Toniolo and Linder, Equation (10+)

Average Error: 33.1 → 6.1

Time: 1.2min

Precision: binary64

Cost: 92304

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ t_3 := \frac{k}{\ell} \cdot t\\ t_4 := \sin k \cdot \tan k\\ t_5 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_6 := \sqrt[3]{t_5 \cdot t_4}\\ t_7 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;k \leq -6 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_2 \cdot \frac{k}{\ell}\right) \cdot t_3}\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{t_1}{t_6}}{{\left(t \cdot \frac{t_6}{t_1}\right)}^{2}}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_7}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_7}\right)\\ \mathbf{elif}\;k \leq 1.18 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{t_5}} \cdot \frac{\frac{t_1}{t}}{\sqrt[3]{t_4}}}{{\left(t_6 \cdot \frac{t}{t_1}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_2 \cdot t_3\right)}\\ \end{array} \]

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 (pow (cbrt l) 2.0))
        (t_2 (pow (sin k) 2.0))
        (t_3 (* (/ k l) t))
        (t_4 (* (sin k) (tan k)))
        (t_5 (+ 2.0 (pow (/ k t) 2.0)))
        (t_6 (cbrt (* t_5 t_4)))
        (t_7 (* t (pow (cbrt k) 2.0))))
   (if (<= k -6e+134)
     (* 2.0 (/ (cos k) (* (* t_2 (/ k l)) t_3)))
     (if (<= k -3.2e-155)
       (/ (* (/ 2.0 t) (/ t_1 t_6)) (pow (* t (/ t_6 t_1)) 2.0))
       (if (<= k 6e-120)
         (* (/ 1.0 (pow (/ t_7 (cbrt l)) 2.0)) (* (cbrt l) (/ l t_7)))
         (if (<= k 1.18e+51)
           (/
            (* (/ 2.0 (cbrt t_5)) (/ (/ t_1 t) (cbrt t_4)))
            (pow (* t_6 (/ t t_1)) 2.0))
           (* 2.0 (/ (cos k) (* (/ k l) (* t_2 t_3))))))))))

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 = pow(cbrt(l), 2.0);
	double t_2 = pow(sin(k), 2.0);
	double t_3 = (k / l) * t;
	double t_4 = sin(k) * tan(k);
	double t_5 = 2.0 + pow((k / t), 2.0);
	double t_6 = cbrt((t_5 * t_4));
	double t_7 = t * pow(cbrt(k), 2.0);
	double tmp;
	if (k <= -6e+134) {
		tmp = 2.0 * (cos(k) / ((t_2 * (k / l)) * t_3));
	} else if (k <= -3.2e-155) {
		tmp = ((2.0 / t) * (t_1 / t_6)) / pow((t * (t_6 / t_1)), 2.0);
	} else if (k <= 6e-120) {
		tmp = (1.0 / pow((t_7 / cbrt(l)), 2.0)) * (cbrt(l) * (l / t_7));
	} else if (k <= 1.18e+51) {
		tmp = ((2.0 / cbrt(t_5)) * ((t_1 / t) / cbrt(t_4))) / pow((t_6 * (t / t_1)), 2.0);
	} else {
		tmp = 2.0 * (cos(k) / ((k / l) * (t_2 * t_3)));
	}
	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.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = (k / l) * t;
	double t_4 = Math.sin(k) * Math.tan(k);
	double t_5 = 2.0 + Math.pow((k / t), 2.0);
	double t_6 = Math.cbrt((t_5 * t_4));
	double t_7 = t * Math.pow(Math.cbrt(k), 2.0);
	double tmp;
	if (k <= -6e+134) {
		tmp = 2.0 * (Math.cos(k) / ((t_2 * (k / l)) * t_3));
	} else if (k <= -3.2e-155) {
		tmp = ((2.0 / t) * (t_1 / t_6)) / Math.pow((t * (t_6 / t_1)), 2.0);
	} else if (k <= 6e-120) {
		tmp = (1.0 / Math.pow((t_7 / Math.cbrt(l)), 2.0)) * (Math.cbrt(l) * (l / t_7));
	} else if (k <= 1.18e+51) {
		tmp = ((2.0 / Math.cbrt(t_5)) * ((t_1 / t) / Math.cbrt(t_4))) / Math.pow((t_6 * (t / t_1)), 2.0);
	} else {
		tmp = 2.0 * (Math.cos(k) / ((k / l) * (t_2 * t_3)));
	}
	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 = cbrt(l) ^ 2.0
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(Float64(k / l) * t)
	t_4 = Float64(sin(k) * tan(k))
	t_5 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_6 = cbrt(Float64(t_5 * t_4))
	t_7 = Float64(t * (cbrt(k) ^ 2.0))
	tmp = 0.0
	if (k <= -6e+134)
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(t_2 * Float64(k / l)) * t_3)));
	elseif (k <= -3.2e-155)
		tmp = Float64(Float64(Float64(2.0 / t) * Float64(t_1 / t_6)) / (Float64(t * Float64(t_6 / t_1)) ^ 2.0));
	elseif (k <= 6e-120)
		tmp = Float64(Float64(1.0 / (Float64(t_7 / cbrt(l)) ^ 2.0)) * Float64(cbrt(l) * Float64(l / t_7)));
	elseif (k <= 1.18e+51)
		tmp = Float64(Float64(Float64(2.0 / cbrt(t_5)) * Float64(Float64(t_1 / t) / cbrt(t_4))) / (Float64(t_6 * Float64(t / t_1)) ^ 2.0));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(k / l) * Float64(t_2 * t_3))));
	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[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(t$95$5 * t$95$4), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$7 = N[(t * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -6e+134], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(t$95$2 * N[(k / l), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -3.2e-155], N[(N[(N[(2.0 / t), $MachinePrecision] * N[(t$95$1 / t$95$6), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t * N[(t$95$6 / t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 6e-120], N[(N[(1.0 / N[Power[N[(t$95$7 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] * N[(l / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.18e+51], N[(N[(N[(2.0 / N[Power[t$95$5, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / t), $MachinePrecision] / N[Power[t$95$4, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$6 * N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(t$95$2 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if k < -5.99999999999999993e134

   1. Initial program 33.5

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

   2. Simplified 33.5

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

      [Start] 33.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 33.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 33.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 23.7

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

   4. Simplified 6.3

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

      [Start] 23.7	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 23.7	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      *-commutative [=>] 23.7	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]
      associate-/l* [=>] 23.7	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]
      unpow2 [=>] 23.7	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]
      unpow2 [=>] 23.7	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}} \]
      times-frac [=>] 6.3	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]

   5. Applied egg-rr 2.7

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

   if -5.99999999999999993e134 < k < -3.20000000000000013e-155

   1. Initial program 31.1

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

   2. Simplified 30.9

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

      [Start] 31.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 31.1	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 31.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 31.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 31.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 30.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 30.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 30.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 30.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 10.0

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

   4. Simplified 10.0

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

      [Start] 10.0	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 10.0	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]

   5. Applied egg-rr 10.0

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

   6. Applied egg-rr 10.0

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

   7. Simplified 10.0

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

      [Start] 10.0	\[ \frac{\frac{\frac{2}{t}}{1} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}{{\left(\frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{2}} \]
      associate-/r/ [=>] 10.0	\[ \frac{\frac{\frac{2}{t}}{1} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}{{\color{blue}{\left(\frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot t\right)}}^{2}} \]

   if -3.20000000000000013e-155 < k < 6.00000000000000022e-120

   1. Initial program 37.3

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

   2. Simplified 58.6

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

      [Start] 37.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 37.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 37.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 37.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 37.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 58.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 58.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 58.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 58.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in k around 0 58.5

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

   4. Simplified 35.9

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

      [Start] 58.5	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 58.5	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 58.2	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 58.2	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      associate-*l* [=>] 35.9	\[ \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   5. Applied egg-rr 3.7

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

   if 6.00000000000000022e-120 < k < 1.18e51

   1. Initial program 28.8

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

   2. Simplified 28.7

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

      [Start] 28.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 28.8	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 28.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 28.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 28.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 28.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 28.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 8.2

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

   4. Simplified 8.2

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

      [Start] 8.2	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 8.2	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]

   5. Applied egg-rr 8.2

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

   6. Simplified 8.2

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

      [Start] 8.2	\[ \frac{\frac{1}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\sin k \cdot \tan k}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*r/ [=>] 8.2	\[ \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \left(2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}{\sqrt[3]{\sin k \cdot \tan k}}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*l/ [=>] 8.2	\[ \frac{\frac{\color{blue}{\frac{1 \cdot \left(2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}}}{\sqrt[3]{\sin k \cdot \tan k}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-lft-identity [=>] 8.2	\[ \frac{\frac{\frac{\color{blue}{2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}}{\sqrt[3]{\sin k \cdot \tan k}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*l/ [<=] 8.2	\[ \frac{\frac{\color{blue}{\frac{2}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}}{\sqrt[3]{\sin k \cdot \tan k}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*r/ [<=] 8.2	\[ \frac{\color{blue}{\frac{2}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\sin k \cdot \tan k}}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   if 1.18e51 < k

   1. Initial program 34.2

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

   2. Simplified 34.2

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

      [Start] 34.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 34.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 34.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 20.9

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

   4. Simplified 9.0

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

      [Start] 20.9	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 21.0	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      *-commutative [=>] 21.0	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]
      associate-/l* [=>] 22.9	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]
      unpow2 [=>] 23.0	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]
      unpow2 [=>] 23.0	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}} \]
      times-frac [=>] 9.0	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]

   5. Applied egg-rr 5.3

      \[\leadsto 2 \cdot \frac{\cos k}{\color{blue}{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 6.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}{{\left(t \cdot \frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)\\ \mathbf{elif}\;k \leq 1.18 \cdot 10^{+51}:\\ \;\;\;\;\frac{\frac{2}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\sin k \cdot \tan k}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot t\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.7
Cost	85640

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := {\sin k}^{2}\\ t_4 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\\ t_5 := \sqrt[3]{t_4}\\ t_6 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;k \leq -6.8 \cdot 10^{+134}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_3 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -7.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{\frac{2}{t_2}}{t_5}}{{\left(t_5 \cdot t_2\right)}^{2}}\\ \mathbf{elif}\;k \leq 3.45 \cdot 10^{-112}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_6}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_6}\right)\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{t_4 \cdot {t_2}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_3 \cdot t_1\right)}\\ \end{array} \]

Alternative 2
Error	6.7
Cost	85640

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \frac{t}{t_2}\\ t_4 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_5 := {\sin k}^{2}\\ t_6 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\\ t_7 := \sqrt[3]{t_6}\\ \mathbf{if}\;k \leq -9.2 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_5 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -3.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{t_2}{t_7}}{{\left(t_7 \cdot t_3\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.48 \cdot 10^{-108}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_4}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_4}\right)\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{2}{t_6 \cdot {t_3}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_5 \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	6.7
Cost	85640

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_4 := {\sin k}^{2}\\ t_5 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\\ t_6 := \sqrt[3]{t_5}\\ \mathbf{if}\;k \leq -1.65 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_4 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -3.15 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{2}{t} \cdot \frac{t_2}{t_6}}{{\left(t \cdot \frac{t_6}{t_2}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_3}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_3}\right)\\ \mathbf{elif}\;k \leq 1.02 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{t_5 \cdot {\left(\frac{t}{t_2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_4 \cdot t_1\right)}\\ \end{array} \]

Alternative 4
Error	7.6
Cost	46348

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\\ t_3 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_4 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -4.3 \cdot 10^{+74}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_4 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -1.35 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.25 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_3}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_3}\right)\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+54}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_4 \cdot t_1\right)}\\ \end{array} \]

Alternative 5
Error	6.8
Cost	46348

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{k}{\ell} \cdot t\\ t_3 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_4 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_5 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -9.2 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_5 \cdot \frac{k}{\ell}\right) \cdot t_2}\\ \mathbf{elif}\;k \leq -1.15 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(t_3 \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot t_1\right)}\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.3 \cdot 10^{-113}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_4}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_4}\right)\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{2}{\left(t_1 \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {t_3}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_5 \cdot t_2\right)}\\ \end{array} \]

Alternative 6
Error	6.8
Cost	46348

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\\ t_3 := {\sin k}^{2}\\ t_4 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;k \leq -1.65 \cdot 10^{+135}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_3 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -1.95 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{t_2} \cdot \left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)\right)}^{3}}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_4}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_4}\right)\\ \mathbf{elif}\;k \leq 1.58 \cdot 10^{+53}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_3 \cdot t_1\right)}\\ \end{array} \]

Alternative 7
Error	7.8
Cost	46220

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\\ t_3 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_4 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -3.2 \cdot 10^{+75}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_4 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -1.35 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{-109}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t_3}{\sqrt[3]{\ell}}\right)}^{2}}}{t_3}\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+56}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_4 \cdot t_1\right)}\\ \end{array} \]

Alternative 8
Error	8.6
Cost	40080

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -2.85 \cdot 10^{+75}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_3 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -2.7 \cdot 10^{-156}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-115}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{+56}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_3 \cdot t_1\right)}\\ \end{array} \]

Alternative 9
Error	8.6
Cost	40080

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -9.4 \cdot 10^{+76}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_3 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -1.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{-112}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 9.8 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_3 \cdot t_1\right)}\\ \end{array} \]

Alternative 10
Error	10.1
Cost	27016

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1020:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_3 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -1.26 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\left(2 + t_2\right) \cdot \frac{k \cdot k}{{\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}}}\\ \mathbf{elif}\;k \leq 1.42 \cdot 10^{-114}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{+51}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_3 \cdot t_1\right)}\\ \end{array} \]

Alternative 11
Error	9.9
Cost	26572

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := \frac{2}{\left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -7.3 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_3 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -2.3 \cdot 10^{-77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 2.75 \cdot 10^{-106}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.25 \cdot 10^{+53}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_3 \cdot t_1\right)}\\ \end{array} \]

Alternative 12
Error	11.8
Cost	21268

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := t \cdot \sqrt[3]{k}\\ t_3 := \frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)\right)}\\ t_4 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -2.4 \cdot 10^{+39}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_4 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq -1.26 \cdot 10^{-154}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \frac{-\ell}{t}}{-t}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-108}:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{+52}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_4 \cdot t_1\right)}\\ \end{array} \]

Alternative 13
Error	11.8
Cost	21132

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot t\\ t_2 := t \cdot \sqrt[3]{k}\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -390:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left(t_3 \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{elif}\;k \leq 7.3 \cdot 10^{-106}:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+49}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \left(t_3 \cdot t_1\right)}\\ \end{array} \]

Alternative 14
Error	15.8
Cost	20488

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+59}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{\frac{\ell}{\frac{t}{\ell}}}}{k}}{t}\right)}^{2}\\ \end{array} \]

Alternative 15
Error	11.8
Cost	20488

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;t \leq -1.38 \cdot 10^{-6}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{\frac{\ell}{\frac{t}{\ell}}}}{k}}{t}\right)}^{2}\\ \end{array} \]

Alternative 16
Error	15.3
Cost	20361

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -1500 \lor \neg \left(k \leq 2.7 \cdot 10^{+47}\right):\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot \frac{1 - \cos \left(k + k\right)}{2}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{t_1}^{2} \cdot \frac{k}{\frac{\ell}{t_1}}}\\ \end{array} \]

Alternative 17
Error	15.3
Cost	20361

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -650 \lor \neg \left(k \leq 3.2 \cdot 10^{+47}\right):\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot \frac{1 - \cos \left(k + k\right)}{2}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \end{array} \]

Alternative 18
Error	16.3
Cost	14672

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{\frac{t \cdot \frac{1 - \cos \left(k + k\right)}{2}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{if}\;k \leq -1700:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-233}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \frac{-\ell}{t}}{-t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 19
Error	22.9
Cost	14092

\[\begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{+79}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{1}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)}\right)\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.000185:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{\frac{\ell}{\frac{t}{\ell}}}}{k}}{t}\right)}^{2}\\ \end{array} \]

Alternative 20
Error	22.2
Cost	13836

\[\begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+76}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{1}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)}\right)\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt{\frac{\ell}{\frac{t}{\ell}}}}{k}}{t}\right)}^{2}\\ \end{array} \]

Alternative 21
Error	21.9
Cost	13644

\[\begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+78}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{1}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)}\right)\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 22
Error	24.0
Cost	7436

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \frac{1}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)}\right)\\ \mathbf{if}\;t \leq -2 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-31}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 23
Error	22.7
Cost	7436

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \frac{1}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)}\right)\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 0.00013:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 24
Error	25.6
Cost	1489

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \frac{-\ell}{t}}{-t}\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-163} \lor \neg \left(t \leq 1.4 \cdot 10^{+137}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{k \cdot \frac{t}{\ell}}}{t \cdot t}\\ \end{array} \]

Alternative 25
Error	25.3
Cost	1488

\[\begin{array}{l} t_1 := \ell \cdot \left(\ell \cdot \frac{1}{t \cdot \left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)}\right)\\ \mathbf{if}\;t \leq -3 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}}{t \cdot t}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-163}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot \left(k \cdot t\right)} \cdot \frac{-\ell}{t}}{-t}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 26
Error	32.4
Cost	832

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

Alternative 27
Error	29.4
Cost	832

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

Alternative 28
Error	29.5
Cost	832

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

Reproduce

herbie shell --seed 2023039 
(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))))