Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 7.8

Time: 46.8s

Precision: binary64

Cost: 53328

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;t \leq -5.9 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+15}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+167}:\\ \;\;\;\;\frac{2}{\left(\frac{{t_1}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t_1}{\ell}\right) \cdot \left(2 + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(t_3 \cdot {t_3}^{2}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(t_2 + 1\right)\right)\right)}\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (* t (cbrt (tan k))))
        (t_2 (pow (/ k t) 2.0))
        (t_3 (/ t (pow (cbrt l) 2.0))))
   (if (<= t -5.9e+50)
     (* (/ (/ (/ l (* t k)) t) k) (/ l t))
     (if (<= t -1.4e-29)
       (*
        (/ (/ 2.0 (pow t 3.0)) (tan k))
        (/ l (/ (* (sin k) (+ 2.0 (* k (/ k (* t t))))) l)))
       (if (<= t 5.4e+15)
         (* 2.0 (/ (cos k) (* (/ (pow (sin k) 2.0) (/ l k)) (/ t (/ l k)))))
         (if (<= t 5.8e+167)
           (/
            2.0
            (* (* (/ (pow t_1 2.0) (/ l (sin k))) (/ t_1 l)) (+ 2.0 t_2)))
           (/
            2.0
            (*
             (* (sin k) (* t_3 (pow t_3 2.0)))
             (* (tan k) (+ 1.0 (+ t_2 1.0)))))))))))

C

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

↓

double code(double t, double l, double k) {
	double t_1 = t * cbrt(tan(k));
	double t_2 = pow((k / t), 2.0);
	double t_3 = t / pow(cbrt(l), 2.0);
	double tmp;
	if (t <= -5.9e+50) {
		tmp = (((l / (t * k)) / t) / k) * (l / t);
	} else if (t <= -1.4e-29) {
		tmp = ((2.0 / pow(t, 3.0)) / tan(k)) * (l / ((sin(k) * (2.0 + (k * (k / (t * t))))) / l));
	} else if (t <= 5.4e+15) {
		tmp = 2.0 * (cos(k) / ((pow(sin(k), 2.0) / (l / k)) * (t / (l / k))));
	} else if (t <= 5.8e+167) {
		tmp = 2.0 / (((pow(t_1, 2.0) / (l / sin(k))) * (t_1 / l)) * (2.0 + t_2));
	} else {
		tmp = 2.0 / ((sin(k) * (t_3 * pow(t_3, 2.0))) * (tan(k) * (1.0 + (t_2 + 1.0))));
	}
	return tmp;
}

Java

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

↓

public static double code(double t, double l, double k) {
	double t_1 = t * Math.cbrt(Math.tan(k));
	double t_2 = Math.pow((k / t), 2.0);
	double t_3 = t / Math.pow(Math.cbrt(l), 2.0);
	double tmp;
	if (t <= -5.9e+50) {
		tmp = (((l / (t * k)) / t) / k) * (l / t);
	} else if (t <= -1.4e-29) {
		tmp = ((2.0 / Math.pow(t, 3.0)) / Math.tan(k)) * (l / ((Math.sin(k) * (2.0 + (k * (k / (t * t))))) / l));
	} else if (t <= 5.4e+15) {
		tmp = 2.0 * (Math.cos(k) / ((Math.pow(Math.sin(k), 2.0) / (l / k)) * (t / (l / k))));
	} else if (t <= 5.8e+167) {
		tmp = 2.0 / (((Math.pow(t_1, 2.0) / (l / Math.sin(k))) * (t_1 / l)) * (2.0 + t_2));
	} else {
		tmp = 2.0 / ((Math.sin(k) * (t_3 * Math.pow(t_3, 2.0))) * (Math.tan(k) * (1.0 + (t_2 + 1.0))));
	}
	return tmp;
}

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

↓

function code(t, l, k)
	t_1 = Float64(t * cbrt(tan(k)))
	t_2 = Float64(k / t) ^ 2.0
	t_3 = Float64(t / (cbrt(l) ^ 2.0))
	tmp = 0.0
	if (t <= -5.9e+50)
		tmp = Float64(Float64(Float64(Float64(l / Float64(t * k)) / t) / k) * Float64(l / t));
	elseif (t <= -1.4e-29)
		tmp = Float64(Float64(Float64(2.0 / (t ^ 3.0)) / tan(k)) * Float64(l / Float64(Float64(sin(k) * Float64(2.0 + Float64(k * Float64(k / Float64(t * t))))) / l)));
	elseif (t <= 5.4e+15)
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64((sin(k) ^ 2.0) / Float64(l / k)) * Float64(t / Float64(l / k)))));
	elseif (t <= 5.8e+167)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_1 ^ 2.0) / Float64(l / sin(k))) * Float64(t_1 / l)) * Float64(2.0 + t_2)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64(t_3 * (t_3 ^ 2.0))) * Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0)))));
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(t * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.9e+50], N[(N[(N[(N[(l / N[(t * k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / k), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -1.4e-29], N[(N[(N[(2.0 / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[(k * N[(k / N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.4e+15], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e+167], N[(2.0 / N[(N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(t$95$3 * N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.8999999999999998e50

   1. Initial program 23.8

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

   2. Simplified 29.6

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

      [Start] 23.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 23.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-*l/ [=>] 22.8	\[ \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]
      associate-*l/ [=>] 23.2	\[ \frac{2}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 23.3	\[ \color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 23.3	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-*r* [=>] 23.3	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      *-commutative [<=] 23.3	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\tan k \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r* [=>] 23.2	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   3. Taylor expanded in k around 0 29.9

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

   4. Simplified 20.5

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

      [Start] 29.9	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 29.9	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 26.9	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 26.9	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      associate-*l* [=>] 20.5	\[ \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   5. Applied egg-rr 20.4

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

   6. Applied egg-rr 17.4

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

   7. Applied egg-rr 9.6

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

   if -5.8999999999999998e50 < t < -1.4000000000000001e-29

   1. Initial program 20.3

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

   2. Simplified 14.0

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

      [Start] 20.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 20.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 20.3	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 18.7	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 18.7	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 18.7	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 14.0	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 14.0	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 14.0	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 14.0	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Taylor expanded in l around 0 18.9

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

   4. Simplified 12.8

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

      [Start] 18.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{{\ell}^{2}}{\sin k \cdot \left(2 + \frac{{k}^{2}}{{t}^{2}}\right)} \]
      unpow2 [=>] 18.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{\sin k \cdot \left(2 + \frac{{k}^{2}}{{t}^{2}}\right)} \]
      associate-/l* [=>] 12.8	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \color{blue}{\frac{\ell}{\frac{\sin k \cdot \left(2 + \frac{{k}^{2}}{{t}^{2}}\right)}{\ell}}} \]
      unpow2 [=>] 12.8	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + \frac{\color{blue}{k \cdot k}}{{t}^{2}}\right)}{\ell}} \]
      unpow2 [=>] 12.8	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + \frac{k \cdot k}{\color{blue}{t \cdot t}}\right)}{\ell}} \]

   5. Applied egg-rr 12.1

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

   if -1.4000000000000001e-29 < t < 5.4e15

   1. Initial program 49.6

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

   2. Simplified 50.2

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

      [Start] 49.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 49.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-*l/ [=>] 49.9	\[ \frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \]
      associate-*l/ [=>] 49.6	\[ \frac{2}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 49.7	\[ \color{blue}{\frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 49.7	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-*r* [=>] 49.8	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      *-commutative [<=] 49.8	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\tan k \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r* [=>] 50.2	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{\frac{2}{\tan k \cdot \left({t}^{3} \cdot \sin k\right)}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]

   3. Taylor expanded in k around inf 27.1

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

   4. Simplified 27.0

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

      [Start] 27.1	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 27.2	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      *-commutative [=>] 27.2	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]
      associate-/l* [=>] 28.8	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]
      unpow2 [=>] 28.8	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]
      associate-/l* [=>] 27.0	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\color{blue}{\frac{\ell}{\frac{{k}^{2}}{\ell}}}}} \]
      unpow2 [=>] 27.0	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell}{\frac{\color{blue}{k \cdot k}}{\ell}}}} \]

   5. Applied egg-rr 6.3

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

   if 5.4e15 < t < 5.79999999999999949e167

   1. Initial program 23.8

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

   2. Simplified 20.2

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

      [Start] 23.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 23.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 22.5	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 22.3	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 20.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 20.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 20.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 20.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 6.2

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

   if 5.79999999999999949e167 < t

   1. Initial program 21.3

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

   2. Simplified 21.3

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

      [Start] 21.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 21.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 21.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 8.0

      \[\leadsto \frac{2}{\left(\color{blue}{\left({\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 7.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.9 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+15}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+167}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\ell}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.9
Cost	46800

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+17}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 2.32 \cdot 10^{+169}:\\ \;\;\;\;\frac{2}{\left(\frac{{t_1}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t_1}{\ell}\right) \cdot \left(2 + t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 2
Error	8.0
Cost	40208

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1.16 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 990:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 10^{+94}:\\ \;\;\;\;\frac{\ell \cdot 2}{\left(\left(2 + t_1\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left({t}^{3} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \end{array} \]

Alternative 3
Error	8.7
Cost	33808

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+50}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2900:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell \cdot 2}{\left(\left(2 + t_1\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left({t}^{3} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]

Alternative 4
Error	7.8
Cost	27344

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t \cdot k}}{t}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+50}:\\ \;\;\;\;\frac{t_1}{k} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -1.3 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2100:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot 2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left({t}^{3} \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(t_1 \cdot \frac{\frac{1}{t}}{k}\right)\\ \end{array} \]

Alternative 5
Error	8.3
Cost	21268

\[\begin{array}{l} t_1 := \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ t_2 := \frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1460:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{+167}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	8.3
Cost	21268

\[\begin{array}{l} t_1 := \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ t_2 := \frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+50}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 800:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+195}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	8.2
Cost	21268

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\ell}{t \cdot k}}{t}}{k} \cdot \frac{\ell}{t}\\ \mathbf{if}\;t \leq -6 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\ell}{\frac{\sin k \cdot \left(2 + k \cdot \frac{k}{t \cdot t}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1460:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2}}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell \cdot 2}{\left({t}^{3} \cdot \tan k\right) \cdot \frac{\sin k \cdot \left(2 + \frac{k \cdot k}{t \cdot t}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+196}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	7.6
Cost	20872

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.25 \cdot 10^{+91}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_2}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq -2.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{t \cdot t_2}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 9
Error	10.6
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\cos k}{t \cdot {\sin k}^{2}} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right)\\ t_2 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-44}:\\ \;\;\;\;\frac{t_2}{\frac{t}{t_2}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+103}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t}{\ell} \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	10.8
Cost	20489

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -7 \cdot 10^{-7} \lor \neg \left(k \leq 7.4 \cdot 10^{-44}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{t} \cdot \frac{\cos k}{{\sin k}^{2}}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 11
Error	8.3
Cost	20489

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -0.095 \lor \neg \left(k \leq 7.4 \cdot 10^{-44}\right):\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{t \cdot {\sin k}^{2}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 12
Error	8.3
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -4.6 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_2}{\frac{\ell}{k}} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{t \cdot t_2}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 13
Error	15.6
Cost	20361

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -1.9 \cdot 10^{-8} \lor \neg \left(k \leq 7.4 \cdot 10^{-44}\right):\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{\left(k \cdot \sin k\right)}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 14
Error	15.9
Cost	20360

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := \frac{t_1}{t}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{-39}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{\ell \cdot \frac{\ell}{t}}{{\left(k \cdot \sin k\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(t_2 \cdot \frac{\frac{1}{t}}{k}\right)\\ \end{array} \]

Alternative 15
Error	15.6
Cost	20360

\[\begin{array}{l} t_1 := {\left(k \cdot \sin k\right)}^{2}\\ t_2 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -3.7 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t_1}\right)\right)\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{t_2}{\frac{t}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t}{\ell} \cdot \frac{t_1}{\ell}}\\ \end{array} \]

Alternative 16
Error	19.2
Cost	13832

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := \frac{t_1}{t}\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{-30}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(t_2 \cdot \frac{\frac{1}{t}}{k}\right)\\ \end{array} \]

Alternative 17
Error	20.9
Cost	7753

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{-5} \lor \neg \left(k \leq 2.7 \cdot 10^{-50}\right):\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k \cdot \left(t \cdot k\right)}{\frac{\ell}{\frac{k \cdot k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 18
Error	20.7
Cost	7752

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := \frac{\ell}{\frac{k \cdot k}{\ell}}\\ \mathbf{if}\;k \leq -0.48:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k \cdot \left(t \cdot k\right)}{t_2}}\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot \left(k \cdot k\right)}{t_2}}\\ \end{array} \]

Alternative 19
Error	20.2
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := \frac{t_1}{t}\\ \mathbf{if}\;t \leq -1.42 \cdot 10^{-39}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-62}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(t_2 \cdot \frac{\frac{1}{t}}{k}\right)\\ \end{array} \]

Alternative 20
Error	21.9
Cost	1096

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := \frac{t_1}{t}\\ \mathbf{if}\;\ell \leq -1.5 \cdot 10^{+21}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-60}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{k} \cdot \frac{\ell}{t}\\ \end{array} \]

Alternative 21
Error	21.9
Cost	1096

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;\ell \leq -4 \cdot 10^{+21}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \mathbf{elif}\;\ell \leq 10^{-58}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{t_1}{t}}{k} \cdot \frac{\ell}{t}\\ \end{array} \]

Alternative 22
Error	31.0
Cost	832

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

Alternative 23
Error	24.1
Cost	832

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_1 \cdot \frac{t_1}{t} \end{array} \]

Reproduce

herbie shell --seed 2023011 
(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))))