Toniolo and Linder, Equation (10+)

Average Accuracy: 47.9% → 85.4%

Time: 31.3s

Precision: binary64

Cost: 92372

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\sin k}\\ t_2 := \ell \cdot \sqrt{2}\\ t_3 := \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ t_4 := \frac{\frac{1}{t_1}}{t_3}\\ t_5 := t_3 \cdot t_1\\ t_6 := \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;{\left(\frac{t_2}{t_5}\right)}^{2} \cdot t_4\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-101}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-15}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot {\left(\frac{t_2}{{\left(\sqrt[3]{t_5}\right)}^{3}}\right)}^{2}\\ \end{array} \]

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 (sin k))))
        (t_2 (* l (sqrt 2.0)))
        (t_3 (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
        (t_4 (/ (/ 1.0 t_1) t_3))
        (t_5 (* t_3 t_1))
        (t_6 (* (* (/ l k) (/ l k)) (/ 2.0 (* (sin k) (* t (tan k)))))))
   (if (<= t -8.5e-39)
     (* (pow (/ t_2 t_5) 2.0) t_4)
     (if (<= t -6e-101)
       t_6
       (if (<= t -2.1e-153)
         (/ 2.0 (* (* (/ t l) (/ (* k k) l)) (* (tan k) (sin k))))
         (if (<= t 3.2e-71)
           (/ (/ (* l (* l 2.0)) (* k (* k (* t (sin k))))) (tan k))
           (if (<= t 9.2e-15)
             t_6
             (* t_4 (pow (/ t_2 (pow (cbrt t_5) 3.0)) 2.0)))))))))

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(sin(k));
	double t_2 = l * sqrt(2.0);
	double t_3 = cbrt((tan(k) * (2.0 + pow((k / t), 2.0))));
	double t_4 = (1.0 / t_1) / t_3;
	double t_5 = t_3 * t_1;
	double t_6 = ((l / k) * (l / k)) * (2.0 / (sin(k) * (t * tan(k))));
	double tmp;
	if (t <= -8.5e-39) {
		tmp = pow((t_2 / t_5), 2.0) * t_4;
	} else if (t <= -6e-101) {
		tmp = t_6;
	} else if (t <= -2.1e-153) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (tan(k) * sin(k)));
	} else if (t <= 3.2e-71) {
		tmp = ((l * (l * 2.0)) / (k * (k * (t * sin(k))))) / tan(k);
	} else if (t <= 9.2e-15) {
		tmp = t_6;
	} else {
		tmp = t_4 * pow((t_2 / pow(cbrt(t_5), 3.0)), 2.0);
	}
	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.sin(k));
	double t_2 = l * Math.sqrt(2.0);
	double t_3 = Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))));
	double t_4 = (1.0 / t_1) / t_3;
	double t_5 = t_3 * t_1;
	double t_6 = ((l / k) * (l / k)) * (2.0 / (Math.sin(k) * (t * Math.tan(k))));
	double tmp;
	if (t <= -8.5e-39) {
		tmp = Math.pow((t_2 / t_5), 2.0) * t_4;
	} else if (t <= -6e-101) {
		tmp = t_6;
	} else if (t <= -2.1e-153) {
		tmp = 2.0 / (((t / l) * ((k * k) / l)) * (Math.tan(k) * Math.sin(k)));
	} else if (t <= 3.2e-71) {
		tmp = ((l * (l * 2.0)) / (k * (k * (t * Math.sin(k))))) / Math.tan(k);
	} else if (t <= 9.2e-15) {
		tmp = t_6;
	} else {
		tmp = t_4 * Math.pow((t_2 / Math.pow(Math.cbrt(t_5), 3.0)), 2.0);
	}
	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(sin(k)))
	t_2 = Float64(l * sqrt(2.0))
	t_3 = cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))
	t_4 = Float64(Float64(1.0 / t_1) / t_3)
	t_5 = Float64(t_3 * t_1)
	t_6 = Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(2.0 / Float64(sin(k) * Float64(t * tan(k)))))
	tmp = 0.0
	if (t <= -8.5e-39)
		tmp = Float64((Float64(t_2 / t_5) ^ 2.0) * t_4);
	elseif (t <= -6e-101)
		tmp = t_6;
	elseif (t <= -2.1e-153)
		tmp = Float64(2.0 / Float64(Float64(Float64(t / l) * Float64(Float64(k * k) / l)) * Float64(tan(k) * sin(k))));
	elseif (t <= 3.2e-71)
		tmp = Float64(Float64(Float64(l * Float64(l * 2.0)) / Float64(k * Float64(k * Float64(t * sin(k))))) / tan(k));
	elseif (t <= 9.2e-15)
		tmp = t_6;
	else
		tmp = Float64(t_4 * (Float64(t_2 / (cbrt(t_5) ^ 3.0)) ^ 2.0));
	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[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[(N[(1.0 / t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 * t$95$1), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(t * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.5e-39], N[(N[Power[N[(t$95$2 / t$95$5), $MachinePrecision], 2.0], $MachinePrecision] * t$95$4), $MachinePrecision], If[LessEqual[t, -6e-101], t$95$6, If[LessEqual[t, -2.1e-153], N[(2.0 / N[(N[(N[(t / l), $MachinePrecision] * N[(N[(k * k), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.2e-71], N[(N[(N[(l * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] / N[(k * N[(k * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e-15], t$95$6, N[(t$95$4 * N[Power[N[(t$95$2 / N[Power[N[Power[t$95$5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -8.5000000000000005e-39

   1. Initial program 62.8%

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

   2. Simplified 62.6%

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

      [Start] 62.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 62.9	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 64.0	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 63.3	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 62.7	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 62.7	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 62.6	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 62.6	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 62.6	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 62.6	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 62.6	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Applied egg-rr 71.9%

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

      [Start] 62.6	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \]
      associate-*r/ [=>] 62.8	\[ \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
      add-cube-cbrt [=>] 62.7	\[ \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]
      associate-/r* [=>] 62.7	\[ \color{blue}{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

   4. Applied egg-rr 93.0%

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

      [Start] 71.9	\[ \frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)} \]
      div-inv [=>] 71.9	\[ \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}} \cdot \frac{1}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}} \]

   if -8.5000000000000005e-39 < t < -6.0000000000000006e-101 or 3.1999999999999999e-71 < t < 9.19999999999999961e-15

   1. Initial program 57.0%

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

   2. Simplified 55.2%

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

      [Start] 57.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 57.1	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 55.1	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 51.9	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 54.8	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 54.8	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 54.8	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 55.3	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 55.3	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 55.2	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 55.2	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Taylor expanded in k around inf 58.7%

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

   4. Simplified 58.7%

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

      [Start] 58.7	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
      unpow2 [=>] 58.7	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

   5. Applied egg-rr 58.7%

      \[\leadsto \color{blue}{0 + \frac{\ell \cdot \left(\ell \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)}} \]
      Proof

      [Start] 58.7	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)} \]
      add-log-exp [=>] 38.3	\[ \color{blue}{\log \left(e^{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}}\right)} \]
      *-un-lft-identity [=>] 38.3	\[ \log \color{blue}{\left(1 \cdot e^{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}}\right)} \]
      log-prod [=>] 38.3	\[ \color{blue}{\log 1 + \log \left(e^{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}}\right)} \]
      metadata-eval [=>] 38.3	\[ \color{blue}{0} + \log \left(e^{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}}\right) \]
      add-log-exp [<=] 58.7	\[ 0 + \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}} \]
      associate-*r/ [=>] 58.7	\[ 0 + \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}} \]
      associate-*l* [=>] 58.7	\[ 0 + \frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)} \]
      *-commutative [=>] 58.7	\[ 0 + \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \tan k}} \]
      associate-*l* [=>] 58.7	\[ 0 + \frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{\left(k \cdot k\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)}} \]

   6. Simplified 72.4%

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

      [Start] 58.7	\[ 0 + \frac{\ell \cdot \left(\ell \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)} \]
      +-lft-identity [=>] 58.7	\[ \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\left(k \cdot k\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 58.7	\[ \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot 2}}{\left(k \cdot k\right) \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)} \]
      unpow2 [<=] 58.7	\[ \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{{k}^{2}} \cdot \left(\left(\sin k \cdot t\right) \cdot \tan k\right)} \]
      times-frac [=>] 57.2	\[ \color{blue}{\frac{\ell \cdot \ell}{{k}^{2}} \cdot \frac{2}{\left(\sin k \cdot t\right) \cdot \tan k}} \]
      unpow2 [=>] 57.2	\[ \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{2}{\left(\sin k \cdot t\right) \cdot \tan k} \]
      times-frac [=>] 72.4	\[ \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{2}{\left(\sin k \cdot t\right) \cdot \tan k} \]
      associate-*l* [=>] 72.4	\[ \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\color{blue}{\sin k \cdot \left(t \cdot \tan k\right)}} \]

   if -6.0000000000000006e-101 < t < -2.10000000000000004e-153

   1. Initial program 6.1%

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

   2. Simplified 6.1%

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

      [Start] 6.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 6.1	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l* [=>] 6.1	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-*r* [=>] 6.1	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      +-commutative [=>] 6.1	\[ \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-+r+ [=>] 6.1	\[ \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      metadata-eval [=>] 6.1	\[ \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Taylor expanded in k around inf 60.7%

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

   4. Simplified 68.2%

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

      [Start] 60.7	\[ \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
      *-commutative [=>] 60.7	\[ \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 60.7	\[ \frac{2}{\frac{t \cdot {k}^{2}}{\color{blue}{\ell \cdot \ell}} \cdot \left(\sin k \cdot \tan k\right)} \]
      times-frac [=>] 68.2	\[ \frac{2}{\color{blue}{\left(\frac{t}{\ell} \cdot \frac{{k}^{2}}{\ell}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 68.2	\[ \frac{2}{\left(\frac{t}{\ell} \cdot \frac{\color{blue}{k \cdot k}}{\ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   if -2.10000000000000004e-153 < t < 3.1999999999999999e-71

   1. Initial program 5.5%

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

   2. Simplified 5.7%

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

      [Start] 5.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 5.5	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 5.1	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 5.0	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 5.6	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 5.6	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 5.6	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 5.7	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 5.7	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 5.7	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 5.7	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Taylor expanded in k around inf 57.9%

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

   4. Simplified 57.9%

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

      [Start] 57.9	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left({k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
      unpow2 [=>] 57.9	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot \left(\sin k \cdot t\right)\right)} \]

   5. Applied egg-rr 69.8%

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

      [Start] 57.9	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)} \]
      associate-*r/ [=>] 58.3	\[ \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right)}} \]
      *-commutative [=>] 58.3	\[ \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)\right) \cdot \tan k}} \]
      associate-/r* [=>] 59.2	\[ \color{blue}{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)}}{\tan k}} \]
      associate-*l* [=>] 59.2	\[ \frac{\frac{\color{blue}{\ell \cdot \left(\ell \cdot 2\right)}}{\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)}}{\tan k} \]
      associate-*l* [=>] 69.8	\[ \frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{\color{blue}{k \cdot \left(k \cdot \left(\sin k \cdot t\right)\right)}}}{\tan k} \]

   if 9.19999999999999961e-15 < t

   1. Initial program 63.9%

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

   2. Simplified 64.5%

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

      [Start] 63.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 63.9	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 65.2	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 65.1	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 64.6	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 64.6	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 64.5	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 64.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 64.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 64.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 64.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Applied egg-rr 74.9%

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

      [Start] 64.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \]
      associate-*r/ [=>] 64.5	\[ \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
      add-cube-cbrt [=>] 64.4	\[ \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]
      associate-/r* [=>] 64.4	\[ \color{blue}{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

   4. Applied egg-rr 93.7%

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

      [Start] 74.9	\[ \frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)} \]
      div-inv [=>] 74.9	\[ \color{blue}{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}} \cdot \frac{1}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}} \]

   5. Applied egg-rr 93.5%

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

      [Start] 93.7	\[ {\left(\frac{\ell \cdot \sqrt{2}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}\right)}^{2} \cdot \frac{\frac{1}{t \cdot \sqrt[3]{\sin k}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      add-cube-cbrt [=>] 93.5	\[ {\left(\frac{\ell \cdot \sqrt{2}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)} \cdot \sqrt[3]{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}\right) \cdot \sqrt[3]{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}}}\right)}^{2} \cdot \frac{\frac{1}{t \cdot \sqrt[3]{\sin k}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      pow3 [=>] 93.5	\[ {\left(\frac{\ell \cdot \sqrt{2}}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}\right)}^{3}}}\right)}^{2} \cdot \frac{\frac{1}{t \cdot \sqrt[3]{\sin k}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-39}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}\right)}^{2} \cdot \frac{\frac{1}{t \cdot \sqrt[3]{\sin k}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-101}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \mathbf{elif}\;t \leq -2.1 \cdot 10^{-153}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{t \cdot \sqrt[3]{\sin k}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot {\left(\frac{\ell \cdot \sqrt{2}}{{\left(\sqrt[3]{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}\right)}^{3}}\right)}^{2}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.5%
Cost	79508

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\sin k}\\ t_2 := \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ t_3 := {\left(\frac{\ell \cdot \sqrt{2}}{t_2 \cdot t_1}\right)}^{2} \cdot \frac{\frac{1}{t_1}}{t_2}\\ t_4 := \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-39}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -6 \cdot 10^{-101}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq -9.2 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \left(\tan k \cdot \sin k\right)}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-17}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Accuracy	85.7%
Cost	33036

\[\begin{array}{l} t_1 := \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \mathbf{if}\;k \leq -1.85 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.1 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k}}{t} \cdot \frac{-\ell}{-t}\\ \mathbf{elif}\;k \leq 9.6 \cdot 10^{-34}:\\ \;\;\;\;{\left(\frac{1}{{\left(\sqrt[3]{k}\right)}^{2}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	84.7%
Cost	27080

\[\begin{array}{l} t_1 := \tan k \cdot \sin k\\ t_2 := \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ t_3 := \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{if}\;k \leq -1.12 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.05 \cdot 10^{-137}:\\ \;\;\;\;\frac{t_3 \cdot {\left(\frac{\ell}{t}\right)}^{2}}{t \cdot t_1}\\ \mathbf{elif}\;k \leq 8.6 \cdot 10^{-133}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{t_3}{t_1}}{t} \cdot \frac{-\ell}{-t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	83.2%
Cost	21132

\[\begin{array}{l} t_1 := \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \mathbf{if}\;k \leq -310:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-133}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \mathbf{elif}\;k \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k}}{t} \cdot \frac{-\ell}{-t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	73.9%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-53} \lor \neg \left(t \leq 10^{-17}\right):\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \end{array} \]

Alternative 6
Accuracy	80.1%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -300 \lor \neg \left(k \leq 8 \cdot 10^{-34}\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{2}{\sin k \cdot \left(t \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \end{array} \]

Alternative 7
Accuracy	66.2%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-53} \lor \neg \left(t \leq 8 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot {k}^{-4}}{t}\\ \end{array} \]

Alternative 8
Accuracy	67.6%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-53} \lor \neg \left(t \leq 2.05 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{{\left(\frac{\ell}{t \cdot k}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{k}^{4}} \cdot \frac{\ell}{\frac{t}{2}}\\ \end{array} \]

Alternative 9
Accuracy	61.3%
Cost	6912

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

Alternative 10
Accuracy	61.6%
Cost	1216

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{1}{k}}{t}\\ t_1 \cdot \left(t_1 \cdot \frac{1}{t}\right) \end{array} \]

Alternative 11
Accuracy	48.5%
Cost	1097

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{t \cdot t}\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{-157} \lor \neg \left(k \leq 4.3 \cdot 10^{-160}\right):\\ \;\;\;\;t_1 \cdot \frac{\ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	53.7%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+67} \lor \neg \left(t \leq 2.5 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 13
Accuracy	61.5%
Cost	1088

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{1}{k}}{t}\\ \frac{t_1 \cdot t_1}{t} \end{array} \]

Alternative 14
Accuracy	61.6%
Cost	1088

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{1}{k}}{t}\\ t_1 \cdot \frac{t_1}{t} \end{array} \]

Alternative 15
Accuracy	52.4%
Cost	832

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

Alternative 16
Accuracy	33.8%
Cost	576

\[\frac{1}{t} \cdot \frac{\frac{\ell}{t}}{t} \]

Alternative 17
Accuracy	33.8%
Cost	448

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

Reproduce

herbie shell --seed 2023152 
(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))))