Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 7.6

Time: 36.1s

Precision: binary64

Cost: 86096

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{k \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k}\\ t_3 := \sqrt[3]{\tan k \cdot \left(\sin k \cdot t_1\right)}\\ t_4 := \frac{2}{t_3 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{if}\;k \leq -1.6 \cdot 10^{+140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -6 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{{\left(\frac{\frac{t_3}{\sqrt[3]{\ell}}}{\frac{\sqrt[3]{\ell}}{t}}\right)}^{2}} \cdot t_4\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+80}:\\ \;\;\;\;t_4 \cdot \frac{1}{{\left(\frac{t \cdot \frac{\sqrt[3]{t_1 \cdot \left(\sin k \cdot \tan k\right)}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (pow (/ k t) 2.0)))
        (t_2
         (* (/ l (* k (* (pow (sin k) 2.0) t))) (/ (* l (* 2.0 (cos k))) k)))
        (t_3 (cbrt (* (tan k) (* (sin k) t_1))))
        (t_4 (/ 2.0 (* t_3 (/ t (pow (cbrt l) 2.0))))))
   (if (<= k -1.6e+140)
     t_2
     (if (<= k -6e-152)
       (* (/ 1.0 (pow (/ (/ t_3 (cbrt l)) (/ (cbrt l) t)) 2.0)) t_4)
       (if (<= k 1.45e-152)
         (/ (/ l (* k t)) (* t (* t (/ k l))))
         (if (<= k 1.15e+80)
           (*
            t_4
            (/
             1.0
             (pow
              (/
               (* t (/ (cbrt (* t_1 (* (sin k) (tan k)))) (cbrt l)))
               (cbrt l))
              2.0)))
           t_2))))))

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 + pow((k / t), 2.0);
	double t_2 = (l / (k * (pow(sin(k), 2.0) * t))) * ((l * (2.0 * cos(k))) / k);
	double t_3 = cbrt((tan(k) * (sin(k) * t_1)));
	double t_4 = 2.0 / (t_3 * (t / pow(cbrt(l), 2.0)));
	double tmp;
	if (k <= -1.6e+140) {
		tmp = t_2;
	} else if (k <= -6e-152) {
		tmp = (1.0 / pow(((t_3 / cbrt(l)) / (cbrt(l) / t)), 2.0)) * t_4;
	} else if (k <= 1.45e-152) {
		tmp = (l / (k * t)) / (t * (t * (k / l)));
	} else if (k <= 1.15e+80) {
		tmp = t_4 * (1.0 / pow(((t * (cbrt((t_1 * (sin(k) * tan(k)))) / cbrt(l))) / cbrt(l)), 2.0));
	} else {
		tmp = t_2;
	}
	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 + Math.pow((k / t), 2.0);
	double t_2 = (l / (k * (Math.pow(Math.sin(k), 2.0) * t))) * ((l * (2.0 * Math.cos(k))) / k);
	double t_3 = Math.cbrt((Math.tan(k) * (Math.sin(k) * t_1)));
	double t_4 = 2.0 / (t_3 * (t / Math.pow(Math.cbrt(l), 2.0)));
	double tmp;
	if (k <= -1.6e+140) {
		tmp = t_2;
	} else if (k <= -6e-152) {
		tmp = (1.0 / Math.pow(((t_3 / Math.cbrt(l)) / (Math.cbrt(l) / t)), 2.0)) * t_4;
	} else if (k <= 1.45e-152) {
		tmp = (l / (k * t)) / (t * (t * (k / l)));
	} else if (k <= 1.15e+80) {
		tmp = t_4 * (1.0 / Math.pow(((t * (Math.cbrt((t_1 * (Math.sin(k) * Math.tan(k)))) / Math.cbrt(l))) / Math.cbrt(l)), 2.0));
	} else {
		tmp = t_2;
	}
	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(k / t) ^ 2.0))
	t_2 = Float64(Float64(l / Float64(k * Float64((sin(k) ^ 2.0) * t))) * Float64(Float64(l * Float64(2.0 * cos(k))) / k))
	t_3 = cbrt(Float64(tan(k) * Float64(sin(k) * t_1)))
	t_4 = Float64(2.0 / Float64(t_3 * Float64(t / (cbrt(l) ^ 2.0))))
	tmp = 0.0
	if (k <= -1.6e+140)
		tmp = t_2;
	elseif (k <= -6e-152)
		tmp = Float64(Float64(1.0 / (Float64(Float64(t_3 / cbrt(l)) / Float64(cbrt(l) / t)) ^ 2.0)) * t_4);
	elseif (k <= 1.45e-152)
		tmp = Float64(Float64(l / Float64(k * t)) / Float64(t * Float64(t * Float64(k / l))));
	elseif (k <= 1.15e+80)
		tmp = Float64(t_4 * Float64(1.0 / (Float64(Float64(t * Float64(cbrt(Float64(t_1 * Float64(sin(k) * tan(k)))) / cbrt(l))) / cbrt(l)) ^ 2.0)));
	else
		tmp = t_2;
	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[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l / N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(2.0 * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(2.0 / N[(t$95$3 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.6e+140], t$95$2, If[LessEqual[k, -6e-152], N[(N[(1.0 / N[Power[N[(N[(t$95$3 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[k, 1.45e-152], N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] / N[(t * N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.15e+80], N[(t$95$4 * N[(1.0 / N[Power[N[(N[(t * N[(N[Power[N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -1.60000000000000005e140 or 1.15000000000000002e80 < k

   1. Initial program 33.2

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

   2. Simplified 33.2

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

      [Start] 33.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 33.2	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 33.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 33.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 33.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 33.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 33.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 33.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 33.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in t around 0 22.1

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

   4. Simplified 17.6

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

      [Start] 22.1	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*r/ [=>] 22.1	\[ \color{blue}{\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
      *-commutative [=>] 22.1	\[ \frac{2 \cdot \color{blue}{\left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*r* [=>] 22.1	\[ \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 22.1	\[ \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*r* [=>] 22.1	\[ \frac{\color{blue}{\left(\left(2 \cdot \ell\right) \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 22.1	\[ \frac{\color{blue}{\left(\ell \cdot \left(2 \cdot \ell\right)\right)} \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 22.1	\[ \frac{\left(\ell \cdot \left(2 \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*l* [=>] 17.6	\[ \frac{\left(\ell \cdot \left(2 \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{k \cdot \left(k \cdot \left({\sin k}^{2} \cdot t\right)\right)}} \]

   5. Applied egg-rr 4.5

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

   if -1.60000000000000005e140 < k < -6e-152

   1. Initial program 29.7

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

   2. Simplified 29.6

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

      [Start] 29.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 29.7	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 29.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 29.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 29.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 29.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 29.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 29.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 29.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 10.1

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

   4. Applied egg-rr 10.1

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

   if -6e-152 < k < 1.4500000000000001e-152

   1. Initial program 37.1

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

   2. Simplified 37.1

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

      [Start] 37.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 37.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 37.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in k around 0 61.7

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

   4. Simplified 61.6

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

      [Start] 61.7	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 61.7	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      *-commutative [=>] 61.7	\[ \frac{\ell \cdot \ell}{\color{blue}{{t}^{3} \cdot {k}^{2}}} \]
      times-frac [=>] 61.6	\[ \color{blue}{\frac{\ell}{{t}^{3}} \cdot \frac{\ell}{{k}^{2}}} \]
      unpow2 [=>] 61.6	\[ \frac{\ell}{{t}^{3}} \cdot \frac{\ell}{\color{blue}{k \cdot k}} \]

   5. Applied egg-rr 40.3

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

   6. Applied egg-rr 23.4

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

   7. Applied egg-rr 12.4

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

   if 1.4500000000000001e-152 < k < 1.15000000000000002e80

   1. Initial program 29.6

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

   2. Simplified 29.5

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

      [Start] 29.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 29.6	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 29.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 29.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 29.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 29.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 29.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 29.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 29.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 8.3

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

   4. Applied egg-rr 8.3

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

   5. Applied egg-rr 8.4

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

   6. Simplified 8.3

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

      [Start] 8.4

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

      associate-*r/ [=>] 8.4

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

3. Recombined 4 regimes into one program.

4. Final simplification 7.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.6 \cdot 10^{+140}:\\ \;\;\;\;\frac{\ell}{k \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k}\\ \mathbf{elif}\;k \leq -6 \cdot 10^{-152}:\\ \;\;\;\;\frac{1}{{\left(\frac{\frac{\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\sqrt[3]{\ell}}}{\frac{\sqrt[3]{\ell}}{t}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+80}:\\ \;\;\;\;\frac{2}{\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{1}{{\left(\frac{t \cdot \frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.6
Cost	86096

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{k \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k}\\ t_3 := \frac{2}{\sqrt[3]{\tan k \cdot \left(\sin k \cdot t_1\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \frac{1}{{\left(\frac{t \cdot \frac{\sqrt[3]{t_1 \cdot \left(\sin k \cdot \tan k\right)}}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{2}}\\ \mathbf{if}\;k \leq -5.2 \cdot 10^{+139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -7 \cdot 10^{-152}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+76}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	7.6
Cost	85904

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := \sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ t_3 := \frac{\ell}{k \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k}\\ t_4 := \frac{t_1 \cdot \frac{2}{t \cdot t_2}}{{\left(\frac{t}{t_1} \cdot t_2\right)}^{2}}\\ \mathbf{if}\;k \leq -4 \cdot 10^{+140}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -6 \cdot 10^{-152}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+79}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	7.7
Cost	46480

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k}\\ t_2 := \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{if}\;k \leq -7.5 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -6.4 \cdot 10^{-152}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+76}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	9.8
Cost	40080

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k}\\ t_2 := \frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{if}\;k \leq -5.6 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -4 \cdot 10^{-72}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	12.1
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\\ \mathbf{if}\;k \leq -14000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.05 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{t}{\ell}}{\frac{\frac{1}{t}}{\frac{k}{\ell}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	9.3
Cost	20620

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell \cdot \left(2 \cdot \cos k\right)}{k \cdot \left({\sin k}^{2} \cdot t\right)}\\ \mathbf{if}\;k \leq -1.7:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{t}{\ell}}{\frac{\frac{1}{t}}{\frac{k}{\ell}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-10}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	9.3
Cost	20620

\[\begin{array}{l} t_1 := \ell \cdot \left(2 \cdot \cos k\right)\\ t_2 := k \cdot \left({\sin k}^{2} \cdot t\right)\\ \mathbf{if}\;k \leq -320:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{t_1}{t_2}\\ \mathbf{elif}\;k \leq -9.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{t}{\ell}}{\frac{\frac{1}{t}}{\frac{k}{\ell}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \frac{t_1}{k}\\ \end{array} \]

Alternative 8
Error	16.5
Cost	14728

\[\begin{array}{l} t_1 := \frac{\cos k \cdot \left(\ell \cdot \left(\ell \cdot 2\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)\right)\right)}\\ \mathbf{if}\;k \leq -23000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \frac{t}{\ell}}{\frac{\frac{1}{t}}{\frac{k}{\ell}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{+40}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	17.1
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -130 \lor \neg \left(k \leq 1.45 \cdot 10^{+42}\right):\\ \;\;\;\;\frac{\cos k \cdot \left(\ell \cdot \left(\ell \cdot 2\right)\right)}{k \cdot \left(k \cdot \left(t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 10
Error	20.0
Cost	8336

\[\begin{array}{l} t_1 := \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot t}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ t_2 := \frac{\ell \cdot \ell}{t}\\ t_3 := \frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{if}\;t \leq -1.55 \cdot 10^{-7}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-149}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{{k}^{4}} + \frac{t_2}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Error	21.4
Cost	8073

\[\begin{array}{l} t_1 := \frac{\ell \cdot \ell}{t}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-97} \lor \neg \left(t \leq 9 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{t_1}{{k}^{4}} + \frac{t_1}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \]

Alternative 12
Error	21.5
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-97} \lor \neg \left(t \leq 2.9 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}\\ \end{array} \]

Alternative 13
Error	28.3
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+111} \lor \neg \left(t \leq 1.35 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot \left(t \cdot t\right)}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 14
Error	23.2
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-167} \lor \neg \left(t \leq 2.6 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 15
Error	22.9
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-167} \lor \neg \left(t \leq 4.2 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}}{t \cdot t}\\ \end{array} \]

Alternative 16
Error	29.4
Cost	832

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

Alternative 17
Error	29.0
Cost	832

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

Alternative 18
Error	29.0
Cost	832

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

Alternative 19
Error	28.0
Cost	832

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

Reproduce

herbie shell --seed 2023045 
(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))))