Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 7.1

Time: 1.2min

Precision: binary64

Cost: 46800

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ t_2 := \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t_1}{\ell} \cdot \frac{{t_1}^{2}}{\frac{\ell}{\sin k}}\right)}\\ t_3 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t_3}{\sqrt[3]{\ell}}\right)}^{2}}}{t_3}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]

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 (* t (cbrt (tan k))))
        (t_2
         (/
          2.0
          (*
           (+ 2.0 (pow (/ k t) 2.0))
           (* (/ t_1 l) (/ (pow t_1 2.0) (/ l (sin k)))))))
        (t_3 (* t (pow (cbrt k) 2.0))))
   (if (<= t -1e+246)
     (* (cbrt l) (/ (/ l (pow (/ t_3 (cbrt l)) 2.0)) t_3))
     (if (<= t -1.25e-29)
       t_2
       (if (<= t 3.1e-73)
         (* 2.0 (/ (* (/ l k) (/ (cos k) t)) (* (/ k l) (pow (sin k) 2.0))))
         (if (<= t 2.6e+154)
           t_2
           (* l (/ 2.0 (* (/ t l) (* 2.0 (* (* t k) (* t k))))))))))))

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 = t * cbrt(tan(k));
	double t_2 = 2.0 / ((2.0 + pow((k / t), 2.0)) * ((t_1 / l) * (pow(t_1, 2.0) / (l / sin(k)))));
	double t_3 = t * pow(cbrt(k), 2.0);
	double tmp;
	if (t <= -1e+246) {
		tmp = cbrt(l) * ((l / pow((t_3 / cbrt(l)), 2.0)) / t_3);
	} else if (t <= -1.25e-29) {
		tmp = t_2;
	} else if (t <= 3.1e-73) {
		tmp = 2.0 * (((l / k) * (cos(k) / t)) / ((k / l) * pow(sin(k), 2.0)));
	} else if (t <= 2.6e+154) {
		tmp = t_2;
	} else {
		tmp = l * (2.0 / ((t / l) * (2.0 * ((t * k) * (t * k)))));
	}
	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 = t * Math.cbrt(Math.tan(k));
	double t_2 = 2.0 / ((2.0 + Math.pow((k / t), 2.0)) * ((t_1 / l) * (Math.pow(t_1, 2.0) / (l / Math.sin(k)))));
	double t_3 = t * Math.pow(Math.cbrt(k), 2.0);
	double tmp;
	if (t <= -1e+246) {
		tmp = Math.cbrt(l) * ((l / Math.pow((t_3 / Math.cbrt(l)), 2.0)) / t_3);
	} else if (t <= -1.25e-29) {
		tmp = t_2;
	} else if (t <= 3.1e-73) {
		tmp = 2.0 * (((l / k) * (Math.cos(k) / t)) / ((k / l) * Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 2.6e+154) {
		tmp = t_2;
	} else {
		tmp = l * (2.0 / ((t / l) * (2.0 * ((t * k) * (t * k)))));
	}
	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(t * cbrt(tan(k)))
	t_2 = Float64(2.0 / Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(Float64(t_1 / l) * Float64((t_1 ^ 2.0) / Float64(l / sin(k))))))
	t_3 = Float64(t * (cbrt(k) ^ 2.0))
	tmp = 0.0
	if (t <= -1e+246)
		tmp = Float64(cbrt(l) * Float64(Float64(l / (Float64(t_3 / cbrt(l)) ^ 2.0)) / t_3));
	elseif (t <= -1.25e-29)
		tmp = t_2;
	elseif (t <= 3.1e-73)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(cos(k) / t)) / Float64(Float64(k / l) * (sin(k) ^ 2.0))));
	elseif (t <= 2.6e+154)
		tmp = t_2;
	else
		tmp = Float64(l * Float64(2.0 / Float64(Float64(t / l) * Float64(2.0 * Float64(Float64(t * k) * Float64(t * k))))));
	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[(t * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / l), $MachinePrecision] * N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e+246], N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[(l / N[Power[N[(t$95$3 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.25e-29], t$95$2, If[LessEqual[t, 3.1e-73], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.6e+154], t$95$2, N[(l * N[(2.0 / N[(N[(t / l), $MachinePrecision] * N[(2.0 * N[(N[(t * k), $MachinePrecision] * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.00000000000000007e246

   1. Initial program 17.2

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

   2. Simplified 25.3

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

      [Start] 17.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 17.2	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 17.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 17.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 17.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 25.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 25.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 25.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 25.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in k around 0 25.4

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

   4. Simplified 11.3

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

      [Start] 25.4	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 25.4	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 19.7	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 19.7	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      associate-*l* [=>] 11.3	\[ \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   5. Applied egg-rr 3.9

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

   if -1.00000000000000007e246 < t < -1.24999999999999996e-29 or 3.09999999999999969e-73 < t < 2.59999999999999989e154

   1. Initial program 22.6

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

   2. Simplified 19.4

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

      [Start] 22.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.6	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 21.7	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 21.5	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 19.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 19.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 19.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 19.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 9.4

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

   if -1.24999999999999996e-29 < t < 3.09999999999999969e-73

   1. Initial program 55.5

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

   2. Simplified 55.6

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

      [Start] 55.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 55.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 55.5	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 55.8	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 55.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 55.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 55.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 55.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 27.3

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

   4. Simplified 17.0

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

      [Start] 27.3	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 27.3	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 29.0	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 29.0	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 29.0	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 17.0	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 17.0	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 3.4

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

   if 2.59999999999999989e154 < t

   1. Initial program 22.5

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

   2. Simplified 28.8

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

      [Start] 22.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.5	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 22.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 22.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 22.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 28.8

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

   4. Applied egg-rr 21.3

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

   5. Taylor expanded in k around 0 24.9

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

   6. Simplified 8.8

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

      [Start] 24.9	\[ \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left({k}^{2} \cdot {t}^{2}\right)\right)} \cdot \ell \]
      *-commutative [=>] 24.9	\[ \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \color{blue}{\left({t}^{2} \cdot {k}^{2}\right)}\right)} \cdot \ell \]
      unpow2 [=>] 24.9	\[ \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\color{blue}{\left(t \cdot t\right)} \cdot {k}^{2}\right)\right)} \cdot \ell \]
      unpow2 [=>] 24.9	\[ \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot t\right) \cdot \color{blue}{\left(k \cdot k\right)}\right)\right)} \cdot \ell \]
      unswap-sqr [=>] 8.8	\[ \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \color{blue}{\left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)}\right)} \cdot \ell \]

3. Recombined 4 regimes into one program.

4. Final simplification 7.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+246}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{2}}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot \sqrt[3]{\tan k}}{\ell} \cdot \frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot \sqrt[3]{\tan k}}{\ell} \cdot \frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.5
Cost	85577

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ t_2 := \sqrt[3]{t_1}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-28} \lor \neg \left(t \leq 3.1 \cdot 10^{-73}\right):\\ \;\;\;\;\frac{2}{\left(\frac{\frac{{t_2}^{2}}{\frac{\ell}{t_2}}}{\frac{\frac{1}{\sin k}}{t_1}} \cdot \frac{t_1}{\ell}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 2
Error	4.6
Cost	53257

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := t \cdot t_1\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{-30} \lor \neg \left(t \leq 5.8 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t_2}{\ell} \cdot \left(\left(t_1 \cdot \frac{t}{\ell}\right) \cdot \frac{t_2}{\frac{1}{\sin k}}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 3
Error	4.7
Cost	53256

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := t \cdot t_1\\ t_3 := t_1 \cdot \frac{t}{\ell}\\ t_4 := \frac{1}{\sin k}\\ t_5 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_6 := \frac{t_2}{\ell}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;\frac{2}{t_5 \cdot \left(t_6 \cdot \frac{t_3}{\frac{t_4}{t_2}}\right)}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_5 \cdot \left(t_6 \cdot \left(t_3 \cdot \frac{t_2}{t_4}\right)\right)}\\ \end{array} \]

Alternative 4
Error	8.1
Cost	46348

\[\begin{array}{l} t_1 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_2 := \frac{t_1}{\sqrt[3]{\ell}}\\ t_3 := {t_2}^{2}\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{-28}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{t_3}}{t_1}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t_3} \cdot \frac{\ell}{t_2}\\ \end{array} \]

Alternative 5
Error	8.1
Cost	46220

\[\begin{array}{l} t_1 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_2 := \sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t_1}{\sqrt[3]{\ell}}\right)}^{2}}}{t_1}\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-28}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	8.9
Cost	27212

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t_1}{\frac{\sqrt[3]{\ell}}{t}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-73}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot t_1}{\sqrt[3]{\ell}}\right)}^{3}}\\ \end{array} \]

Alternative 7
Error	10.2
Cost	26440

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.2 \cdot 10^{-31}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot t_1\right)}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot t_1}\\ \end{array} \]

Alternative 8
Error	12.3
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \frac{\ell}{\frac{k}{\ell} \cdot \left(k \cdot \frac{t_1}{\frac{\cos k}{t}}\right)}\\ t_3 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -1 \cdot 10^{+225}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{elif}\;k \leq -5.2 \cdot 10^{-28}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.92 \cdot 10^{-230}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{t_3 \cdot \frac{k}{\frac{\ell}{{t_3}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	14.4
Cost	20620

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -1.6 \cdot 10^{-28}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.08 \cdot 10^{-231}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	11.6
Cost	20620

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ t_2 := 2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{if}\;k \leq -8.3 \cdot 10^{-29}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -8.4 \cdot 10^{-229}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Error	11.6
Cost	20620

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ t_2 := \frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)\\ \mathbf{if}\;k \leq -1.3 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{t_2}\\ \mathbf{elif}\;k \leq -2.15 \cdot 10^{-231}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot t_2}\\ \end{array} \]

Alternative 12
Error	11.5
Cost	20620

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot t_1\right)}\\ \mathbf{elif}\;k \leq -5.7 \cdot 10^{-229}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{-29}:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot t_1}\\ \end{array} \]

Alternative 13
Error	14.6
Cost	20492

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := t \cdot \sqrt[3]{k}\\ t_3 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{t_1}{2}\right)}\right)\\ \mathbf{elif}\;k \leq -8 \cdot 10^{-229}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \frac{\frac{2 \cdot \cos k}{t}}{1 - t_1}\right)\\ \end{array} \]

Alternative 14
Error	15.7
Cost	20172

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -6.2 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{t_1}{2}\right)}\right)\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-232}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-221}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\frac{2 \cdot \cos k}{t}}{1 - t_1}\right)\\ \end{array} \]

Alternative 15
Error	16.4
Cost	14672

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.3 \cdot 10^{-255}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-256}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot {\left(t \cdot \sqrt[3]{k}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	16.4
Cost	14672

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{-9}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{t_1}{2}\right)}\right)\\ \mathbf{elif}\;k \leq -2.15 \cdot 10^{-252}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-256}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot {\left(t \cdot \sqrt[3]{k}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\frac{2 \cdot \cos k}{t}}{1 - t_1}\right)\\ \end{array} \]

Alternative 17
Error	20.5
Cost	13908

\[\begin{array}{l} t_1 := t \cdot \left(k \cdot k\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{+130}:\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq -6.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-171}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell \cdot -0.3333333333333333}}\\ \mathbf{elif}\;t \leq -2.05 \cdot 10^{-175}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{t_1}\right) + \frac{-0.16666666666666666}{t}\right)\right)\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-104}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 18
Error	20.2
Cost	8656

\[\begin{array}{l} t_1 := \ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ t_2 := t \cdot \left(k \cdot k\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.4 \cdot 10^{-106}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-171}:\\ \;\;\;\;\frac{\ell}{\frac{t_2}{\ell \cdot -0.3333333333333333}}\\ \mathbf{elif}\;t \leq -2.05 \cdot 10^{-175}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\mathsf{fma}\left(-0.058333333333333334, \frac{k \cdot k}{t}, \frac{1}{t_2}\right) + \frac{-0.16666666666666666}{t}\right)\right)\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-104}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{+100}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	19.7
Cost	7436

\[\begin{array}{l} t_1 := \ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{if}\;t \leq -9.6 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-104}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+97}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	20.2
Cost	7436

\[\begin{array}{l} t_1 := \ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-104}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{elif}\;t \leq 10^{+99}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 21
Error	27.6
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-27} \lor \neg \left(t \leq 9.2 \cdot 10^{-104}\right):\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \end{array} \]

Alternative 22
Error	20.8
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{-27} \lor \neg \left(t \leq 9.2 \cdot 10^{-104}\right):\\ \;\;\;\;\ell \cdot \frac{2}{\frac{t}{\ell} \cdot \left(2 \cdot \left(\left(t \cdot k\right) \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \end{array} \]

Alternative 23
Error	34.4
Cost	1352

\[\begin{array}{l} t_1 := k \cdot \frac{t}{\ell}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-170}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k}}{t_1}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+189}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{k \cdot t_1}\\ \end{array} \]

Alternative 24
Error	34.4
Cost	1352

\[\begin{array}{l} t_1 := k \cdot \frac{t}{\ell}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-169}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k}}{t_1}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+189}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{k \cdot t_1}\\ \end{array} \]

Alternative 25
Error	32.3
Cost	1352

\[\begin{array}{l} t_1 := k \cdot \frac{t}{\ell}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-169}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k}}{t_1}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{k \cdot t_1}\\ \end{array} \]

Alternative 26
Error	34.8
Cost	704

\[-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k}\right) \]

Alternative 27
Error	33.9
Cost	704

\[-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot \frac{t}{\ell}\right)} \]

Reproduce

herbie shell --seed 2023066 
(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))))