Toniolo and Linder, Equation (10+)

Average Accuracy: 50.8% → 89.3%

Time: 43.3s

Precision: binary64

Cost: 59148

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \cdot \sqrt[3]{\sin k}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{+37}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-181}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\frac{\frac{\ell}{t} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (pow (sin k) 2.0))
        (t_2
         (pow
          (*
           (/
            (cbrt 2.0)
            (* (cbrt (* (+ 2.0 (pow (/ k t) 2.0)) (tan k))) (cbrt (sin k))))
           (/ (pow (cbrt l) 2.0) t))
          3.0)))
   (if (<= t -6.8e+37)
     t_2
     (if (<= t -1e-181)
       (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_1))))
       (if (<= t 4.2e-73)
         (/ 2.0 (/ t_1 (/ (* (/ l t) (* l (/ (cos k) k))) k)))
         t_2)))))

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(sin(k), 2.0);
	double t_2 = pow(((cbrt(2.0) / (cbrt(((2.0 + pow((k / t), 2.0)) * tan(k))) * cbrt(sin(k)))) * (pow(cbrt(l), 2.0) / t)), 3.0);
	double tmp;
	if (t <= -6.8e+37) {
		tmp = t_2;
	} else if (t <= -1e-181) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	} else if (t <= 4.2e-73) {
		tmp = 2.0 / (t_1 / (((l / t) * (l * (cos(k) / k))) / k));
	} else {
		tmp = t_2;
	}
	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.sin(k), 2.0);
	double t_2 = Math.pow(((Math.cbrt(2.0) / (Math.cbrt(((2.0 + Math.pow((k / t), 2.0)) * Math.tan(k))) * Math.cbrt(Math.sin(k)))) * (Math.pow(Math.cbrt(l), 2.0) / t)), 3.0);
	double tmp;
	if (t <= -6.8e+37) {
		tmp = t_2;
	} else if (t <= -1e-181) {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_1)));
	} else if (t <= 4.2e-73) {
		tmp = 2.0 / (t_1 / (((l / t) * (l * (Math.cos(k) / k))) / k));
	} else {
		tmp = t_2;
	}
	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 = sin(k) ^ 2.0
	t_2 = Float64(Float64(cbrt(2.0) / Float64(cbrt(Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * tan(k))) * cbrt(sin(k)))) * Float64((cbrt(l) ^ 2.0) / t)) ^ 3.0
	tmp = 0.0
	if (t <= -6.8e+37)
		tmp = t_2;
	elseif (t <= -1e-181)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * t_1))));
	elseif (t <= 4.2e-73)
		tmp = Float64(2.0 / Float64(t_1 / Float64(Float64(Float64(l / t) * Float64(l * Float64(cos(k) / k))) / k)));
	else
		tmp = t_2;
	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[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / N[(N[Power[N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, If[LessEqual[t, -6.8e+37], t$95$2, If[LessEqual[t, -1e-181], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.2e-73], N[(2.0 / N[(t$95$1 / N[(N[(N[(l / t), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.80000000000000011e37 or 4.1999999999999997e-73 < t

   1. Initial program 65.4%

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

   2. Simplified 62.1%

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

      [Start] 65.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 65.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/l/ [<=] 65.4	\[ \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}} \]
      *-commutative [=>] 65.4	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]
      associate-*r/ [=>] 66.6	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k \cdot {t}^{3}}{\ell \cdot \ell}}} \]
      associate-/l* [=>] 65.5	\[ \frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\color{blue}{\frac{\sin k}{\frac{\ell \cdot \ell}{{t}^{3}}}}} \]
      associate-/r/ [=>] 56.4	\[ \color{blue}{\frac{\frac{2}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}}{\sin k} \cdot \frac{\ell \cdot \ell}{{t}^{3}}} \]

   3. Applied egg-rr 76.4%

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

      [Start] 62.1	\[ \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right) \]
      add-cube-cbrt [=>] 62.0	\[ \color{blue}{\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)} \cdot \sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)}\right) \cdot \sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)}} \]
      pow3 [=>] 62.0	\[ \color{blue}{{\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\frac{\ell}{{t}^{3}} \cdot \ell\right)}\right)}^{3}} \]
      cbrt-prod [=>] 62.0	\[ {\color{blue}{\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt[3]{\frac{\ell}{{t}^{3}} \cdot \ell}\right)}}^{3} \]
      associate-*l/ [=>] 56.3	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \sqrt[3]{\color{blue}{\frac{\ell \cdot \ell}{{t}^{3}}}}\right)}^{3} \]
      cbrt-div [=>] 57.3	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \color{blue}{\frac{\sqrt[3]{\ell \cdot \ell}}{\sqrt[3]{{t}^{3}}}}\right)}^{3} \]
      cbrt-unprod [<=] 64.3	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\color{blue}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]
      pow2 [=>] 64.3	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\color{blue}{{\left(\sqrt[3]{\ell}\right)}^{2}}}{\sqrt[3]{{t}^{3}}}\right)}^{3} \]
      rem-cbrt-cube [=>] 76.4	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\color{blue}{t}}\right)}^{3} \]

   4. Applied egg-rr 77.3%

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

      [Start] 76.4	\[ {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      cbrt-div [=>] 77.3	\[ {\left(\color{blue}{\frac{\sqrt[3]{2}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      div-inv [=>] 77.3	\[ {\left(\color{blue}{\left(\sqrt[3]{2} \cdot \frac{1}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}\right)} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]

   5. Simplified 77.3%

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

      [Start] 77.3	\[ {\left(\left(\sqrt[3]{2} \cdot \frac{1}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}\right) \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      associate-*r/ [=>] 77.3	\[ {\left(\color{blue}{\frac{\sqrt[3]{2} \cdot 1}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      *-rgt-identity [=>] 77.3	\[ {\left(\frac{\color{blue}{\sqrt[3]{2}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      *-commutative [=>] 77.3	\[ {\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      associate-*l* [=>] 77.3	\[ {\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\color{blue}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]

   6. Applied egg-rr 95.8%

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

      [Start] 77.3	\[ {\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      *-commutative [=>] 77.3	\[ {\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      cbrt-prod [=>] 95.8	\[ {\left(\frac{\sqrt[3]{2}}{\color{blue}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]

   7. Simplified 95.8%

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

      [Start] 95.8	\[ {\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt[3]{\sin k}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]
      *-commutative [=>] 95.8	\[ {\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}} \cdot \sqrt[3]{\sin k}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3} \]

   if -6.80000000000000011e37 < t < -1.00000000000000005e-181

   1. Initial program 45.4%

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

   2. Simplified 43.2%

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

      [Start] 45.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 45.4	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l/ [=>] 44.8	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k\right)} \]
      associate-*l/ [=>] 42.3	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-*r/ [=>] 43.9	\[ \frac{2}{\color{blue}{\frac{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right)}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 43.7	\[ \color{blue}{\frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right)} \cdot \left(\ell \cdot \ell\right)} \]

   3. Taylor expanded in t around 0 56.5%

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

   4. Simplified 74.6%

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

      [Start] 56.5	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 56.5	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 56.1	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 56.1	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 56.1	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 74.6	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]

   if -1.00000000000000005e-181 < t < 4.1999999999999997e-73

   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 5.6%

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

      [Start] 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* [=>] 5.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 5.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 21.0%

      \[\leadsto \frac{2}{\left(\color{blue}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      Proof

      [Start] 5.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      add-sqr-sqrt [=>] 5.6	\[ \frac{2}{\left(\color{blue}{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell}} \cdot \sqrt{\frac{{t}^{3}}{\ell \cdot \ell}}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      pow2 [=>] 5.6	\[ \frac{2}{\left(\color{blue}{{\left(\sqrt{\frac{{t}^{3}}{\ell \cdot \ell}}\right)}^{2}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      sqrt-div [=>] 5.6	\[ \frac{2}{\left({\color{blue}{\left(\frac{\sqrt{{t}^{3}}}{\sqrt{\ell \cdot \ell}}\right)}}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      sqrt-pow1 [=>] 17.2	\[ \frac{2}{\left({\left(\frac{\color{blue}{{t}^{\left(\frac{3}{2}\right)}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      metadata-eval [=>] 17.2	\[ \frac{2}{\left({\left(\frac{{t}^{\color{blue}{1.5}}}{\sqrt{\ell \cdot \ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      sqrt-prod [=>] 10.1	\[ \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\sqrt{\ell} \cdot \sqrt{\ell}}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      add-sqr-sqrt [<=] 21.0	\[ \frac{2}{\left({\left(\frac{{t}^{1.5}}{\color{blue}{\ell}}\right)}^{2} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. Taylor expanded in t around 0 58.8%

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

   5. Simplified 57.2%

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

      [Start] 58.8	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 56.0	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 56.0	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 57.2	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\frac{{\sin k}^{2}}{\frac{{\ell}^{2}}{t}}}} \]
      unpow2 [=>] 57.2	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{{\sin k}^{2}}{\frac{\color{blue}{\ell \cdot \ell}}{t}}} \]

   6. Taylor expanded in k around inf 58.8%

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

   7. Simplified 79.0%

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

      [Start] 58.8	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      unpow2 [=>] 58.8	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}} \]
      *-commutative [=>] 58.8	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\color{blue}{\left(\ell \cdot \ell\right) \cdot \cos k}}} \]
      unpow2 [=>] 58.8	\[ \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)}{\left(\ell \cdot \ell\right) \cdot \cos k}} \]
      *-commutative [=>] 58.8	\[ \frac{2}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot \left(k \cdot k\right)}}{\left(\ell \cdot \ell\right) \cdot \cos k}} \]
      times-frac [=>] 56.0	\[ \frac{2}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\ell \cdot \ell} \cdot \frac{k \cdot k}{\cos k}}} \]
      associate-/l* [=>] 57.2	\[ \frac{2}{\color{blue}{\frac{{\sin k}^{2}}{\frac{\ell \cdot \ell}{t}}} \cdot \frac{k \cdot k}{\cos k}} \]
      associate-*l/ [<=] 66.3	\[ \frac{2}{\frac{{\sin k}^{2}}{\color{blue}{\frac{\ell}{t} \cdot \ell}} \cdot \frac{k \cdot k}{\cos k}} \]
      associate-/l* [=>] 66.3	\[ \frac{2}{\frac{{\sin k}^{2}}{\frac{\ell}{t} \cdot \ell} \cdot \color{blue}{\frac{k}{\frac{\cos k}{k}}}} \]
      times-frac [<=] 70.3	\[ \frac{2}{\color{blue}{\frac{{\sin k}^{2} \cdot k}{\left(\frac{\ell}{t} \cdot \ell\right) \cdot \frac{\cos k}{k}}}} \]
      associate-/l* [=>] 71.9	\[ \frac{2}{\color{blue}{\frac{{\sin k}^{2}}{\frac{\left(\frac{\ell}{t} \cdot \ell\right) \cdot \frac{\cos k}{k}}{k}}}} \]
      associate-*l* [=>] 79.0	\[ \frac{2}{\frac{{\sin k}^{2}}{\frac{\color{blue}{\frac{\ell}{t} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}{k}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+37}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \cdot \sqrt[3]{\sin k}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -1 \cdot 10^{-181}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{{\sin k}^{2}}{\frac{\frac{\ell}{t} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}{k}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{2}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k} \cdot \sqrt[3]{\sin k}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.3%
Cost	52744

\[\begin{array}{l} t_1 := \tan k \cdot \sin k\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_4 := \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\\ \mathbf{if}\;k \leq -6.5 \cdot 10^{+127}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -7.5 \cdot 10^{-155}:\\ \;\;\;\;{\left(t_4 \cdot \left(\sqrt[3]{2} \cdot \sqrt[3]{\frac{\frac{1}{t_1}}{t_2}}\right)\right)}^{3}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{+117}:\\ \;\;\;\;{\left(t_4 \cdot \sqrt[3]{\frac{2}{t_2 \cdot t_1}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Accuracy	85.4%
Cost	46480

\[\begin{array}{l} t_1 := {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t} \cdot \sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \sin k\right)}}\right)}^{3}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -1.85 \cdot 10^{+126}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -9 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{-129}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	82.6%
Cost	40212

\[\begin{array}{l} t_1 := \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \sin k\right)} \cdot {\left(\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -2.2 \cdot 10^{+147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -4.8 \cdot 10^{+52}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \mathbf{elif}\;k \leq -2.75 \cdot 10^{-152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-118}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	80.5%
Cost	33940

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_2 := {\sin k}^{2}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -3 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-189}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_2}\right)\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\frac{\ell}{t} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}{k}}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+79}:\\ \;\;\;\;\ell \cdot \left(\ell \cdot \frac{\frac{2}{\tan k}}{\left(2 + t_3\right) \cdot \left(\sin k \cdot {t}^{3}\right)}\right)\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{+179}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(1 + \left(t_3 + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	79.4%
Cost	33808

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-182}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_2}\right)\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\frac{\ell}{t} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}{k}}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+60}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_2} \cdot \left(\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \frac{\ell}{{t}^{3}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	79.8%
Cost	27344

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -8.4 \cdot 10^{-184}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_2}\right)\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\frac{\ell}{t} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}{k}}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot 0.5\right)\right)} \cdot \left(\ell \cdot {t}^{-3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	77.8%
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -7 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.8 \cdot 10^{+52}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \mathbf{elif}\;k \leq -8.4 \cdot 10^{-156}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(\tan k \cdot \sin k\right) \cdot \left(t \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	74.8%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -2450 \lor \neg \left(t \leq 9.5 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \end{array} \]

Alternative 9
Accuracy	69.1%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-17} \lor \neg \left(t \leq 1.75 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \left(\frac{1}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right) + \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \]

Alternative 10
Accuracy	64.6%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \left(\frac{1}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right) + \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot {t}^{-3}\right)\\ \end{array} \]

Alternative 11
Accuracy	64.5%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-76}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \left(\frac{1}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right) + \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 12
Accuracy	64.9%
Cost	2121

\[\begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-14} \lor \neg \left(t \leq 1.7 \cdot 10^{-76}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \left(\frac{1}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right) + \frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k}\right)\\ \end{array} \]

Alternative 13
Accuracy	62.1%
Cost	1992

\[\begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{-14}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-99}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \frac{-0.16666666666666666}{k} + \frac{\frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	55.0%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{+15} \lor \neg \left(t \leq 2.1 \cdot 10^{-86}\right):\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot t}\\ \end{array} \]

Alternative 15
Accuracy	53.5%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{-214}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{k}\right)\\ \end{array} \]

Alternative 16
Accuracy	53.8%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3 \cdot 10^{-214}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 17
Accuracy	54.4%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 18
Accuracy	54.4%
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot t}}{t \cdot \frac{k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 19
Accuracy	46.1%
Cost	832

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

Reproduce

herbie shell --seed 2023146 
(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))))