Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 9.6

Time: 52.1s

Precision: binary64

Cost: 40476

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ t_3 := \frac{2}{t \cdot \left(t_2 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ t_4 := \frac{2}{\left(t_1 \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ t_5 := \sin k \cdot \left(t_1 \cdot \tan k\right)\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{+58}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -1.25 \cdot 10^{+17}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -900:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_2 \cdot \frac{t}{\ell}\right)}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{elif}\;k \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t \cdot t\right) \cdot t_5\right)}{\ell}}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{t_5 \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+116}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

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 (pow (sin k) 2.0))
        (t_3 (/ 2.0 (* t (* t_2 (* (/ k l) (/ k (* l (cos k))))))))
        (t_4
         (/
          2.0
          (* (* t_1 (* (sin k) (tan k))) (pow (* t (pow (cbrt l) -2.0)) 3.0))))
        (t_5 (* (sin k) (* t_1 (tan k)))))
   (if (<= k -4.5e+58)
     t_3
     (if (<= k -1.25e+17)
       t_4
       (if (<= k -900.0)
         (/ 2.0 (/ (* k (* t_2 (/ t l))) (* l (/ (cos k) k))))
         (if (<= k -7.5e-67)
           (/ 2.0 (/ (* (/ t l) (* (* t t) t_5)) l))
           (if (<= k -4.5e-107)
             (/ 2.0 (* t_5 (pow (/ t (pow (cbrt l) 2.0)) 3.0)))
             (if (<= k 4.8e-110)
               (* (/ (/ l t) (* k t)) (/ l (* k t)))
               (if (<= k 1.55e+116) t_4 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 = 2.0 + pow((k / t), 2.0);
	double t_2 = pow(sin(k), 2.0);
	double t_3 = 2.0 / (t * (t_2 * ((k / l) * (k / (l * cos(k))))));
	double t_4 = 2.0 / ((t_1 * (sin(k) * tan(k))) * pow((t * pow(cbrt(l), -2.0)), 3.0));
	double t_5 = sin(k) * (t_1 * tan(k));
	double tmp;
	if (k <= -4.5e+58) {
		tmp = t_3;
	} else if (k <= -1.25e+17) {
		tmp = t_4;
	} else if (k <= -900.0) {
		tmp = 2.0 / ((k * (t_2 * (t / l))) / (l * (cos(k) / k)));
	} else if (k <= -7.5e-67) {
		tmp = 2.0 / (((t / l) * ((t * t) * t_5)) / l);
	} else if (k <= -4.5e-107) {
		tmp = 2.0 / (t_5 * pow((t / pow(cbrt(l), 2.0)), 3.0));
	} else if (k <= 4.8e-110) {
		tmp = ((l / t) / (k * t)) * (l / (k * t));
	} else if (k <= 1.55e+116) {
		tmp = t_4;
	} else {
		tmp = 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 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = 2.0 / (t * (t_2 * ((k / l) * (k / (l * Math.cos(k))))));
	double t_4 = 2.0 / ((t_1 * (Math.sin(k) * Math.tan(k))) * Math.pow((t * Math.pow(Math.cbrt(l), -2.0)), 3.0));
	double t_5 = Math.sin(k) * (t_1 * Math.tan(k));
	double tmp;
	if (k <= -4.5e+58) {
		tmp = t_3;
	} else if (k <= -1.25e+17) {
		tmp = t_4;
	} else if (k <= -900.0) {
		tmp = 2.0 / ((k * (t_2 * (t / l))) / (l * (Math.cos(k) / k)));
	} else if (k <= -7.5e-67) {
		tmp = 2.0 / (((t / l) * ((t * t) * t_5)) / l);
	} else if (k <= -4.5e-107) {
		tmp = 2.0 / (t_5 * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0));
	} else if (k <= 4.8e-110) {
		tmp = ((l / t) / (k * t)) * (l / (k * t));
	} else if (k <= 1.55e+116) {
		tmp = t_4;
	} else {
		tmp = 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 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(2.0 / Float64(t * Float64(t_2 * Float64(Float64(k / l) * Float64(k / Float64(l * cos(k)))))))
	t_4 = Float64(2.0 / Float64(Float64(t_1 * Float64(sin(k) * tan(k))) * (Float64(t * (cbrt(l) ^ -2.0)) ^ 3.0)))
	t_5 = Float64(sin(k) * Float64(t_1 * tan(k)))
	tmp = 0.0
	if (k <= -4.5e+58)
		tmp = t_3;
	elseif (k <= -1.25e+17)
		tmp = t_4;
	elseif (k <= -900.0)
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t_2 * Float64(t / l))) / Float64(l * Float64(cos(k) / k))));
	elseif (k <= -7.5e-67)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(t * t) * t_5)) / l));
	elseif (k <= -4.5e-107)
		tmp = Float64(2.0 / Float64(t_5 * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)));
	elseif (k <= 4.8e-110)
		tmp = Float64(Float64(Float64(l / t) / Float64(k * t)) * Float64(l / Float64(k * t)));
	elseif (k <= 1.55e+116)
		tmp = t_4;
	else
		tmp = 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[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 / N[(t * N[(t$95$2 * N[(N[(k / l), $MachinePrecision] * N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 / N[(N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t * N[Power[N[Power[l, 1/3], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[k], $MachinePrecision] * N[(t$95$1 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -4.5e+58], t$95$3, If[LessEqual[k, -1.25e+17], t$95$4, If[LessEqual[k, -900.0], N[(2.0 / N[(N[(k * N[(t$95$2 * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -7.5e-67], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -4.5e-107], N[(2.0 / N[(t$95$5 * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.8e-110], N[(N[(N[(l / t), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.55e+116], t$95$4, t$95$3]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if k < -4.4999999999999998e58 or 1.54999999999999998e116 < k

   1. Initial program 34.0

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

   2. Simplified 34.0

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

      [Start] 34.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 34.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 34.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 22.2

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

   4. Simplified 21.7

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

      [Start] 22.2	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 23.5	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 23.5	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 23.5	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 23.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]
      unpow2 [=>] 23.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{t}}} \]
      associate-/l* [=>] 21.7	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}} \]
      associate-/r/ [=>] 21.7	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\color{blue}{\frac{\ell}{t} \cdot \ell}}} \]

   5. Applied egg-rr 17.7

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

   6. Taylor expanded in k around inf 22.2

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

   7. Simplified 8.4

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

      [Start] 22.2	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      *-commutative [=>] 22.2	\[ \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 23.5	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}}} \]
      unpow2 [=>] 23.5	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}} \]
      associate-*r/ [<=] 23.5	\[ \frac{2}{\color{blue}{\left(k \cdot \frac{k}{\cos k}\right)} \cdot \frac{t \cdot {\sin k}^{2}}{{\ell}^{2}}} \]
      unpow2 [=>] 23.5	\[ \frac{2}{\left(k \cdot \frac{k}{\cos k}\right) \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      associate-*r/ [=>] 22.2	\[ \frac{2}{\color{blue}{\frac{\left(k \cdot \frac{k}{\cos k}\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\ell \cdot \ell}}} \]
      associate-*l/ [<=] 23.4	\[ \frac{2}{\color{blue}{\frac{k \cdot \frac{k}{\cos k}}{\ell \cdot \ell} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
      *-commutative [=>] 23.4	\[ \frac{2}{\color{blue}{\left(t \cdot {\sin k}^{2}\right) \cdot \frac{k \cdot \frac{k}{\cos k}}{\ell \cdot \ell}}} \]
      associate-*l* [=>] 23.4	\[ \frac{2}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot \frac{k \cdot \frac{k}{\cos k}}{\ell \cdot \ell}\right)}} \]
      times-frac [=>] 8.4	\[ \frac{2}{t \cdot \left({\sin k}^{2} \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{\frac{k}{\cos k}}{\ell}\right)}\right)} \]
      associate-/l/ [=>] 8.4	\[ \frac{2}{t \cdot \left({\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \color{blue}{\frac{k}{\ell \cdot \cos k}}\right)\right)} \]

   if -4.4999999999999998e58 < k < -1.25e17 or 4.80000000000000013e-110 < k < 1.54999999999999998e116

   1. Initial program 28.9

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

   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] 28.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 28.9	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 28.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 28.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 28.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 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 14.9

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

   4. Simplified 14.9

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

      [Start] 14.9	\[ \frac{2}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow2 [=>] 14.9	\[ \frac{2}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      cube-mult [<=] 14.9	\[ \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Applied egg-rr 10.5

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

   6. Simplified 14.9

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

      [Start] 10.5	\[ \frac{2}{{\left(\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}\right)}^{3}} \]
      cube-prod [=>] 14.9	\[ \frac{2}{\color{blue}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot {\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}\right)}^{3}}} \]
      *-commutative [=>] 14.9	\[ \frac{2}{\color{blue}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}\right)}^{3} \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}} \]
      rem-cube-cbrt [=>] 14.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\right)} \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \]
      *-commutative [=>] 14.9	\[ \frac{2}{\color{blue}{\left(\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k\right)} \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \]
      associate-*l* [=>] 14.9	\[ \frac{2}{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}} \]

   if -1.25e17 < k < -900

   1. Initial program 22.4

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

   2. Simplified 22.4

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

      [Start] 22.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 22.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 22.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 23.0

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

   4. Simplified 22.3

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

      [Start] 23.0	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 25.7	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 25.7	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 25.7	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 25.7	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\frac{{\sin k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]
      unpow2 [=>] 25.7	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{t}}} \]
      associate-/l* [=>] 22.3	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}} \]
      associate-/r/ [=>] 22.3	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{{\sin k}^{2}}{\color{blue}{\frac{\ell}{t} \cdot \ell}}} \]

   5. Applied egg-rr 22.3

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

   if -900 < k < -7.5000000000000005e-67

   1. Initial program 26.4

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

   2. Simplified 26.1

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

      [Start] 26.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 26.4	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 26.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 26.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 26.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 26.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 26.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 26.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 26.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 23.2

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

   4. Applied egg-rr 13.4

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

   if -7.5000000000000005e-67 < k < -4.50000000000000016e-107

   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 29.6

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

      [Start] 29.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-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* [=>] 29.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      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 13.1

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

   4. Simplified 13.1

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

      [Start] 13.1	\[ \frac{2}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow2 [=>] 13.1	\[ \frac{2}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      cube-mult [<=] 13.1	\[ \frac{2}{\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if -4.50000000000000016e-107 < k < 4.80000000000000013e-110

   1. Initial program 36.7

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

   2. Simplified 30.2

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

      [Start] 36.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 36.7	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 34.9	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 35.0	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 30.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 30.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 30.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 30.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around 0 53.3

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

   4. Simplified 52.4

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

      [Start] 53.3	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 53.3	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 52.4	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 52.4	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

   5. Applied egg-rr 49.6

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

   6. Taylor expanded in k around 0 52.4

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

   7. Simplified 17.8

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

      [Start] 52.4	\[ \frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]
      unpow2 [=>] 52.4	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      unpow3 [=>] 52.4	\[ \frac{\ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{\ell}} \]
      associate-*r* [=>] 50.7	\[ \frac{\ell}{\frac{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}}{\ell}} \]
      swap-sqr [<=] 22.3	\[ \frac{\ell}{\frac{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot t}{\ell}} \]
      unpow2 [<=] 22.3	\[ \frac{\ell}{\frac{\color{blue}{{\left(k \cdot t\right)}^{2}} \cdot t}{\ell}} \]
      *-commutative [=>] 22.3	\[ \frac{\ell}{\frac{\color{blue}{t \cdot {\left(k \cdot t\right)}^{2}}}{\ell}} \]
      *-commutative [<=] 22.3	\[ \frac{\ell}{\frac{\color{blue}{{\left(k \cdot t\right)}^{2} \cdot t}}{\ell}} \]
      associate-*l/ [<=] 17.8	\[ \frac{\ell}{\color{blue}{\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot t}} \]
      *-commutative [=>] 17.8	\[ \frac{\ell}{\color{blue}{t \cdot \frac{{\left(k \cdot t\right)}^{2}}{\ell}}} \]

   8. Applied egg-rr 4.4

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -4.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{2}{t \cdot \left({\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{elif}\;k \leq -1.25 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{elif}\;k \leq -900:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{elif}\;k \leq -7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+116}:\\ \;\;\;\;\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left({\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.7
Cost	46480

\[\begin{array}{l} t_1 := \frac{2}{t \cdot \left({\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ t_2 := \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \tan k}\right)}^{3}}\\ \mathbf{if}\;k \leq -2.8 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.3 \cdot 10^{-107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+116}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	9.6
Cost	40476

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ t_3 := \frac{2}{t \cdot \left(t_2 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ t_4 := \frac{2}{\left(t_1 \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}}\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{+58}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{+16}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -560:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_2 \cdot \frac{t}{\ell}\right)}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{elif}\;k \leq -8.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(t_1 \cdot \tan k\right)\right)\right)}{\ell}}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-107}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+115}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	10.8
Cost	39560

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{2}{t \cdot \left(t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{if}\;k \leq -1.48 \cdot 10^{+54}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.05 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \frac{2}{\frac{\cos k}{t_1}}}\\ \mathbf{elif}\;k \leq -3.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\frac{\ell}{t} \cdot \frac{\cos k}{k}}{\frac{t_1}{\ell}}}}\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+115}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	12.2
Cost	33744

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{t_1 \cdot \frac{\sin k}{\ell}}\\ \mathbf{if}\;t \leq -1.9 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-61}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+105}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(t_1 \cdot \tan k\right)\right) \cdot {\left(t \cdot \frac{\sqrt{t}}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 5
Error	10.3
Cost	27344

\[\begin{array}{l} t_1 := \frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+111}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot t_2\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}{\ell \cdot \frac{\cos k}{k}}}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{k \cdot t}{t_2}}\\ \end{array} \]

Alternative 6
Error	11.1
Cost	21004

\[\begin{array}{l} t_1 := \frac{2}{t \cdot \left({\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{if}\;k \leq -2.3 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{+115}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	10.8
Cost	21004

\[\begin{array}{l} t_1 := \frac{2}{t \cdot \left({\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{if}\;k \leq -1.42 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{-72}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.28 \cdot 10^{+116}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	11.5
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -2.4 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 11800000000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{0.5 + \cos \left(k + k\right) \cdot -0.5}{\ell \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 8.6 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(t \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\tan k}{\ell \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	11.5
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -3.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{t \cdot \left(t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{elif}\;k \leq 6000000000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{0.5 + \cos \left(k + k\right) \cdot -0.5}{\ell \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(t \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\tan k}{\ell \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \]

Alternative 10
Error	11.4
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -3.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{t \cdot \left(t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{elif}\;k \leq 1600000000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t_1}{\ell}\right)}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(t \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\tan k}{\ell \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \]

Alternative 11
Error	11.5
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -3 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{t \cdot \left(t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{elif}\;k \leq 185000000000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.4 \cdot 10^{+95}:\\ \;\;\;\;\frac{2}{k \cdot \frac{\frac{k}{\cos k}}{\left(\ell \cdot \frac{\ell}{t}\right) \cdot {\sin k}^{-2}}}\\ \mathbf{elif}\;k \leq 7.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(t \cdot t\right) \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\tan k}{\ell \cdot 0.5}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \end{array} \]

Alternative 12
Error	11.2
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.45 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{t \cdot \left(t_1 \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell \cdot \cos k}\right)\right)}\\ \mathbf{elif}\;k \leq 4000000000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_1 \cdot \frac{t}{\ell}\right)}{\ell \cdot \frac{\cos k}{k}}}\\ \end{array} \]

Alternative 13
Error	17.9
Cost	14672

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := \frac{2}{\frac{k}{t_1} \cdot \frac{0.5 + \cos \left(k + k\right) \cdot -0.5}{\ell \cdot \frac{\ell}{t}}}\\ t_3 := \frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-297}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{\frac{k}{t_1 \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)}}\\ \mathbf{elif}\;t \leq 4.55 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 14
Error	18.0
Cost	8009

\[\begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{-62} \lor \neg \left(t \leq 3.4 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)}}\\ \end{array} \]

Alternative 15
Error	18.8
Cost	7753

\[\begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-64} \lor \neg \left(t \leq 2.15 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{k \cdot k}{\ell \cdot \frac{\ell}{t}}}\\ \end{array} \]

Alternative 16
Error	18.3
Cost	7753

\[\begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-60} \lor \neg \left(t \leq 9 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right)}}\\ \end{array} \]

Alternative 17
Error	19.9
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{-61} \lor \neg \left(t \leq 8 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{k}^{4}}{\ell \cdot \ell}}\\ \end{array} \]

Alternative 18
Error	19.2
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-63} \lor \neg \left(t \leq 9.5 \cdot 10^{-69}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{\ell \cdot \frac{\ell}{t}}}\\ \end{array} \]

Alternative 19
Error	21.5
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -1.3 \cdot 10^{+134} \lor \neg \left(k \leq 5 \cdot 10^{+50}\right):\\ \;\;\;\;\frac{\ell}{t \cdot \frac{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 20
Error	21.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{+70} \lor \neg \left(k \leq 7.7 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 21
Error	34.0
Cost	832

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

Alternative 22
Error	24.1
Cost	832

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

Reproduce

herbie shell --seed 2023059 
(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))))