Toniolo and Linder, Equation (10+)

Average Error: 32.9 → 8.1

Time: 54.7s

Precision: binary64

Cost: 46540

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ t_3 := t \cdot \sqrt[3]{\tan k}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{2}{\left(\frac{{t_3}^{2}}{\ell} \cdot \left(\frac{t_3}{\ell} \cdot \sin k\right)\right) \cdot t_2}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_2 \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)) (t_2 (+ 2.0 t_1)) (t_3 (* t (cbrt (tan k)))))
   (if (<= t -5.2e-32)
     (/ 2.0 (* (* (/ (pow t_3 2.0) l) (* (/ t_3 l) (sin k))) t_2))
     (if (<= t 1.45e-58)
       (* 2.0 (/ (cos k) (* (/ k l) (/ (* (* t k) (pow (sin k) 2.0)) l))))
       (if (<= t 6.8e+102)
         (/ (* l (* 2.0 (/ (pow t -3.0) (tan k)))) (* t_2 (/ (sin k) l)))
         (/
          2.0
          (*
           (pow (/ (cbrt (sin k)) (* (cbrt l) (/ (cbrt l) t))) 3.0)
           (* (tan k) (+ 1.0 (+ t_1 1.0))))))))))

C

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

↓

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double t_2 = 2.0 + t_1;
	double t_3 = t * cbrt(tan(k));
	double tmp;
	if (t <= -5.2e-32) {
		tmp = 2.0 / (((pow(t_3, 2.0) / l) * ((t_3 / l) * sin(k))) * t_2);
	} else if (t <= 1.45e-58) {
		tmp = 2.0 * (cos(k) / ((k / l) * (((t * k) * pow(sin(k), 2.0)) / l)));
	} else if (t <= 6.8e+102) {
		tmp = (l * (2.0 * (pow(t, -3.0) / tan(k)))) / (t_2 * (sin(k) / l));
	} else {
		tmp = 2.0 / (pow((cbrt(sin(k)) / (cbrt(l) * (cbrt(l) / t))), 3.0) * (tan(k) * (1.0 + (t_1 + 1.0))));
	}
	return tmp;
}

Java

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

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double t_2 = 2.0 + t_1;
	double t_3 = t * Math.cbrt(Math.tan(k));
	double tmp;
	if (t <= -5.2e-32) {
		tmp = 2.0 / (((Math.pow(t_3, 2.0) / l) * ((t_3 / l) * Math.sin(k))) * t_2);
	} else if (t <= 1.45e-58) {
		tmp = 2.0 * (Math.cos(k) / ((k / l) * (((t * k) * Math.pow(Math.sin(k), 2.0)) / l)));
	} else if (t <= 6.8e+102) {
		tmp = (l * (2.0 * (Math.pow(t, -3.0) / Math.tan(k)))) / (t_2 * (Math.sin(k) / l));
	} else {
		tmp = 2.0 / (Math.pow((Math.cbrt(Math.sin(k)) / (Math.cbrt(l) * (Math.cbrt(l) / t))), 3.0) * (Math.tan(k) * (1.0 + (t_1 + 1.0))));
	}
	return tmp;
}

Julia

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

↓

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(2.0 + t_1)
	t_3 = Float64(t * cbrt(tan(k)))
	tmp = 0.0
	if (t <= -5.2e-32)
		tmp = Float64(2.0 / Float64(Float64(Float64((t_3 ^ 2.0) / l) * Float64(Float64(t_3 / l) * sin(k))) * t_2));
	elseif (t <= 1.45e-58)
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(k / l) * Float64(Float64(Float64(t * k) * (sin(k) ^ 2.0)) / l))));
	elseif (t <= 6.8e+102)
		tmp = Float64(Float64(l * Float64(2.0 * Float64((t ^ -3.0) / tan(k)))) / Float64(t_2 * Float64(sin(k) / l)));
	else
		tmp = Float64(2.0 / Float64((Float64(cbrt(sin(k)) / Float64(cbrt(l) * Float64(cbrt(l) / t))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(t_1 + 1.0)))));
	end
	return tmp
end

Wolfram

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

↓

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.2e-32], N[(2.0 / N[(N[(N[(N[Power[t$95$3, 2.0], $MachinePrecision] / l), $MachinePrecision] * N[(N[(t$95$3 / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e-58], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(N[(N[(t * k), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e+102], N[(N[(l * N[(2.0 * N[(N[Power[t, -3.0], $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.1999999999999995e-32

   1. Initial program 23.0

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

   2. Simplified 20.7

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

      [Start] 23.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 23.0	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 22.4	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 22.1	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 20.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 20.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 20.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 20.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 11.0

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

   4. Simplified 9.4

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

      [Start] 11.0	\[ \frac{2}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\ell} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\frac{\ell}{\sin k}}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      associate-/r/ [=>] 9.4	\[ \frac{2}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\ell} \cdot \color{blue}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\ell} \cdot \sin k\right)}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   if -5.1999999999999995e-32 < t < 1.44999999999999995e-58

   1. Initial program 54.5

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

   2. Simplified 55.3

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

      [Start] 54.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 54.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 54.5	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 54.5	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 55.2	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 55.5	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 55.3	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 55.3	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 55.3	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 55.3	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Taylor expanded in t around 0 25.7

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

   4. Simplified 25.8

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

      [Start] 25.7	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 25.8	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      unpow2 [=>] 25.8	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}} \]
      unpow2 [=>] 25.8	\[ 2 \cdot \frac{\cos k}{\frac{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot t\right)}{\color{blue}{\ell \cdot \ell}}} \]

   5. Applied egg-rr 8.3

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

   if 1.44999999999999995e-58 < t < 6.8000000000000001e102

   1. Initial program 21.3

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

   2. Simplified 15.6

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

      [Start] 21.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 21.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 21.3	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 19.7	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 19.5	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 19.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 15.6	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 15.6	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 15.6	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 15.6	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Applied egg-rr 8.1

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

   if 6.8000000000000001e102 < t

   1. Initial program 24.7

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

   2. Simplified 24.7

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

      [Start] 24.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 24.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 24.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 5.5

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

   4. Applied egg-rr 5.5

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

   5. Simplified 5.5

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

      [Start] 5.5	\[ \frac{2}{{\left(\frac{\frac{\sqrt[3]{\sin k}}{\frac{\sqrt[3]{\ell}}{t}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-/l/ [=>] 5.5	\[ \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-32}:\\ \;\;\;\;\frac{2}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\ell} \cdot \left(\frac{t \cdot \sqrt[3]{\tan k}}{\ell} \cdot \sin k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.0
Cost	46540

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ t_3 := \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -4.4 \cdot 10^{-32}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{t_3} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \sqrt[3]{\tan k \cdot t_2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 9.4 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_2 \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \end{array} \]

Alternative 2
Error	6.9
Cost	46476

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ t_3 := \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{t_3} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \sqrt[3]{\tan k \cdot t_2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_2 \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Alternative 3
Error	7.0
Cost	46476

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ t_3 := \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{t_3} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \sqrt[3]{\tan k \cdot t_2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_2 \cdot t_3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \end{array} \]

Alternative 4
Error	7.0
Cost	46281

\[\begin{array}{l} \mathbf{if}\;t \leq -1.55 \cdot 10^{-32} \lor \neg \left(t \leq 1.6 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{2}{{\left(\left(\sqrt[3]{\frac{\sin k}{\ell}} \cdot \frac{t}{\sqrt[3]{\ell}}\right) \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \end{array} \]

Alternative 5
Error	9.3
Cost	40268

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\sin k}{\ell}\\ t_3 := \frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot {\left(\sqrt[3]{t_2} \cdot \frac{1}{\frac{\sqrt[3]{\ell}}{t}}\right)}^{3}}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-31}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\left(2 + t_1\right) \cdot t_2}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Error	11.0
Cost	33676

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := 2 + t_2\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t_3\right) \cdot \frac{{\left(t \cdot \sqrt[3]{t_1}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 5.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_3 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]

Alternative 7
Error	11.1
Cost	33612

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{2}{\left(\tan k \cdot t_2\right) \cdot \frac{{\left(t \cdot \sqrt[3]{t_1}\right)}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-30}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_2 \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Error	12.3
Cost	27344

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ t_3 := \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-33}:\\ \;\;\;\;\frac{2 \cdot {t}^{-3}}{\frac{\tan k}{\frac{\ell}{t_2 \cdot t_3}}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+91}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{t_3}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 9
Error	11.3
Cost	27344

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \left(2 + t_1\right) \cdot \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+86}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq -5 \cdot 10^{-32}:\\ \;\;\;\;\frac{2 \cdot {t}^{-3}}{\frac{\tan k}{\frac{\ell}{t_2}}}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_2}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 10
Error	13.0
Cost	27212

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.32 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-58}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 11
Error	17.2
Cost	20624

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{\frac{t}{\ell} \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\ell}}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-24}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-209}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-287}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{\frac{1 - \cos \left(k + k\right)}{2} \cdot \left(k \cdot \left(t \cdot k\right)\right)}{\ell \cdot \ell}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-58}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 12
Error	13.3
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{t \cdot \left({\sin k}^{2} \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{if}\;k \leq -65:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{-32}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{+175}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t}{\ell} \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	10.7
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -11500 \lor \neg \left(k \leq 2.1 \cdot 10^{-32}\right):\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{k}{\ell} \cdot \frac{\left(t \cdot k\right) \cdot {\sin k}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \end{array} \]

Alternative 14
Error	18.1
Cost	19912

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{\frac{\frac{1 - \cos \left(k + k\right)}{2} \cdot \left(k \cdot \left(t \cdot k\right)\right)}{\ell \cdot \ell}}\\ \mathbf{if}\;k \leq -8500:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{\sqrt{k}}}{t \cdot \sqrt{k}}\right)\\ \end{array} \]

Alternative 15
Error	20.6
Cost	14672

\[\begin{array}{l} t_1 := 2 \cdot \frac{\cos k}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \frac{1 - \cos \left(k + k\right)}{2}\right)}{\ell \cdot \ell}}\\ \mathbf{if}\;k \leq -52:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-32}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{\ell}{k}}}{t}\right)}^{3} \cdot \frac{-\ell}{-k}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+16}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+155}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{\sqrt{k}}}{t \cdot \sqrt{k}}\right)\\ \end{array} \]

Alternative 16
Error	18.6
Cost	14408

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{t \cdot t}}{t \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-61}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{\frac{1 - \cos \left(k + k\right)}{2} \cdot \left(k \cdot \left(t \cdot k\right)\right)}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 17
Error	21.0
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{-26}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{t \cdot t}}{t \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{0.3333333333333333 + \frac{-2}{k \cdot k}}{\frac{k}{\ell} \cdot \left(t \cdot \frac{-k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 18
Error	22.0
Cost	1416

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{t \cdot t}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{t_1}{t \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{-59}:\\ \;\;\;\;\frac{0.3333333333333333 + \frac{-2}{k \cdot k}}{\frac{k}{\ell} \cdot \left(t \cdot \frac{-k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{-k} \cdot \left(t_1 \cdot \frac{-1}{t}\right)\\ \end{array} \]

Alternative 19
Error	24.0
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -1.18 \cdot 10^{-27} \lor \neg \left(t \leq 3.4 \cdot 10^{-81}\right):\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{t \cdot t}}{t \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 + \frac{\frac{2}{k}}{k}}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]

Alternative 20
Error	24.1
Cost	1352

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{t \cdot t}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{t_1}{t \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-72}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 + \frac{\frac{2}{k}}{k}}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{-k} \cdot \left(t_1 \cdot \frac{-1}{t}\right)\\ \end{array} \]

Alternative 21
Error	23.5
Cost	1352

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{t \cdot t}\\ \mathbf{if}\;t \leq -1.06 \cdot 10^{-26}:\\ \;\;\;\;\frac{t_1}{t \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-59}:\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{-0.3333333333333333 + \frac{2}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{-k} \cdot \left(t_1 \cdot \frac{-1}{t}\right)\\ \end{array} \]

Alternative 22
Error	22.3
Cost	1352

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{t \cdot t}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{t_1}{t \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-58}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{-k} \cdot \left(t_1 \cdot \frac{-1}{t}\right)\\ \end{array} \]

Alternative 23
Error	24.1
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-28} \lor \neg \left(t \leq 1.75 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{t \cdot t}}{t \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot k} \cdot \frac{2}{k \cdot \left(t \cdot k\right)}\\ \end{array} \]

Alternative 24
Error	24.2
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -4.2 \cdot 10^{-34} \lor \neg \left(t \leq 5.8 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{\frac{k}{\ell} \cdot \left(t \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k}}{t \cdot k}\\ \end{array} \]

Alternative 25
Error	24.3
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-34} \lor \neg \left(t \leq 4.5 \cdot 10^{-59}\right):\\ \;\;\;\;\frac{\frac{\frac{\ell}{k}}{t \cdot t}}{t \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k}}{t \cdot k}\\ \end{array} \]

Alternative 26
Error	35.2
Cost	704

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

Alternative 27
Error	34.4
Cost	704

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

Reproduce

herbie shell --seed 2023083 
(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))))