Toniolo and Linder, Equation (10+)

Average Error: 32.7 → 5.8

Time: 1.5min

Precision: binary64

Cost: 98832

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := 2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ t_3 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}\\ t_4 := \sqrt[3]{\tan k}\\ t_5 := \frac{1}{{\left(\frac{t_4}{\frac{t_1}{t \cdot t_3}}\right)}^{2}} \cdot \frac{2}{\frac{t_3}{\frac{\frac{t_1}{t}}{t_4}}}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-25}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(\sin k \cdot \sqrt{t}\right)} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_5\\ \end{array} \]

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
         (* 2.0 (* (/ (/ (cos k) k) t) (/ (* l (/ l k)) (pow (sin k) 2.0)))))
        (t_3 (cbrt (* (+ 2.0 (pow (/ k t) 2.0)) (sin k))))
        (t_4 (cbrt (tan k)))
        (t_5
         (*
          (/ 1.0 (pow (/ t_4 (/ t_1 (* t t_3))) 2.0))
          (/ 2.0 (/ t_3 (/ (/ t_1 t) t_4))))))
   (if (<= t -4.8e-25)
     t_5
     (if (<= t -5e-310)
       t_2
       (if (<= t 6.2e-180)
         (*
          2.0
          (*
           (/ (cos k) (* k (* (sin k) (sqrt t))))
           (* (/ (/ l k) (sin k)) (/ l (sqrt t)))))
         (if (<= t 1.15e-21) t_2 t_5))))))

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 = 2.0 * (((cos(k) / k) / t) * ((l * (l / k)) / pow(sin(k), 2.0)));
	double t_3 = cbrt(((2.0 + pow((k / t), 2.0)) * sin(k)));
	double t_4 = cbrt(tan(k));
	double t_5 = (1.0 / pow((t_4 / (t_1 / (t * t_3))), 2.0)) * (2.0 / (t_3 / ((t_1 / t) / t_4)));
	double tmp;
	if (t <= -4.8e-25) {
		tmp = t_5;
	} else if (t <= -5e-310) {
		tmp = t_2;
	} else if (t <= 6.2e-180) {
		tmp = 2.0 * ((cos(k) / (k * (sin(k) * sqrt(t)))) * (((l / k) / sin(k)) * (l / sqrt(t))));
	} else if (t <= 1.15e-21) {
		tmp = t_2;
	} else {
		tmp = t_5;
	}
	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 = 2.0 * (((Math.cos(k) / k) / t) * ((l * (l / k)) / Math.pow(Math.sin(k), 2.0)));
	double t_3 = Math.cbrt(((2.0 + Math.pow((k / t), 2.0)) * Math.sin(k)));
	double t_4 = Math.cbrt(Math.tan(k));
	double t_5 = (1.0 / Math.pow((t_4 / (t_1 / (t * t_3))), 2.0)) * (2.0 / (t_3 / ((t_1 / t) / t_4)));
	double tmp;
	if (t <= -4.8e-25) {
		tmp = t_5;
	} else if (t <= -5e-310) {
		tmp = t_2;
	} else if (t <= 6.2e-180) {
		tmp = 2.0 * ((Math.cos(k) / (k * (Math.sin(k) * Math.sqrt(t)))) * (((l / k) / Math.sin(k)) * (l / Math.sqrt(t))));
	} else if (t <= 1.15e-21) {
		tmp = t_2;
	} else {
		tmp = t_5;
	}
	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 = Float64(2.0 * Float64(Float64(Float64(cos(k) / k) / t) * Float64(Float64(l * Float64(l / k)) / (sin(k) ^ 2.0))))
	t_3 = cbrt(Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * sin(k)))
	t_4 = cbrt(tan(k))
	t_5 = Float64(Float64(1.0 / (Float64(t_4 / Float64(t_1 / Float64(t * t_3))) ^ 2.0)) * Float64(2.0 / Float64(t_3 / Float64(Float64(t_1 / t) / t_4))))
	tmp = 0.0
	if (t <= -4.8e-25)
		tmp = t_5;
	elseif (t <= -5e-310)
		tmp = t_2;
	elseif (t <= 6.2e-180)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * Float64(sin(k) * sqrt(t)))) * Float64(Float64(Float64(l / k) / sin(k)) * Float64(l / sqrt(t)))));
	elseif (t <= 1.15e-21)
		tmp = t_2;
	else
		tmp = t_5;
	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[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] / t), $MachinePrecision] * N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$5 = N[(N[(1.0 / N[Power[N[(t$95$4 / N[(t$95$1 / N[(t * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(t$95$3 / N[(N[(t$95$1 / t), $MachinePrecision] / t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.8e-25], t$95$5, If[LessEqual[t, -5e-310], t$95$2, If[LessEqual[t, 6.2e-180], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * N[(N[Sin[k], $MachinePrecision] * N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / k), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(l / N[Sqrt[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.15e-21], t$95$2, t$95$5]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.80000000000000018e-25 or 1.15e-21 < t

   1. Initial program 22.2

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

   2. Simplified 27.1

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

      [Start] 22.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.2	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 22.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 22.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 22.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 27.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 27.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 27.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 27.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 14.3

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

   4. Applied egg-rr 14.3

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

   5. Applied egg-rr 2.5

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

   if -4.80000000000000018e-25 < t < -4.999999999999985e-310 or 6.1999999999999998e-180 < t < 1.15e-21

   1. Initial program 49.5

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

   2. Simplified 49.9

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

      [Start] 49.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 49.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 49.5	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 49.5	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 50.1	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 50.3	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 49.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 49.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 49.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 49.9	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Taylor expanded in t around 0 27.2

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

   4. Simplified 28.2

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

      [Start] 27.2	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/r* [=>] 28.2	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]
      unpow2 [=>] 28.2	\[ 2 \cdot \frac{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 28.2	\[ 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

   5. Applied egg-rr 13.0

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

   if -4.999999999999985e-310 < t < 6.1999999999999998e-180

   1. Initial program 64.0

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

   2. Simplified 64.0

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

      [Start] 64.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 64.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 64.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 64.0	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 64.0	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 64.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 64.0	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 64.0	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 64.0	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 64.0	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Taylor expanded in t around 0 27.0

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

   4. Simplified 30.5

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

      [Start] 27.0	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/r* [=>] 30.5	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]
      unpow2 [=>] 30.5	\[ 2 \cdot \frac{\frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 30.5	\[ 2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]

   5. Applied egg-rr 15.3

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

   6. Simplified 9.2

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

      [Start] 15.3	\[ 2 \cdot \left(\frac{\frac{\cos k}{k}}{\sin k \cdot \sqrt{t}} \cdot \frac{\frac{\ell}{k} \cdot \ell}{\sin k \cdot \sqrt{t}}\right) \]
      associate-/l/ [=>] 15.3	\[ 2 \cdot \left(\color{blue}{\frac{\cos k}{\left(\sin k \cdot \sqrt{t}\right) \cdot k}} \cdot \frac{\frac{\ell}{k} \cdot \ell}{\sin k \cdot \sqrt{t}}\right) \]
      times-frac [=>] 9.2	\[ 2 \cdot \left(\frac{\cos k}{\left(\sin k \cdot \sqrt{t}\right) \cdot k} \cdot \color{blue}{\left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 5.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{{\left(\frac{\sqrt[3]{\tan k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}}\right)}^{2}} \cdot \frac{2}{\frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\tan k}}}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(\sin k \cdot \sqrt{t}\right)} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{\sqrt[3]{\tan k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}}\right)}^{2}} \cdot \frac{2}{\frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\tan k}}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.0
Cost	52880

\[\begin{array}{l} t_1 := \frac{2}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k} \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot \sqrt[3]{\tan k}\right)\right)\right)}^{3}}\\ t_2 := 2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-179}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(\sin k \cdot \sqrt{t}\right)} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)\right)\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	6.0
Cost	52880

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}\\ t_3 := 2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{if}\;t \leq -8 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \left(t \cdot t_1\right)\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-308}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-181}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(\sin k \cdot \sqrt{t}\right)} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)\right)\\ \mathbf{elif}\;t \leq 1.16 \cdot 10^{-21}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t_1}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{t_2}}\right)}^{3}}\\ \end{array} \]

Alternative 3
Error	10.9
Cost	46932

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{2}{t_2 \cdot \left(\frac{{t_1}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t_1}{\ell}\right)}\\ t_4 := 2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-26}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{-306}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-180}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(\sin k \cdot \sqrt{t}\right)} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)\right)\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-21}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+166}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+287}:\\ \;\;\;\;\frac{2}{{\left(\frac{\frac{t}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\tan k \cdot \left(t_2 \cdot \sin k\right)}}}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)}\\ \end{array} \]

Alternative 4
Error	12.0
Cost	46216

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+149}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left({\left(\sqrt[3]{\ell}\right)}^{-2} \cdot \sqrt[3]{\tan k \cdot \left(t_2 \cdot \sin k\right)}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-179}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(\sin k \cdot \sqrt{t}\right)} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)\right)\\ \mathbf{elif}\;t \leq 8.4 \cdot 10^{-22}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{t_2 \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\tan k \cdot t_2}\right)\right)}^{3}}\\ \end{array} \]

Alternative 5
Error	12.8
Cost	40276

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ t_3 := 2 + t_1\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t_1\right)\right)\right)}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 10^{-181}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(\sin k \cdot \sqrt{t}\right)} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)\right)\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-22}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{t_3 \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\frac{k}{\ell}} \cdot \sqrt[3]{\tan k \cdot t_3}\right)\right)}^{3}}\\ \end{array} \]

Alternative 6
Error	12.9
Cost	33612

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := 2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t_2\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-308}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-183}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot \left(\sin k \cdot \sqrt{t}\right)} \cdot \left(\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\ell}{\sqrt{t}}\right)\right)\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{\left(2 + t_2\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 7
Error	13.2
Cost	27212

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t_2\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+106}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{\left(2 + t_2\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 8
Error	13.8
Cost	20872

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-24}:\\ \;\;\;\;\frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 9
Error	14.2
Cost	20868

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-21}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 10
Error	13.7
Cost	20752

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.9 \cdot 10^{+118}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_2}\right)\\ \mathbf{elif}\;k \leq -3.25 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell \cdot \cos k}{\left(k \cdot k\right) \cdot t_2}\right)\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 2.75:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \frac{1}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{t_1}{t_2}\right)\\ \end{array} \]

Alternative 11
Error	14.9
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \left(\cos k \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{+14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-166}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 2.35:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \frac{1}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	14.9
Cost	20620

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -27000000000000:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_2}\right)\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-165}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 1.05:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \frac{1}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{t_1}{t_2}\right)\\ \end{array} \]

Alternative 13
Error	14.1
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+50}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\cos k}{k}}{t} \cdot \frac{\ell \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 14
Error	21.4
Cost	19908

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 15
Error	21.9
Cost	14596

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-102}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\left(t \cdot t\right) \cdot \left(t \cdot \frac{1}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 16
Error	23.0
Cost	14408

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{2 \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 7.8 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k \cdot k}}{t \cdot \left(0.5 + \cos \left(k + k\right) \cdot -0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 17
Error	24.3
Cost	13960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\ell}{k \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 18
Error	24.3
Cost	13960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{2 \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-22}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 19
Error	24.1
Cost	13896

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -13500000:\\ \;\;\;\;\frac{\ell}{k \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 20
Error	24.6
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -13500000:\\ \;\;\;\;\frac{\ell}{k \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 21
Error	26.4
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -13500000 \lor \neg \left(t \leq 1.75 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \end{array} \]

Alternative 22
Error	26.9
Cost	7304

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;t \leq -13500000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t_1}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{t}\\ \end{array} \]

Alternative 23
Error	26.0
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -13500000:\\ \;\;\;\;\frac{\ell}{k \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot \frac{k \cdot {t}^{3}}{\ell}}\\ \end{array} \]

Alternative 24
Error	26.4
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -13500000:\\ \;\;\;\;\frac{\ell}{k \cdot \left(k \cdot \frac{{t}^{3}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \end{array} \]

Alternative 25
Error	27.9
Cost	1360

\[\begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{+32}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{t \cdot k}\right)\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{-248}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{t} \cdot \frac{1}{k}\right)\right)\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{+17}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.3333333333333333}{k \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 26
Error	28.1
Cost	1360

\[\begin{array}{l} \mathbf{if}\;k \leq -6.8 \cdot 10^{+35}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{t \cdot k}\right)\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{\ell}{t \cdot t} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-248}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{t} \cdot \frac{1}{k}\right)\right)\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \frac{t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.3333333333333333}{k \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 27
Error	27.6
Cost	1352

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;t \leq -13500000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t_1}\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 + \frac{2}{k \cdot k}}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{t}\\ \end{array} \]

Alternative 28
Error	27.4
Cost	1352

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;t \leq -13500000:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t_1}\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{-34}:\\ \;\;\;\;\frac{-0.3333333333333333 + \frac{2}{k \cdot k}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{t}\\ \end{array} \]

Alternative 29
Error	27.7
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{-69} \lor \neg \left(t \leq 8 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \end{array} \]

Alternative 30
Error	27.4
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-70} \lor \neg \left(t \leq 3.15 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \end{array} \]

Alternative 31
Error	27.4
Cost	1096

\[\begin{array}{l} t_1 := \left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\ell}\right)\\ \mathbf{if}\;t \leq -1 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t_1}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-34}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{t}\\ \end{array} \]

Alternative 32
Error	35.0
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot k}\right)\\ \end{array} \]

Alternative 33
Error	36.1
Cost	704

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

Alternative 34
Error	34.7
Cost	704

\[-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{t \cdot k}}{k} \]

Reproduce

herbie shell --seed 2023060 
(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))))