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]
      associate-*r* [=>] 20.1	\[ \frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \sqrt{\color{blue}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}}} \]
      sqrt-prod [=>] 21.9	\[ \frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \color{blue}{\left(\sqrt{k \cdot k} \cdot \sqrt{t \cdot t}\right)}} \]
      sqrt-unprod [<=] 6.7	\[ \frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(\color{blue}{\left(\sqrt{k} \cdot \sqrt{k}\right)} \cdot \sqrt{t \cdot t}\right)} \]
      add-sqr-sqrt [<=] 50.9	\[ \frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(\color{blue}{k} \cdot \sqrt{t \cdot t}\right)} \]
      sqrt-prod [=>] 42.5	\[ \frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)}\right)} \]
      add-sqr-sqrt [<=] 91.4	\[ \frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot \color{blue}{t}\right)} \]

   if 3.8000000000000003e-260 < k < 7.79999999999999947e-161

   1. Initial program 42.5%

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

   2. Simplified 54.3%

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

      [Start] 42.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 42.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 46.5	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 46.6	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 54.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 54.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 54.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 54.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 68.9%

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

      [Start] 54.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      add-cube-cbrt [=>] 54.0	\[ \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right) \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      *-commutative [=>] 54.0	\[ \frac{2}{\color{blue}{\left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      cbrt-div [=>] 54.0	\[ \frac{2}{\left(\color{blue}{\frac{\sqrt[3]{\tan k \cdot {t}^{3}}}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}}} \cdot \left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      *-commutative [=>] 54.0	\[ \frac{2}{\left(\frac{\sqrt[3]{\color{blue}{{t}^{3} \cdot \tan k}}}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      cbrt-prod [=>] 54.0	\[ \frac{2}{\left(\frac{\color{blue}{\sqrt[3]{{t}^{3}} \cdot \sqrt[3]{\tan k}}}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      rem-cbrt-cube [=>] 54.0	\[ \frac{2}{\left(\frac{\color{blue}{t} \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      associate-/r/ [=>] 54.0	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\color{blue}{\frac{\ell}{\sin k} \cdot \ell}}} \cdot \left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      *-commutative [=>] 54.0	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\color{blue}{\ell \cdot \frac{\ell}{\sin k}}}} \cdot \left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      pow2 [=>] 54.0	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot \color{blue}{{\left(\sqrt[3]{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\right)}^{2}}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   4. Simplified 69.0%

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

      [Start] 68.9	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{2}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      unpow2 [=>] 68.9	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot \color{blue}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      cube-mult [<=] 69.0	\[ \frac{2}{\color{blue}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied egg-rr 82.7%

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

      [Start] 69.0	\[ \frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      cbrt-prod [=>] 82.7	\[ \frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}}}\right)}^{3} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      *-commutative [=>] 82.7	\[ \frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\color{blue}{\sqrt[3]{\frac{\ell}{\sin k}} \cdot \sqrt[3]{\ell}}}\right)}^{3} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   if 7.79999999999999947e-161 < k < 2.3499999999999998e108

   1. Initial program 55.8%

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

   2. Simplified 56.0%

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

      [Start] 55.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 55.8	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 55.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 55.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 55.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 56.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 56.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 56.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 56.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 69.5%

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

      [Start] 56.0	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow3 [=>] 56.0	\[ \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 69.5	\[ \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. Applied egg-rr 84.6%

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

      [Start] 69.5	\[ \frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      *-commutative [=>] 69.5	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      associate-/r* [=>] 68.7	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}}} \]
      clear-num [=>] 68.7	\[ \frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{t \cdot t}{\ell} \cdot \color{blue}{\frac{1}{\frac{\ell}{t}}}} \]
      un-div-inv [=>] 68.7	\[ \frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\color{blue}{\frac{\frac{t \cdot t}{\ell}}{\frac{\ell}{t}}}} \]
      associate-/r/ [=>] 73.5	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{t \cdot t}{\ell}} \cdot \frac{\ell}{t}} \]
      *-commutative [=>] 73.5	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k}}}{\frac{t \cdot t}{\ell}} \cdot \frac{\ell}{t} \]
      associate-*l* [=>] 73.4	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}}{\frac{t \cdot t}{\ell}} \cdot \frac{\ell}{t} \]
      div-inv [=>] 73.4	\[ \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{\color{blue}{\left(t \cdot t\right) \cdot \frac{1}{\ell}}} \cdot \frac{\ell}{t} \]
      associate-*l* [=>] 84.5	\[ \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{\color{blue}{t \cdot \left(t \cdot \frac{1}{\ell}\right)}} \cdot \frac{\ell}{t} \]
      div-inv [<=] 84.6	\[ \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t \cdot \color{blue}{\frac{t}{\ell}}} \cdot \frac{\ell}{t} \]

   5. Applied egg-rr 86.8%

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

      [Start] 84.6	\[ \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t \cdot \frac{t}{\ell}} \cdot \frac{\ell}{t} \]
      *-commutative [=>] 84.6	\[ \color{blue}{\frac{\ell}{t} \cdot \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t \cdot \frac{t}{\ell}}} \]
      associate-/l/ [=>] 85.6	\[ \frac{\ell}{t} \cdot \color{blue}{\frac{2}{\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\right)}} \]
      associate-*r/ [=>] 85.8	\[ \color{blue}{\frac{\frac{\ell}{t} \cdot 2}{\left(t \cdot \frac{t}{\ell}\right) \cdot \left(\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\right)}} \]
      associate-*l* [=>] 86.8	\[ \frac{\frac{\ell}{t} \cdot 2}{\color{blue}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\right)\right)}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5.4 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-260}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+108}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.5%
Cost	52616

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -1.36 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot t_1}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+108}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	84.8%
Cost	21136

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \left(\ell \cdot \frac{\frac{2}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{t \cdot \left(t \cdot \sin k\right)}\right)\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -5.4 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.05 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.36 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	87.7%
Cost	21136

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -4.9 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	87.7%
Cost	21136

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{if}\;k \leq -3.6 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -6.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 2.25 \cdot 10^{+108}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	87.7%
Cost	21136

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)\\ \mathbf{if}\;k \leq -5.4 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -7.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{+108}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	70.5%
Cost	20492

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{t}\\ \mathbf{if}\;k \leq -5.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot \sqrt[3]{k \cdot k}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right)\right)\right)}\\ \mathbf{elif}\;k \leq -1.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\cos k}{{\sin k}^{2}}}{t \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+20} \lor \neg \left(k \leq 5.5 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{t_1}{t \cdot \left({k}^{4} \cdot \left(2 \cdot \frac{t \cdot -0.16666666666666666 + t \cdot 0.3333333333333333}{\ell} + \frac{1}{\ell \cdot t}\right) + 2 \cdot \frac{t \cdot {k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(2 \cdot \sin k\right)\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	82.0%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -1.12 \cdot 10^{-39} \lor \neg \left(k \leq 1.12 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 8
Accuracy	70.5%
Cost	15961

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{t}\\ t_2 := \tan k \cdot \left(2 \cdot \sin k\right)\\ \mathbf{if}\;k \leq -6.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot \sqrt[3]{k \cdot k}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq -1.38 \cdot 10^{+69}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right)\right)\right)}\\ \mathbf{elif}\;k \leq -1.65 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2}{t_2}}{t \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 5.1 \cdot 10^{+20} \lor \neg \left(k \leq 5.2 \cdot 10^{+89}\right):\\ \;\;\;\;\frac{t_1}{t \cdot \left({k}^{4} \cdot \left(2 \cdot \frac{t \cdot -0.16666666666666666 + t \cdot 0.3333333333333333}{\ell} + \frac{1}{\ell \cdot t}\right) + 2 \cdot \frac{t \cdot {k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(\frac{t}{\ell} \cdot t_2\right)}\\ \end{array} \]

Alternative 9
Accuracy	69.3%
Cost	14812

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{t}\\ t_2 := \tan k \cdot \left(2 \cdot \sin k\right)\\ t_3 := t \cdot \frac{t}{\ell}\\ t_4 := \frac{\ell}{t} \cdot \frac{\frac{2}{t_2}}{t_3}\\ \mathbf{if}\;k \leq -5.8 \cdot 10^{+108}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot \sqrt[3]{k \cdot k}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq -1.15 \cdot 10^{+69}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right)}}{t_3}\\ \mathbf{elif}\;k \leq -1.02 \cdot 10^{+14}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -78000000000:\\ \;\;\;\;\frac{\frac{2}{t \cdot t}}{t \cdot \left(\frac{\tan k}{\ell \cdot 0.5} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{elif}\;k \leq -2.1 \cdot 10^{-49}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+88}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(\frac{t}{\ell} \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \end{array} \]

Alternative 10
Accuracy	70.6%
Cost	14536

\[\begin{array}{l} t_1 := \frac{t}{\ell} \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{-40}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t_1}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{k \cdot k}{t \cdot t}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{k \cdot t}\\ \end{array} \]

Alternative 11
Accuracy	69.6%
Cost	14416

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\frac{2}{\tan k \cdot \left(2 \cdot \sin k\right)}}{t \cdot \frac{t}{\ell}}\\ \mathbf{if}\;k \leq -4.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot \sqrt[3]{k \cdot k}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq -2 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 4.9 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t \cdot \left(2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \end{array} \]

Alternative 12
Accuracy	69.6%
Cost	14416

\[\begin{array}{l} t_1 := \tan k \cdot \left(2 \cdot \sin k\right)\\ t_2 := 2 \cdot \frac{\ell}{t}\\ \mathbf{if}\;k \leq -5.1 \cdot 10^{+123}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot \sqrt[3]{k \cdot k}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq -1.85 \cdot 10^{-49}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2}{t_1}}{t \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+89}:\\ \;\;\;\;\frac{t_2}{t \cdot \left(\frac{t}{\ell} \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t \cdot \left(2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \end{array} \]

Alternative 13
Accuracy	66.9%
Cost	13900

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{-38}:\\ \;\;\;\;\frac{t_1}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-96}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t \cdot \left(2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+26}:\\ \;\;\;\;\frac{{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}^{-1}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{k \cdot t}{t_1}}\\ \end{array} \]

Alternative 14
Accuracy	66.7%
Cost	7564

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{t_1}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-179}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t \cdot \left(2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \mathbf{elif}\;t \leq 490:\\ \;\;\;\;\frac{1}{t \cdot t} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{k \cdot t}{t_1}}\\ \end{array} \]

Alternative 15
Accuracy	65.9%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-43} \lor \neg \left(t \leq 5.8 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{{\left(k \cdot t\right)}^{2}}}{t}\\ \end{array} \]

Alternative 16
Accuracy	66.1%
Cost	1352

\[\begin{array}{l} t_1 := \frac{t}{\ell} \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t_1}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{t}}{t \cdot \left(2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{k \cdot t}\\ \end{array} \]

Alternative 17
Accuracy	63.3%
Cost	1224

\[\begin{array}{l} t_1 := \frac{t}{\ell} \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t_1}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-22}:\\ \;\;\;\;\frac{\ell \cdot \frac{-\frac{\ell}{t}}{k \cdot k}}{t \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{k \cdot t}\\ \end{array} \]

Alternative 18
Accuracy	64.2%
Cost	1224

\[\begin{array}{l} t_1 := \frac{t}{\ell} \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t_1}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-21}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k} \cdot \left(-\frac{\ell}{t}\right)}{t \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{k \cdot t}\\ \end{array} \]

Alternative 19
Accuracy	64.4%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-38} \lor \neg \left(t \leq 8 \cdot 10^{-125}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)}}{t \cdot \frac{t}{\ell}}\\ \end{array} \]

Alternative 20
Accuracy	53.0%
Cost	832

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

Alternative 21
Accuracy	56.5%
Cost	832

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

Alternative 22
Accuracy	63.0%
Cost	832

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

Reproduce

herbie shell --seed 2023137 
(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))))