Toniolo and Linder, Equation (10+)

Average Error: 50.53% → 11.92%

Time: 51.4s

Precision: binary64

Cost: 98440

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := \frac{\ell}{k \cdot t}\\ t_4 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_5 := \sqrt[3]{\sin k \cdot \left(t_4 \cdot \tan k\right)}\\ t_6 := \left(\sqrt[3]{t_4} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot t_2\\ t_7 := {\sin k}^{2} \cdot t\\ \mathbf{if}\;k \leq -4.4 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_7}{t_1}}\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{2}{t_6}}{{t_6}^{2}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-147}:\\ \;\;\;\;t_3 \cdot \left(\frac{1}{t} \cdot t_3\right)\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{2}{t_2 \cdot t_5}}{{\left(\frac{\frac{t \cdot t_5}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_7}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* (/ l k) (/ l k)))
        (t_2 (/ t (pow (cbrt l) 2.0)))
        (t_3 (/ l (* k t)))
        (t_4 (+ 2.0 (pow (/ k t) 2.0)))
        (t_5 (cbrt (* (sin k) (* t_4 (tan k)))))
        (t_6 (* (* (cbrt t_4) (cbrt (* (sin k) (tan k)))) t_2))
        (t_7 (* (pow (sin k) 2.0) t)))
   (if (<= k -4.4e+52)
     (* 2.0 (/ (cos k) (/ t_7 t_1)))
     (if (<= k -3.5e-131)
       (/ (/ 2.0 t_6) (pow t_6 2.0))
       (if (<= k 2.7e-147)
         (* t_3 (* (/ 1.0 t) t_3))
         (if (<= k 7.2e+54)
           (/
            (/ 2.0 (* t_2 t_5))
            (pow (/ (/ (* t t_5) (cbrt l)) (cbrt l)) 2.0))
           (* 2.0 (* t_1 (/ (cos k) t_7)))))))))

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 = (l / k) * (l / k);
	double t_2 = t / pow(cbrt(l), 2.0);
	double t_3 = l / (k * t);
	double t_4 = 2.0 + pow((k / t), 2.0);
	double t_5 = cbrt((sin(k) * (t_4 * tan(k))));
	double t_6 = (cbrt(t_4) * cbrt((sin(k) * tan(k)))) * t_2;
	double t_7 = pow(sin(k), 2.0) * t;
	double tmp;
	if (k <= -4.4e+52) {
		tmp = 2.0 * (cos(k) / (t_7 / t_1));
	} else if (k <= -3.5e-131) {
		tmp = (2.0 / t_6) / pow(t_6, 2.0);
	} else if (k <= 2.7e-147) {
		tmp = t_3 * ((1.0 / t) * t_3);
	} else if (k <= 7.2e+54) {
		tmp = (2.0 / (t_2 * t_5)) / pow((((t * t_5) / cbrt(l)) / cbrt(l)), 2.0);
	} else {
		tmp = 2.0 * (t_1 * (cos(k) / t_7));
	}
	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 = (l / k) * (l / k);
	double t_2 = t / Math.pow(Math.cbrt(l), 2.0);
	double t_3 = l / (k * t);
	double t_4 = 2.0 + Math.pow((k / t), 2.0);
	double t_5 = Math.cbrt((Math.sin(k) * (t_4 * Math.tan(k))));
	double t_6 = (Math.cbrt(t_4) * Math.cbrt((Math.sin(k) * Math.tan(k)))) * t_2;
	double t_7 = Math.pow(Math.sin(k), 2.0) * t;
	double tmp;
	if (k <= -4.4e+52) {
		tmp = 2.0 * (Math.cos(k) / (t_7 / t_1));
	} else if (k <= -3.5e-131) {
		tmp = (2.0 / t_6) / Math.pow(t_6, 2.0);
	} else if (k <= 2.7e-147) {
		tmp = t_3 * ((1.0 / t) * t_3);
	} else if (k <= 7.2e+54) {
		tmp = (2.0 / (t_2 * t_5)) / Math.pow((((t * t_5) / Math.cbrt(l)) / Math.cbrt(l)), 2.0);
	} else {
		tmp = 2.0 * (t_1 * (Math.cos(k) / t_7));
	}
	return tmp;
}

Julia

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

↓

function code(t, l, k)
	t_1 = Float64(Float64(l / k) * Float64(l / k))
	t_2 = Float64(t / (cbrt(l) ^ 2.0))
	t_3 = Float64(l / Float64(k * t))
	t_4 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_5 = cbrt(Float64(sin(k) * Float64(t_4 * tan(k))))
	t_6 = Float64(Float64(cbrt(t_4) * cbrt(Float64(sin(k) * tan(k)))) * t_2)
	t_7 = Float64((sin(k) ^ 2.0) * t)
	tmp = 0.0
	if (k <= -4.4e+52)
		tmp = Float64(2.0 * Float64(cos(k) / Float64(t_7 / t_1)));
	elseif (k <= -3.5e-131)
		tmp = Float64(Float64(2.0 / t_6) / (t_6 ^ 2.0));
	elseif (k <= 2.7e-147)
		tmp = Float64(t_3 * Float64(Float64(1.0 / t) * t_3));
	elseif (k <= 7.2e+54)
		tmp = Float64(Float64(2.0 / Float64(t_2 * t_5)) / (Float64(Float64(Float64(t * t_5) / cbrt(l)) / cbrt(l)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(cos(k) / t_7)));
	end
	return tmp
end

Wolfram

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

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[Sin[k], $MachinePrecision] * N[(t$95$4 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[Power[t$95$4, 1/3], $MachinePrecision] * N[Power[N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$7 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k, -4.4e+52], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(t$95$7 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -3.5e-131], N[(N[(2.0 / t$95$6), $MachinePrecision] / N[Power[t$95$6, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.7e-147], N[(t$95$3 * N[(N[(1.0 / t), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.2e+54], N[(N[(2.0 / N[(t$95$2 * t$95$5), $MachinePrecision]), $MachinePrecision] / N[Power[N[(N[(N[(t * t$95$5), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -4.4e52

   1. Initial program 51.83

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

   2. Simplified 51.83

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

      [Start] 51.83	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 51.83	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 51.83	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 32.53

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

   4. Simplified 14.02

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

      [Start] 32.53	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 32.54	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      *-commutative [=>] 32.54	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]
      associate-/l* [=>] 35.17	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]
      unpow2 [=>] 35.17	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]
      unpow2 [=>] 35.17	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}} \]
      times-frac [=>] 14.02	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]

   if -4.4e52 < k < -3.5000000000000002e-131

   1. Initial program 44.52

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

   2. Simplified 44.2

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

      [Start] 44.52	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 44.52	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 44.52	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 44.52	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 44.24	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 44.19	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 44.19	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 44.19	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 44.19	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 13.54

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

   4. Simplified 13.55

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

      [Start] 13.54	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 13.54	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]
      associate-*r/ [=>] 13.54	\[ \frac{\color{blue}{\frac{1 \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      metadata-eval [=>] 13.54	\[ \frac{\frac{\color{blue}{2}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*r* [=>] 13.55	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-commutative [<=] 13.55	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   5. Applied egg-rr 13.55

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

   6. Simplified 13.55

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

      [Start] 13.55	\[ \frac{\frac{2}{\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-commutative [=>] 13.55	\[ \frac{\frac{2}{\color{blue}{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   7. Applied egg-rr 13.65

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

   8. Simplified 13.65

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

      [Start] 13.65	\[ \frac{\frac{2}{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\left(\sqrt[3]{\sin k \cdot \tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-commutative [=>] 13.65	\[ \frac{\frac{2}{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\color{blue}{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   if -3.5000000000000002e-131 < k < 2.6999999999999999e-147

   1. Initial program 58.7

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

   2. Simplified 90.14

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

      [Start] 58.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 58.7	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 58.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 58.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 58.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 89.21	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 89.21	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 89.21	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 89.21	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in k around 0 90.63

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

   4. Simplified 56.85

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

      [Start] 90.63	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 90.63	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 90.26	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 90.26	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      associate-*l* [=>] 56.85	\[ \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   5. Applied egg-rr 62.1

      \[\leadsto \color{blue}{{\left({\left(\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. Applied egg-rr 34.74

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

   7. Applied egg-rr 3.64

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

   if 2.6999999999999999e-147 < k < 7.2000000000000003e54

   1. Initial program 42.86

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

   2. Simplified 42.68

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

      [Start] 42.86	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 42.86	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 42.86	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 42.86	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 42.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 42.67	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 42.67	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 42.67	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 42.68	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 12.49

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

   4. Simplified 12.48

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

      [Start] 12.49	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 12.48	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]
      associate-*r/ [=>] 12.48	\[ \frac{\color{blue}{\frac{1 \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      metadata-eval [=>] 12.48	\[ \frac{\frac{\color{blue}{2}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*r* [=>] 12.48	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-commutative [<=] 12.48	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   5. Applied egg-rr 12.49

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

   if 7.2000000000000003e54 < k

   1. Initial program 51.62

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

   2. Simplified 51.62

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

      [Start] 51.62	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 51.62	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 51.62	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 51.62	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 51.62	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 51.62	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 51.62	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 51.62	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 30.86

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

   4. Simplified 13.65

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

      [Start] 30.86	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 30.86	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 33.9	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 33.9	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 33.9	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 13.65	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 13.65	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 11.92

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.4 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{2}{\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\left(\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-147}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \left(\frac{1}{t} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}}}{{\left(\frac{\frac{t \cdot \sqrt[3]{\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	11.93%
Cost	92040

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \sqrt[3]{\sin k \cdot \left(t_3 \cdot \tan k\right)}\\ t_5 := t_2 \cdot t_4\\ t_6 := {\sin k}^{2} \cdot t\\ t_7 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -9 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_6}{t_1}}\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{\frac{2}{\left(\sqrt[3]{t_3} \cdot \sqrt[3]{\sin k \cdot \tan k}\right) \cdot t_2}}{{t_5}^{2}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-150}:\\ \;\;\;\;t_7 \cdot \left(\frac{1}{t} \cdot t_7\right)\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{2}{t_5}}{{\left(\frac{\frac{t \cdot t_4}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_6}\right)\\ \end{array} \]

Alternative 2
Error	11.86%
Cost	85968

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := {\sin k}^{2} \cdot t\\ t_4 := \frac{\ell}{k \cdot t}\\ t_5 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_6 := \sqrt[3]{\sin k \cdot \left(t_5 \cdot \tan k\right)}\\ t_7 := \frac{t}{t_2} \cdot t_6\\ \mathbf{if}\;k \leq -4 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \mathbf{elif}\;k \leq -4.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{t_2 \cdot \frac{2}{t \cdot \sqrt[3]{t_5 \cdot \left(\sin k \cdot \tan k\right)}}}{{t_7}^{2}}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{-147}:\\ \;\;\;\;t_4 \cdot \left(\frac{1}{t} \cdot t_4\right)\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{\frac{2}{t_7}}{{\left(\frac{\frac{t \cdot t_6}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \end{array} \]

Alternative 3
Error	11.86%
Cost	85640

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \frac{t}{t_2}\\ t_4 := {\sin k}^{2} \cdot t\\ t_5 := \frac{\ell}{k \cdot t}\\ t_6 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -1.6 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_1}}\\ \mathbf{elif}\;k \leq -3.9 \cdot 10^{-131}:\\ \;\;\;\;\frac{t_2 \cdot \frac{2}{t \cdot \sqrt[3]{t_6 \cdot \left(\sin k \cdot \tan k\right)}}}{{\left(t_3 \cdot \sqrt[3]{\sin k \cdot \left(t_6 \cdot \tan k\right)}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.16 \cdot 10^{-154}:\\ \;\;\;\;t_5 \cdot \left(\frac{1}{t} \cdot t_5\right)\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{{\left(t_3 \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot t_6\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_4}\right)\\ \end{array} \]

Alternative 4
Error	11.93%
Cost	46480

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ t_3 := \frac{\ell}{k \cdot t}\\ t_4 := {\sin k}^{2} \cdot t\\ \mathbf{if}\;k \leq -1.1 \cdot 10^{+52}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_1}}\\ \mathbf{elif}\;k \leq -6 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{-154}:\\ \;\;\;\;t_3 \cdot \left(\frac{1}{t} \cdot t_3\right)\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_4}\right)\\ \end{array} \]

Alternative 5
Error	16.27%
Cost	27212

\[\begin{array}{l} t_1 := \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\left(2 + \frac{k \cdot k}{t \cdot t}\right) \cdot \frac{\sin k}{\ell}}\\ t_2 := \frac{\ell}{k \cdot t}\\ t_3 := t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ t_4 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+108}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-33}:\\ \;\;\;\;\frac{\frac{2 \cdot \ell}{\frac{\tan k}{\frac{\frac{\ell}{\sin k}}{{t}^{3}}}}}{t_4}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-48}:\\ \;\;\;\;\frac{\cos k}{\frac{\frac{k \cdot t}{\ell}}{\ell}} \cdot \frac{2}{k \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+144}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(t_4 \cdot \tan k\right)\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	16.48%
Cost	21396

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{\ell}{k \cdot t}\\ t_3 := {\sin k}^{2}\\ t_4 := t_3 \cdot t\\ t_5 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -1.55 \cdot 10^{+25}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_1}}\\ \mathbf{elif}\;k \leq -7 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot t_5\right)}}{t} \cdot \frac{\ell \cdot \ell}{t \cdot t}\\ \mathbf{elif}\;k \leq 10^{-64}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-23}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+48}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{t_5}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+160}:\\ \;\;\;\;\frac{\cos k}{\frac{\frac{k \cdot t}{\ell}}{\ell}} \cdot \frac{2}{k \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_4}\right)\\ \end{array} \]

Alternative 7
Error	16.07%
Cost	21268

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := {\sin k}^{2}\\ t_3 := t_2 \cdot t\\ t_4 := \frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t} \cdot \frac{\ell \cdot \ell}{t \cdot t}\\ t_5 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -1.06 \cdot 10^{+27}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-30}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 10^{-64}:\\ \;\;\;\;t_5 \cdot \left(\frac{1}{t} \cdot t_5\right)\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 2.25 \cdot 10^{+45}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+159}:\\ \;\;\;\;\frac{\cos k}{\frac{\frac{k \cdot t}{\ell}}{\ell}} \cdot \frac{2}{k \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \end{array} \]

Alternative 8
Error	17.72%
Cost	21016

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \ell \cdot \frac{\ell}{t}\\ t_3 := \frac{\ell}{k \cdot t}\\ t_4 := {\sin k}^{2}\\ t_5 := t_4 \cdot t\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_5}{t_1}}\\ \mathbf{elif}\;k \leq 10^{-64}:\\ \;\;\;\;t_3 \cdot \left(\frac{1}{t} \cdot t_3\right)\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-26}:\\ \;\;\;\;2 \cdot \left(\frac{t_2 \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 3800000000:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \left(0.5 \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+14}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{t_4}\right)\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{k \cdot \frac{k}{\cos k}} \cdot \frac{t_2}{t_4}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_5}\right)\\ \end{array} \]

Alternative 9
Error	16.27%
Cost	20884

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{\ell}{k \cdot t}\\ t_3 := {\sin k}^{2}\\ t_4 := t_3 \cdot t\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_1}}\\ \mathbf{elif}\;k \leq 10^{-64}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 255:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \left(0.5 \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{elif}\;k \leq 5.3 \cdot 10^{+160}:\\ \;\;\;\;\frac{\cos k}{\frac{\frac{k \cdot t}{\ell}}{\ell}} \cdot \frac{2}{k \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_4}\right)\\ \end{array} \]

Alternative 10
Error	17.34%
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-64}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 92000000:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \left(0.5 \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	17.28%
Cost	20752

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{\ell}{k \cdot t}\\ t_3 := {\sin k}^{2} \cdot t\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \mathbf{elif}\;k \leq 10^{-64}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{elif}\;k \leq 8 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 2700000:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \left(0.5 \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \end{array} \]

Alternative 12
Error	26.49%
Cost	20624

\[\begin{array}{l} t_1 := \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\frac{\ell}{\frac{t}{\ell}}}}\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -5.8 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-64}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{elif}\;k \leq 1.12 \cdot 10^{-20}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 1760000:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \left(0.5 \cdot \left(\ell \cdot \frac{\ell}{\sin k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	26.33%
Cost	14672

\[\begin{array}{l} t_1 := \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\frac{\ell}{\frac{t}{\ell}}}}\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -2.55 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-64}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{\frac{{k}^{4}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 550:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{\frac{2}{k}}{k}}{{t}^{3}}}{2 + \frac{\frac{k}{t}}{\frac{t}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	27.79%
Cost	1993

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-33} \lor \neg \left(t \leq 2.45 \cdot 10^{-14}\right):\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\left(\ell \cdot \frac{\ell}{t}\right) \cdot -0.16666666666666666}{k \cdot k} + \frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \end{array} \]

Alternative 15
Error	29.05%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-40} \lor \neg \left(t \leq 2.45 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(k \cdot t\right) \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k}}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]

Alternative 16
Error	28%
Cost	1225

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{-40} \lor \neg \left(t \leq 2.45 \cdot 10^{-14}\right):\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k \cdot k}}{\left(k \cdot k\right) \cdot \frac{t}{\ell}}\\ \end{array} \]

Alternative 17
Error	36%
Cost	1097

\[\begin{array}{l} t_1 := k \cdot \frac{t}{\ell}\\ \mathbf{if}\;k \leq -4.4 \cdot 10^{-168} \lor \neg \left(k \leq 5 \cdot 10^{-177}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]

Alternative 18
Error	38.27%
Cost	1096

\[\begin{array}{l} t_1 := k \cdot \frac{t}{\ell}\\ \mathbf{if}\;\ell \leq 5 \cdot 10^{-232}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t_1\right)}\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-37}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(k \cdot t\right) \cdot t_1}\\ \end{array} \]

Alternative 19
Error	36.39%
Cost	1096

\[\begin{array}{l} t_1 := k \cdot \frac{t}{\ell}\\ \mathbf{if}\;t \leq 2.7 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t_1\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+153}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\left(t \cdot t\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(k \cdot t\right) \cdot t_1}\\ \end{array} \]

Alternative 20
Error	45.34%
Cost	832

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

Alternative 21
Error	41.65%
Cost	832

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

Alternative 22
Error	41.56%
Cost	832

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

Reproduce

herbie shell --seed 2023088 
(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))))