Toniolo and Linder, Equation (10+)

Average Error: 47.8% → 87.1%

Time: 1.0min

Precision: binary64

Cost: 92304.00

Math

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

↓

\[\begin{array}{l} t_1 := \sin k \cdot \tan k\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_5 := \sqrt[3]{t_3 \cdot t_1}\\ t_6 := t_5 \cdot \frac{t}{t_2}\\ \mathbf{if}\;k \leq -4.6 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{t_4 \cdot \left(\cos k \cdot {\sin k}^{-2}\right)}{t}\\ \mathbf{elif}\;k \leq -1.45 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{\frac{2}{t_5}}{t} \cdot t_2}{{t_6}^{2}}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{2}{t_6}}{{\left(\frac{\sqrt[3]{t_3}}{\frac{t_2}{t \cdot \sqrt[3]{t_1}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_4 \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

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 (* (sin k) (tan k)))
        (t_2 (pow (cbrt l) 2.0))
        (t_3 (+ 2.0 (pow (/ k t) 2.0)))
        (t_4 (* (/ l k) (/ l k)))
        (t_5 (cbrt (* t_3 t_1)))
        (t_6 (* t_5 (/ t t_2))))
   (if (<= k -4.6e+127)
     (* 2.0 (/ (* t_4 (* (cos k) (pow (sin k) -2.0))) t))
     (if (<= k -1.45e-145)
       (/ (* (/ (/ 2.0 t_5) t) t_2) (pow t_6 2.0))
       (if (<= k 1.95e-92)
         (/ (* (/ l (* k t)) (/ l t)) (* k t))
         (if (<= k 1.02e+128)
           (/ (/ 2.0 t_6) (pow (/ (cbrt t_3) (/ t_2 (* t (cbrt t_1)))) 2.0))
           (* 2.0 (* t_4 (/ (cos k) (* t (pow (sin k) 2.0)))))))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

↓

double code(double t, double l, double k) {
	double t_1 = sin(k) * tan(k);
	double t_2 = pow(cbrt(l), 2.0);
	double t_3 = 2.0 + pow((k / t), 2.0);
	double t_4 = (l / k) * (l / k);
	double t_5 = cbrt((t_3 * t_1));
	double t_6 = t_5 * (t / t_2);
	double tmp;
	if (k <= -4.6e+127) {
		tmp = 2.0 * ((t_4 * (cos(k) * pow(sin(k), -2.0))) / t);
	} else if (k <= -1.45e-145) {
		tmp = (((2.0 / t_5) / t) * t_2) / pow(t_6, 2.0);
	} else if (k <= 1.95e-92) {
		tmp = ((l / (k * t)) * (l / t)) / (k * t);
	} else if (k <= 1.02e+128) {
		tmp = (2.0 / t_6) / pow((cbrt(t_3) / (t_2 / (t * cbrt(t_1)))), 2.0);
	} else {
		tmp = 2.0 * (t_4 * (cos(k) / (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.sin(k) * Math.tan(k);
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double t_3 = 2.0 + Math.pow((k / t), 2.0);
	double t_4 = (l / k) * (l / k);
	double t_5 = Math.cbrt((t_3 * t_1));
	double t_6 = t_5 * (t / t_2);
	double tmp;
	if (k <= -4.6e+127) {
		tmp = 2.0 * ((t_4 * (Math.cos(k) * Math.pow(Math.sin(k), -2.0))) / t);
	} else if (k <= -1.45e-145) {
		tmp = (((2.0 / t_5) / t) * t_2) / Math.pow(t_6, 2.0);
	} else if (k <= 1.95e-92) {
		tmp = ((l / (k * t)) * (l / t)) / (k * t);
	} else if (k <= 1.02e+128) {
		tmp = (2.0 / t_6) / Math.pow((Math.cbrt(t_3) / (t_2 / (t * Math.cbrt(t_1)))), 2.0);
	} else {
		tmp = 2.0 * (t_4 * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	}
	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(sin(k) * tan(k))
	t_2 = cbrt(l) ^ 2.0
	t_3 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_4 = Float64(Float64(l / k) * Float64(l / k))
	t_5 = cbrt(Float64(t_3 * t_1))
	t_6 = Float64(t_5 * Float64(t / t_2))
	tmp = 0.0
	if (k <= -4.6e+127)
		tmp = Float64(2.0 * Float64(Float64(t_4 * Float64(cos(k) * (sin(k) ^ -2.0))) / t));
	elseif (k <= -1.45e-145)
		tmp = Float64(Float64(Float64(Float64(2.0 / t_5) / t) * t_2) / (t_6 ^ 2.0));
	elseif (k <= 1.95e-92)
		tmp = Float64(Float64(Float64(l / Float64(k * t)) * Float64(l / t)) / Float64(k * t));
	elseif (k <= 1.02e+128)
		tmp = Float64(Float64(2.0 / t_6) / (Float64(cbrt(t_3) / Float64(t_2 / Float64(t * cbrt(t_1)))) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(t_4 * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	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[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(t$95$3 * t$95$1), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[(t / t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -4.6e+127], N[(2.0 * N[(N[(t$95$4 * N[(N[Cos[k], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1.45e-145], N[(N[(N[(N[(2.0 / t$95$5), $MachinePrecision] / t), $MachinePrecision] * t$95$2), $MachinePrecision] / N[Power[t$95$6, 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.95e-92], N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.02e+128], N[(N[(2.0 / t$95$6), $MachinePrecision] / N[Power[N[(N[Power[t$95$3, 1/3], $MachinePrecision] / N[(t$95$2 / N[(t * N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$4 * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -4.6000000000000003e127

   1. Initial program 46.0

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

   2. Simplified 45.1

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

      [Start] 46.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 46.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-*l/ [=>] 46.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]
      associate-*l/ [=>] 45.5	\[ \frac{2}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 45.6	\[ \color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 45.6	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-*r* [=>] 45.6	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      *-commutative [<=] 45.6	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\tan k \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r* [=>] 45.1	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   3. Applied egg-rr 60.6

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

   4. Taylor expanded in k around inf 63.7

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

   5. Simplified 62.7

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

      [Start] 63.7	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/r* [=>] 62.7	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]
      associate-/r* [=>] 62.7	\[ 2 \cdot \color{blue}{\frac{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2}}}{t}} \]
      *-commutative [=>] 62.7	\[ 2 \cdot \frac{\frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{{\sin k}^{2}}}{t} \]
      associate-/l* [=>] 62.7	\[ 2 \cdot \frac{\frac{\color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\cos k}}}}{{\sin k}^{2}}}{t} \]
      unpow2 [=>] 62.7	\[ 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{{k}^{2}}{\cos k}}}{{\sin k}^{2}}}{t} \]
      unpow2 [=>] 62.7	\[ 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\cos k}}}{{\sin k}^{2}}}{t} \]

   6. Applied egg-rr 89.4

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

   if -4.6000000000000003e127 < k < -1.44999999999999992e-145

   1. Initial program 52.2

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

   2. Simplified 52.3

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

      [Start] 52.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 52.2	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 52.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 52.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 52.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 52.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 52.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 52.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 52.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 85.9

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

   4. Simplified 85.9

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

      [Start] 85.9	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 85.9	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]
      associate-*r/ [=>] 85.9	\[ \frac{\color{blue}{\frac{1 \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      metadata-eval [=>] 85.9	\[ \frac{\frac{\color{blue}{2}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-commutative [=>] 85.9	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*l* [=>] 85.9	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   5. Applied egg-rr 85.9

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

   if -1.44999999999999992e-145 < k < 1.9499999999999998e-92

   1. Initial program 42.7

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

   2. Simplified 52.3

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

      [Start] 42.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 42.7	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 46.0	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 45.7	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 52.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 52.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 52.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 52.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around 0 14.8

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

   4. Simplified 16.1

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

      [Start] 14.8	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 14.8	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 16.1	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 16.1	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

   5. Applied egg-rr 20.4

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

   6. Applied egg-rr 21.1

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

   7. Applied egg-rr 88.9

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

   if 1.9499999999999998e-92 < k < 1.02000000000000008e128

   1. Initial program 53.3

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

   2. Simplified 53.4

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

      [Start] 53.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 53.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 53.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 53.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 53.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 53.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 53.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 53.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 53.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 81.1

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

   4. Simplified 81.1

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

      [Start] 81.1	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 81.1	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]
      associate-*r/ [=>] 81.1	\[ \frac{\color{blue}{\frac{1 \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      metadata-eval [=>] 81.1	\[ \frac{\frac{\color{blue}{2}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-commutative [=>] 81.1	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*l* [=>] 81.1	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   5. Applied egg-rr 81.1

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

   if 1.02000000000000008e128 < k

   1. Initial program 44.5

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

   2. Simplified 44.5

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

      [Start] 44.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 44.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 44.5	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 44.5	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 44.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 44.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 44.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 44.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 62.5

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

   4. Simplified 90.0

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

      [Start] 62.5	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 62.5	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 62.0	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 62.0	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 62.0	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 90.0	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 90.0	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 87.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.6 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {\sin k}^{-2}\right)}{t}\\ \mathbf{elif}\;k \leq -1.45 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}{t} \cdot {\left(\sqrt[3]{\ell}\right)}^{2}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\frac{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\sin k \cdot \tan k}}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	87.1%
Cost	85904.00

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_3 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ t_4 := \frac{\frac{\frac{2}{t_3}}{t} \cdot t_1}{{\left(t_3 \cdot \frac{t}{t_1}\right)}^{2}}\\ \mathbf{if}\;k \leq -8.5 \cdot 10^{+121}:\\ \;\;\;\;2 \cdot \frac{t_2 \cdot \left(\cos k \cdot {\sin k}^{-2}\right)}{t}\\ \mathbf{elif}\;k \leq -8 \cdot 10^{-145}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+128}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2
Error	87.1%
Cost	85904.00

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_3 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ t_4 := t_3 \cdot \frac{t}{t_1}\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{+127}:\\ \;\;\;\;2 \cdot \frac{t_2 \cdot \left(\cos k \cdot {\sin k}^{-2}\right)}{t}\\ \mathbf{elif}\;k \leq -2.4 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{\frac{2}{t_3}}{t} \cdot t_1}{{t_4}^{2}}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+128}:\\ \;\;\;\;\frac{\frac{2}{t_4}}{{\left(\frac{t}{\frac{t_1}{t_3}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3
Error	83.9%
Cost	46348.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{\frac{t_1}{\sin k} \cdot \frac{\cos k}{\sin k}}{t}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{-92}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4
Error	81.8%
Cost	20489.00

\[\begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{-6} \lor \neg \left(k \leq 4.5 \cdot 10^{-27}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 5
Error	81.8%
Cost	20488.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -1.56 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot \left(\cos k \cdot {\sin k}^{-2}\right)}{t}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6
Error	81.8%
Cost	20488.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{\frac{t_1}{\sin k} \cdot \frac{\cos k}{\sin k}}{t}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 7
Error	71.0%
Cost	14664.00

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{if}\;t \leq -4200:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-31}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{0.5 + -0.5 \cdot \cos \left(k + k\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	70.1%
Cost	14409.00

\[\begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{-6} \lor \neg \left(k \leq 3.4 \cdot 10^{-8}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{0.5 - 0.5 \cdot \cos \left(k + k\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 9
Error	70.4%
Cost	13833.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-65} \lor \neg \left(t \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\frac{\ell}{k}}{\sin k}\right)}^{2} \cdot \frac{1}{t}\right)\\ \end{array} \]

Alternative 10
Error	69.5%
Cost	7433.00

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-64} \lor \neg \left(t \leq 1.22 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{1}{t} \cdot {\left(\frac{\ell}{k \cdot k}\right)}^{2}\right)\\ \end{array} \]

Alternative 11
Error	67.1%
Cost	7305.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-69} \lor \neg \left(t \leq 2 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\\ \end{array} \]

Alternative 12
Error	68.5%
Cost	7305.00

\[\begin{array}{l} \mathbf{if}\;t \leq -1.66 \cdot 10^{-69} \lor \neg \left(t \leq 1.22 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \]

Alternative 13
Error	68.5%
Cost	7305.00

\[\begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-72} \lor \neg \left(t \leq 3.6 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{k}^{4}}\\ \end{array} \]

Alternative 14
Error	56.8%
Cost	1097.00

\[\begin{array}{l} \mathbf{if}\;k \leq -1.5 \cdot 10^{-145} \lor \neg \left(k \leq 1.55 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{\ell}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 15
Error	58.3%
Cost	1097.00

\[\begin{array}{l} \mathbf{if}\;k \leq -1.55 \cdot 10^{-145} \lor \neg \left(k \leq 1.15 \cdot 10^{-149}\right):\\ \;\;\;\;\frac{\ell}{\left(t \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}\\ \end{array} \]

Alternative 16
Error	59.1%
Cost	1097.00

\[\begin{array}{l} \mathbf{if}\;k \leq -1.9 \cdot 10^{-146} \lor \neg \left(k \leq 1.3 \cdot 10^{-149}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}\\ \end{array} \]

Alternative 17
Error	59.2%
Cost	1097.00

\[\begin{array}{l} \mathbf{if}\;k \leq -7.5 \cdot 10^{-146} \lor \neg \left(k \leq 3.1 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 18
Error	64.6%
Cost	1097.00

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-64} \lor \neg \left(t \leq 2.95 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]

Alternative 19
Error	46.0%
Cost	832.00

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

Alternative 20
Error	46.6%
Cost	832.00

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

Alternative 21
Error	53.1%
Cost	832.00

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

Alternative 22
Error	53.5%
Cost	832.00

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

Reproduce

herbie shell --seed 2023093 
(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))))