Toniolo and Linder, Equation (10+)

Average Error: 32.2 → 9.7

Time: 51.8s

Precision: binary64

Cost: 52808

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_4 := \sqrt[3]{\tan k}\\ t_5 := t_4 \cdot \left(t \cdot \sin k\right)\\ t_6 := t \cdot t_4\\ \mathbf{if}\;t \leq -6.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{t_5 \cdot \left({\left(\frac{t_4}{\frac{\ell}{t}}\right)}^{2} \cdot t_2\right)}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-290}:\\ \;\;\;\;\frac{2}{t_5 \cdot \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \sqrt[3]{\frac{t_1}{{\cos k}^{2}}}\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t_1}{\ell}\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{t_6}{\ell} \cdot \frac{{t_6}^{2}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t_3}{\sqrt[3]{\ell}}\right)}^{2}}}{t_3}\\ \end{array} \]

FPCore

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

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0))
        (t_2 (+ 2.0 (pow (/ k t) 2.0)))
        (t_3 (* t (pow (cbrt k) 2.0)))
        (t_4 (cbrt (tan k)))
        (t_5 (* t_4 (* t (sin k))))
        (t_6 (* t t_4)))
   (if (<= t -6.4e-11)
     (/ 2.0 (* t_5 (* (pow (/ t_4 (/ l t)) 2.0) t_2)))
     (if (<= t -8e-290)
       (/ 2.0 (* t_5 (* (* (/ k l) (/ k l)) (cbrt (/ t_1 (pow (cos k) 2.0))))))
       (if (<= t 4e-157)
         (/ 2.0 (* (/ (* k k) (cos k)) (* (/ t l) (/ t_1 l))))
         (if (<= t 4.6e-29)
           (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_1))))
           (if (<= t 5.5e+143)
             (/ 2.0 (* t_2 (* (/ t_6 l) (/ (pow t_6 2.0) (/ l (sin k))))))
             (* (cbrt l) (/ (/ l (pow (/ t_3 (cbrt l)) 2.0)) t_3)))))))))

C

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

↓

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = 2.0 + pow((k / t), 2.0);
	double t_3 = t * pow(cbrt(k), 2.0);
	double t_4 = cbrt(tan(k));
	double t_5 = t_4 * (t * sin(k));
	double t_6 = t * t_4;
	double tmp;
	if (t <= -6.4e-11) {
		tmp = 2.0 / (t_5 * (pow((t_4 / (l / t)), 2.0) * t_2));
	} else if (t <= -8e-290) {
		tmp = 2.0 / (t_5 * (((k / l) * (k / l)) * cbrt((t_1 / pow(cos(k), 2.0)))));
	} else if (t <= 4e-157) {
		tmp = 2.0 / (((k * k) / cos(k)) * ((t / l) * (t_1 / l)));
	} else if (t <= 4.6e-29) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	} else if (t <= 5.5e+143) {
		tmp = 2.0 / (t_2 * ((t_6 / l) * (pow(t_6, 2.0) / (l / sin(k)))));
	} else {
		tmp = cbrt(l) * ((l / pow((t_3 / cbrt(l)), 2.0)) / t_3);
	}
	return tmp;
}

Java

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

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double t_2 = 2.0 + Math.pow((k / t), 2.0);
	double t_3 = t * Math.pow(Math.cbrt(k), 2.0);
	double t_4 = Math.cbrt(Math.tan(k));
	double t_5 = t_4 * (t * Math.sin(k));
	double t_6 = t * t_4;
	double tmp;
	if (t <= -6.4e-11) {
		tmp = 2.0 / (t_5 * (Math.pow((t_4 / (l / t)), 2.0) * t_2));
	} else if (t <= -8e-290) {
		tmp = 2.0 / (t_5 * (((k / l) * (k / l)) * Math.cbrt((t_1 / Math.pow(Math.cos(k), 2.0)))));
	} else if (t <= 4e-157) {
		tmp = 2.0 / (((k * k) / Math.cos(k)) * ((t / l) * (t_1 / l)));
	} else if (t <= 4.6e-29) {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_1)));
	} else if (t <= 5.5e+143) {
		tmp = 2.0 / (t_2 * ((t_6 / l) * (Math.pow(t_6, 2.0) / (l / Math.sin(k)))));
	} else {
		tmp = Math.cbrt(l) * ((l / Math.pow((t_3 / Math.cbrt(l)), 2.0)) / t_3);
	}
	return tmp;
}

Julia

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

↓

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_3 = Float64(t * (cbrt(k) ^ 2.0))
	t_4 = cbrt(tan(k))
	t_5 = Float64(t_4 * Float64(t * sin(k)))
	t_6 = Float64(t * t_4)
	tmp = 0.0
	if (t <= -6.4e-11)
		tmp = Float64(2.0 / Float64(t_5 * Float64((Float64(t_4 / Float64(l / t)) ^ 2.0) * t_2)));
	elseif (t <= -8e-290)
		tmp = Float64(2.0 / Float64(t_5 * Float64(Float64(Float64(k / l) * Float64(k / l)) * cbrt(Float64(t_1 / (cos(k) ^ 2.0))))));
	elseif (t <= 4e-157)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * k) / cos(k)) * Float64(Float64(t / l) * Float64(t_1 / l))));
	elseif (t <= 4.6e-29)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * t_1))));
	elseif (t <= 5.5e+143)
		tmp = Float64(2.0 / Float64(t_2 * Float64(Float64(t_6 / l) * Float64((t_6 ^ 2.0) / Float64(l / sin(k))))));
	else
		tmp = Float64(cbrt(l) * Float64(Float64(l / (Float64(t_3 / cbrt(l)) ^ 2.0)) / t_3));
	end
	return tmp
end

Wolfram

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

↓

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t * t$95$4), $MachinePrecision]}, If[LessEqual[t, -6.4e-11], N[(2.0 / N[(t$95$5 * N[(N[Power[N[(t$95$4 / N[(l / t), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -8e-290], N[(2.0 / N[(t$95$5 * N[(N[(N[(k / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[Power[N[(t$95$1 / N[Power[N[Cos[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4e-157], N[(2.0 / N[(N[(N[(k * k), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.6e-29], 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, 5.5e+143], N[(2.0 / N[(t$95$2 * N[(N[(t$95$6 / l), $MachinePrecision] * N[(N[Power[t$95$6, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[(l / N[Power[N[(t$95$3 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if t < -6.39999999999999987e-11

   1. Initial program 21.6

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

   2. Simplified 20.1

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

      [Start] 21.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 21.6	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 21.4	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 21.1	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 20.1	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 20.1	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 20.1	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 20.1	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 10.8

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

   4. Applied egg-rr 3.6

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

   5. Applied egg-rr 20.7

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

   6. Simplified 4.9

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

      [Start] 20.7	\[ \frac{2}{\left(\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{\sqrt[3]{\tan k}}{\ell} \cdot t\right)}^{2}\right) \cdot 2 + \left(\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{\sqrt[3]{\tan k}}{\ell} \cdot t\right)}^{2}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \]
      distribute-lft-in [<=] 5.7	\[ \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{\sqrt[3]{\tan k}}{\ell} \cdot t\right)}^{2}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 5.6	\[ \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left({\left(\frac{\sqrt[3]{\tan k}}{\ell} \cdot t\right)}^{2} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-/r/ [<=] 4.9	\[ \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left({\color{blue}{\left(\frac{\sqrt[3]{\tan k}}{\frac{\ell}{t}}\right)}}^{2} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   if -6.39999999999999987e-11 < t < -8.0000000000000006e-290

   1. Initial program 49.1

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

   2. Simplified 49.0

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

      [Start] 49.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 49.1	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 49.1	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 49.6	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 49.0	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 49.0	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 49.0	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 49.0	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 41.3

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

   4. Applied egg-rr 35.5

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

   5. Applied egg-rr 36.5

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

   6. Simplified 32.9

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

      [Start] 36.5	\[ \frac{2}{\left(\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{\sqrt[3]{\tan k}}{\ell} \cdot t\right)}^{2}\right) \cdot 2 + \left(\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{\sqrt[3]{\tan k}}{\ell} \cdot t\right)}^{2}\right) \cdot {\left(\frac{k}{t}\right)}^{2}} \]
      distribute-lft-in [<=] 36.5	\[ \frac{2}{\color{blue}{\left(\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot {\left(\frac{\sqrt[3]{\tan k}}{\ell} \cdot t\right)}^{2}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 32.9	\[ \frac{2}{\color{blue}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left({\left(\frac{\sqrt[3]{\tan k}}{\ell} \cdot t\right)}^{2} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-/r/ [<=] 32.9	\[ \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left({\color{blue}{\left(\frac{\sqrt[3]{\tan k}}{\frac{\ell}{t}}\right)}}^{2} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

   7. Taylor expanded in k around inf 26.5

      \[\leadsto \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \color{blue}{\left(\frac{{k}^{2}}{{\ell}^{2}} \cdot {\left(\frac{1 \cdot {\sin k}^{2}}{{\cos k}^{2}}\right)}^{0.3333333333333333}\right)}} \]

   8. Simplified 15.3

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

      [Start] 26.5	\[ \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{{k}^{2}}{{\ell}^{2}} \cdot {\left(\frac{1 \cdot {\sin k}^{2}}{{\cos k}^{2}}\right)}^{0.3333333333333333}\right)} \]
      unpow2 [=>] 26.5	\[ \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot {\left(\frac{1 \cdot {\sin k}^{2}}{{\cos k}^{2}}\right)}^{0.3333333333333333}\right)} \]
      unpow2 [=>] 26.5	\[ \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot {\left(\frac{1 \cdot {\sin k}^{2}}{{\cos k}^{2}}\right)}^{0.3333333333333333}\right)} \]
      times-frac [=>] 15.4	\[ \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot {\left(\frac{1 \cdot {\sin k}^{2}}{{\cos k}^{2}}\right)}^{0.3333333333333333}\right)} \]
      unpow1/3 [=>] 15.3	\[ \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\sqrt[3]{\frac{1 \cdot {\sin k}^{2}}{{\cos k}^{2}}}}\right)} \]
      *-lft-identity [=>] 15.3	\[ \frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \sqrt[3]{\frac{\color{blue}{{\sin k}^{2}}}{{\cos k}^{2}}}\right)} \]

   if -8.0000000000000006e-290 < t < 3.99999999999999977e-157

   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

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

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

   3. Taylor expanded in k around inf 28.2

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

   4. Simplified 21.1

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

      [Start] 28.2	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 31.0	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 31.0	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 31.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 31.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 21.1	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   if 3.99999999999999977e-157 < t < 4.59999999999999982e-29

   1. Initial program 42.0

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

   2. Simplified 42.2

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

      [Start] 42.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 42.0	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 42.0	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 43.0	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 42.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 42.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 42.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 42.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 26.9

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

   4. Simplified 14.0

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

      [Start] 26.9	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 26.9	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 27.0	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 27.0	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 27.0	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 14.0	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 14.0	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if 4.59999999999999982e-29 < t < 5.4999999999999997e143

   1. Initial program 22.5

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

   2. Simplified 17.5

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

      [Start] 22.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 20.9	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 20.3	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 17.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 17.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 17.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 17.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 7.6

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

   if 5.4999999999999997e143 < t

   1. Initial program 21.9

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

   2. Simplified 28.5

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

      [Start] 21.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 21.9	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 21.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 21.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 21.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 28.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 28.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 28.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 28.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in k around 0 28.5

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

   4. Simplified 18.4

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

      [Start] 28.5	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 28.5	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 25.0	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 25.0	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      associate-*l* [=>] 18.4	\[ \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   5. Applied egg-rr 6.6

      \[\leadsto \color{blue}{\frac{\frac{\ell}{{\left(\frac{{\left(\sqrt[3]{k}\right)}^{2} \cdot t}{\sqrt[3]{\ell}}\right)}^{2}}}{{\left(\sqrt[3]{k}\right)}^{2} \cdot t} \cdot \sqrt[3]{\ell}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 9.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left({\left(\frac{\sqrt[3]{\tan k}}{\frac{\ell}{t}}\right)}^{2} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-290}:\\ \;\;\;\;\frac{2}{\left(\sqrt[3]{\tan k} \cdot \left(t \cdot \sin k\right)\right) \cdot \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \sqrt[3]{\frac{{\sin k}^{2}}{{\cos k}^{2}}}\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-157}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t \cdot \sqrt[3]{\tan k}}{\ell} \cdot \frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{2}}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.0
Cost	66320

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \sqrt[3]{\tan k}\\ t_4 := t \cdot t_3\\ t_5 := t_3 \cdot \left(t \cdot \sin k\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{t_5 \cdot \left({\left(\frac{t_3}{\frac{\ell}{t}}\right)}^{2} \cdot t_2\right)}\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-291}:\\ \;\;\;\;\frac{2}{t_5 \cdot \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \sqrt[3]{\frac{t_1}{{\cos k}^{2}}}\right)}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t_1}{\ell}\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\left(\frac{\sqrt{t}}{\frac{\frac{\ell}{t_3}}{\sqrt{t}}} \cdot \left(\sin k \cdot t_4\right)\right) \cdot \frac{t_4}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	9.0
Cost	53392

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \sqrt[3]{\tan k}\\ t_4 := t_3 \cdot \left(t \cdot \sin k\right)\\ t_5 := t \cdot t_3\\ \mathbf{if}\;t \leq -1.36 \cdot 10^{-10}:\\ \;\;\;\;\frac{2}{t_4 \cdot \left({\left(\frac{t_3}{\frac{\ell}{t}}\right)}^{2} \cdot t_2\right)}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-289}:\\ \;\;\;\;\frac{2}{t_4 \cdot \left(\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \sqrt[3]{\frac{t_1}{{\cos k}^{2}}}\right)}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t_1}{\ell}\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{t_5}{\ell} \cdot \left(\left(\sin k \cdot t_5\right) \cdot \left(t \cdot \frac{t_3}{\ell}\right)\right)\right)}\\ \end{array} \]

Alternative 3
Error	10.5
Cost	46672

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\tan k}\\ t_3 := \frac{2}{\left(t_2 \cdot \left(t \cdot \sin k\right)\right) \cdot \left({\left(\frac{t_2}{\frac{\ell}{t}}\right)}^{2} \cdot t_1\right)}\\ t_4 := \frac{\ell}{\sin k}\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{-12}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\frac{{t}^{3}}{t_4} \cdot \frac{\tan k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+218}:\\ \;\;\;\;\frac{2}{t_1 \cdot {\left(\frac{t \cdot t_2}{\sqrt[3]{\ell \cdot t_4}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	9.7
Cost	46668

\[\begin{array}{l} t_1 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \sqrt[3]{\tan k}\\ t_4 := t \cdot t_3\\ \mathbf{if}\;t \leq -8 \cdot 10^{-14}:\\ \;\;\;\;\frac{2}{\left(t_3 \cdot \left(t \cdot \sin k\right)\right) \cdot \left({\left(\frac{t_3}{\frac{\ell}{t}}\right)}^{2} \cdot t_2\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-28}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{t_4}{\ell} \cdot \frac{{t_4}^{2}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t_1}{\sqrt[3]{\ell}}\right)}^{2}}}{t_1}\\ \end{array} \]

Alternative 5
Error	9.9
Cost	46348

\[\begin{array}{l} t_1 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_2 := \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{if}\;k \leq -2.9 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-160}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t_1}{\sqrt[3]{\ell}}\right)}^{2}}}{t_1}\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6
Error	8.5
Cost	46220

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := \tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ t_3 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;k \leq -1.45 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.85 \cdot 10^{-68}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{-\ell}{t}}{-t}}{t_2 \cdot \frac{t}{2}}\\ \mathbf{elif}\;k \leq 3.25 \cdot 10^{-102}:\\ \;\;\;\;\sqrt[3]{\ell} \cdot \frac{\frac{\ell}{{\left(\frac{t_3}{\sqrt[3]{\ell}}\right)}^{2}}}{t_3}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t \cdot t_2}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	9.2
Cost	26572

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := \tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ \mathbf{if}\;k \leq -2.15 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.9 \cdot 10^{-93}:\\ \;\;\;\;\frac{\frac{2}{t_2}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+57}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t \cdot t_2}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	10.5
Cost	21264

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.8 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(2 + t_2\right)\right)}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-261}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(\sqrt[3]{k} \cdot \left(t \cdot \sqrt[3]{k}\right)\right)}^{3}}{\ell}}\\ \mathbf{elif}\;k \leq 2.35 \cdot 10^{+61}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \frac{t \cdot \left(\sin k \cdot \frac{t}{\ell}\right)}{\frac{\ell}{t}}\right) \cdot \left(1 + \left(t_2 + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	10.8
Cost	21136

\[\begin{array}{l} t_1 := \frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_3 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -2.1 \cdot 10^{+75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -7 \cdot 10^{-103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 8.6 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\ell} \cdot {t_3}^{2}}}{t_3}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	10.7
Cost	21136

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := \tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ t_3 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -2.35 \cdot 10^{+73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -9.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{2}{t_2}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-150}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\ell} \cdot {t_3}^{2}}}{t_3}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+58}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{t \cdot t_2}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	15.1
Cost	20752

\[\begin{array}{l} t_1 := \frac{\cos k}{t \cdot {\sin k}^{2}}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot t_1\right)\\ t_3 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -2.7 \cdot 10^{+71}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -175000000000:\\ \;\;\;\;t_1 \cdot \frac{\ell \cdot \ell}{t \cdot t}\\ \mathbf{elif}\;k \leq -1.15 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{\frac{t}{\ell}} \cdot \left(\frac{2}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot k}\right)\\ \mathbf{elif}\;k \leq 0.0013:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\ell} \cdot {t_3}^{2}}}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	14.8
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := t \cdot \sqrt[3]{k}\\ t_3 := \frac{\cos k}{t \cdot t_1}\\ t_4 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot t_3\right)\\ \mathbf{if}\;k \leq -2.7 \cdot 10^{+71}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -350000000000:\\ \;\;\;\;t_3 \cdot \frac{\ell \cdot \ell}{t \cdot t}\\ \mathbf{elif}\;k \leq -5.3 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{t_1}{\ell}\right)}\\ \mathbf{elif}\;k \leq 0.00155:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\ell} \cdot {t_2}^{2}}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 13
Error	21.9
Cost	20360

\[\begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{-52}:\\ \;\;\;\;\frac{\ell}{k} \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 14
Error	21.4
Cost	20360

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 15
Error	21.7
Cost	20360

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{k}{\ell} \cdot {t_1}^{2}}}{t_1}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 16
Error	22.2
Cost	13572

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\ell}{k} \cdot {\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 17
Error	22.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\frac{k}{\ell} \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-84}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{\ell}{k}}{{t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 18
Error	22.9
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-53} \lor \neg \left(t \leq 3.2 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\frac{k}{\ell} \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{{k}^{4}}{\ell}}}{t}\\ \end{array} \]

Alternative 19
Error	23.7
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-52} \lor \neg \left(t \leq 3.3 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\frac{k}{\ell} \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 + \frac{2}{k \cdot k}}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]

Alternative 20
Error	25.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -5.9 \cdot 10^{-53} \lor \neg \left(t \leq 2.05 \cdot 10^{-67}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \end{array} \]

Alternative 21
Error	24.0
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-53} \lor \neg \left(t \leq 3.1 \cdot 10^{-84}\right):\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\frac{k}{\ell} \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right)\\ \end{array} \]

Alternative 22
Error	33.4
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+64}:\\ \;\;\;\;0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{t} \cdot \frac{-1}{k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]

Alternative 23
Error	33.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;\ell \leq -2 \cdot 10^{+63}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{-188}:\\ \;\;\;\;\frac{-0.3333333333333333}{k \cdot \frac{t \cdot k}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot k}\right)\\ \end{array} \]

Alternative 24
Error	34.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5 \cdot 10^{+63}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{-188}:\\ \;\;\;\;\frac{-0.3333333333333333}{k \cdot \frac{t \cdot k}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot -0.3333333333333333}{\frac{k}{\ell} \cdot \left(t \cdot k\right)}\\ \end{array} \]

Alternative 25
Error	35.4
Cost	704

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

Alternative 26
Error	35.4
Cost	704

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

Reproduce

herbie shell --seed 2023057 
(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))))