Toniolo and Linder, Equation (10+)

Average Error: 32.9 → 6.7

Time: 40.5s

Precision: binary64

Cost: 86032

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ t_3 := \frac{1}{{t_2}^{2}} \cdot \frac{2}{t_2}\\ t_4 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -9 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{t_4}{t_1}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-95}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.12 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 3200:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_4}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

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
         (*
          (cbrt (* (tan k) (* (sin k) (+ 2.0 (pow (/ k t) 2.0)))))
          (/ t (pow (cbrt l) 2.0))))
        (t_3 (* (/ 1.0 (pow t_2 2.0)) (/ 2.0 t_2)))
        (t_4 (/ (cos k) t)))
   (if (<= k -9e+100)
     (* 2.0 (/ (* (/ l k) (/ t_4 t_1)) (/ k l)))
     (if (<= k -3.2e-95)
       t_3
       (if (<= k 1.12e-130)
         (/ (* (/ l (* k t)) (/ l t)) (* k t))
         (if (<= k 3200.0)
           t_3
           (* 2.0 (/ (* (/ l k) t_4) (* t_1 (/ k l))))))))))

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 = cbrt((tan(k) * (sin(k) * (2.0 + pow((k / t), 2.0))))) * (t / pow(cbrt(l), 2.0));
	double t_3 = (1.0 / pow(t_2, 2.0)) * (2.0 / t_2);
	double t_4 = cos(k) / t;
	double tmp;
	if (k <= -9e+100) {
		tmp = 2.0 * (((l / k) * (t_4 / t_1)) / (k / l));
	} else if (k <= -3.2e-95) {
		tmp = t_3;
	} else if (k <= 1.12e-130) {
		tmp = ((l / (k * t)) * (l / t)) / (k * t);
	} else if (k <= 3200.0) {
		tmp = t_3;
	} else {
		tmp = 2.0 * (((l / k) * t_4) / (t_1 * (k / l)));
	}
	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.cbrt((Math.tan(k) * (Math.sin(k) * (2.0 + Math.pow((k / t), 2.0))))) * (t / Math.pow(Math.cbrt(l), 2.0));
	double t_3 = (1.0 / Math.pow(t_2, 2.0)) * (2.0 / t_2);
	double t_4 = Math.cos(k) / t;
	double tmp;
	if (k <= -9e+100) {
		tmp = 2.0 * (((l / k) * (t_4 / t_1)) / (k / l));
	} else if (k <= -3.2e-95) {
		tmp = t_3;
	} else if (k <= 1.12e-130) {
		tmp = ((l / (k * t)) * (l / t)) / (k * t);
	} else if (k <= 3200.0) {
		tmp = t_3;
	} else {
		tmp = 2.0 * (((l / k) * t_4) / (t_1 * (k / l)));
	}
	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(cbrt(Float64(tan(k) * Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))) * Float64(t / (cbrt(l) ^ 2.0)))
	t_3 = Float64(Float64(1.0 / (t_2 ^ 2.0)) * Float64(2.0 / t_2))
	t_4 = Float64(cos(k) / t)
	tmp = 0.0
	if (k <= -9e+100)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(t_4 / t_1)) / Float64(k / l)));
	elseif (k <= -3.2e-95)
		tmp = t_3;
	elseif (k <= 1.12e-130)
		tmp = Float64(Float64(Float64(l / Float64(k * t)) * Float64(l / t)) / Float64(k * t));
	elseif (k <= 3200.0)
		tmp = t_3;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * t_4) / Float64(t_1 * Float64(k / l))));
	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[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(1.0 / N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[k, -9e+100], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(t$95$4 / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -3.2e-95], t$95$3, If[LessEqual[k, 1.12e-130], N[(N[(N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3200.0], t$95$3, N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * t$95$4), $MachinePrecision] / N[(t$95$1 * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -9.00000000000000073e100

   1. Initial program 34.4

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

   2. Simplified 34.4

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

      [Start] 34.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 34.4	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 34.4	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 34.4	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 34.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 34.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 34.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 34.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 22.3

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

   4. Simplified 6.6

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

      [Start] 22.3	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 22.3	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 23.0	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 23.0	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 23.0	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 6.6	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 6.6	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 3.2

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

   if -9.00000000000000073e100 < k < -3.1999999999999997e-95 or 1.12e-130 < k < 3200

   1. Initial program 28.8

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

   2. Simplified 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] 28.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 28.8	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 28.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 28.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      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. Applied egg-rr 8.8

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

   if -3.1999999999999997e-95 < k < 1.12e-130

   1. Initial program 36.8

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

   2. Simplified 30.6

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

      [Start] 36.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 36.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 35.3	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 35.1	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 30.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 30.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 30.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 30.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around 0 54.3

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

   4. Simplified 53.7

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

      [Start] 54.3	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 54.3	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 53.7	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 53.7	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

   5. Applied egg-rr 51.3

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

   6. Applied egg-rr 50.9

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

   7. Applied egg-rr 7.2

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

   if 3200 < k

   1. Initial program 32.6

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

   2. Simplified 32.6

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

      [Start] 32.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 32.6	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 32.6	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 32.6	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 32.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 32.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 32.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 32.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 21.1

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

   4. Simplified 11.4

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

      [Start] 21.1	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 21.1	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 23.0	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 23.0	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 23.0	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 11.4	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 11.4	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 7.0

      \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-95}:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{elif}\;k \leq 1.12 \cdot 10^{-130}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 3200:\\ \;\;\;\;\frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{{\sin k}^{2} \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.6
Cost	85640

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ t_3 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -1.5 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{t_3}{t_1}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq -2.9 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{2}{t_2}}{{t_2}^{2}}\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{-152}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 3200:\\ \;\;\;\;\frac{2}{{t_2}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_3}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 2
Error	6.5
Cost	46480

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{3}}\\ t_3 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -2.1 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{t_3}{t_1}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq -9.5 \cdot 10^{-118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{elif}\;k \leq 3200:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_3}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 3
Error	8.5
Cost	39944

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -2.7 \cdot 10^{+100}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{t_2}{t_1}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-85}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;k \leq 0.00032:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 4
Error	8.5
Cost	39816

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -6.4 \cdot 10^{+99}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{t_2}{t_1}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-91}:\\ \;\;\;\;\frac{2}{{\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}\\ \mathbf{elif}\;k \leq 0.00165:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 5
Error	9.7
Cost	21000

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ t_2 := \sin k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+18}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t_2} \cdot \frac{\frac{1}{t}}{t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	8.7
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -125000000 \lor \neg \left(k \leq 7.5\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{{\sin k}^{2} \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 7
Error	8.7
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -16000000:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{t_2}{t_1}}{\frac{k}{\ell}}\\ \mathbf{elif}\;k \leq 0.00085:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 8
Error	11.3
Cost	14540

\[\begin{array}{l} t_1 := \cos \left(k \cdot -2\right)\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot \frac{\frac{-2}{-1 + t_1}}{t}\right)\right)\\ \mathbf{if}\;k \leq -4.1 \cdot 10^{+162}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -23000000:\\ \;\;\;\;2 \cdot \left(2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \left(k \cdot \left(1 - t_1\right)\right)\right)}\right)\\ \mathbf{elif}\;k \leq 0.58:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	11.3
Cost	14540

\[\begin{array}{l} t_1 := \cos \left(k \cdot -2\right)\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -4.1 \cdot 10^{+162}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{elif}\;k \leq -145000000:\\ \;\;\;\;2 \cdot \left(2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \left(k \cdot \left(1 - t_1\right)\right)\right)}\right)\\ \mathbf{elif}\;k \leq 0.7:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(\cos k \cdot \frac{\frac{-2}{-1 + t_1}}{t}\right)\right)\\ \end{array} \]

Alternative 10
Error	11.3
Cost	14540

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -4.1 \cdot 10^{+162}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{t_1}{2}\right)}\right)\\ \mathbf{elif}\;k \leq -8200000:\\ \;\;\;\;2 \cdot \left(2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \left(k \cdot \left(1 - \cos \left(k \cdot -2\right)\right)\right)\right)}\right)\\ \mathbf{elif}\;k \leq 0.002:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\frac{2 \cdot \cos k}{t}}{1 - t_1}\right)\\ \end{array} \]

Alternative 11
Error	17.5
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -220000000 \lor \neg \left(k \leq 14.2\right):\\ \;\;\;\;2 \cdot \left(2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t \cdot \left(k \cdot \left(k \cdot \left(1 - \cos \left(k \cdot -2\right)\right)\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 12
Error	8.8
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -9500000 \lor \neg \left(k \leq 0.021\right):\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{\frac{-1 + \cos \left(k + k\right)}{\frac{\cos k \cdot -2}{t}}}}{\frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 13
Error	18.6
Cost	14088

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{-22}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{k \cdot t}{t_1}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-68}:\\ \;\;\;\;2 \cdot \frac{\frac{1}{t}}{\frac{k}{\ell} \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 14
Error	19.2
Cost	7752

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{-99}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{k \cdot t}{t_1}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{t}}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 15
Error	19.9
Cost	1608

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{k \cdot t}{t_1}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-69}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{-0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 16
Error	20.2
Cost	1352

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{-24}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{k \cdot t}{t_1}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 17
Error	28.7
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{-14} \lor \neg \left(t \leq 8 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array} \]

Alternative 18
Error	26.3
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -2.8 \cdot 10^{-137} \lor \neg \left(k \leq 0.0056\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{k \cdot t}}}\\ \end{array} \]

Alternative 19
Error	26.5
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -1.22 \cdot 10^{-138} \lor \neg \left(k \leq 5 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot t}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 20
Error	23.2
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -700000 \lor \neg \left(t \leq 5.3 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]

Alternative 21
Error	23.2
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -1.12 \cdot 10^{+31} \lor \neg \left(t \leq 5.3 \cdot 10^{-154}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t} \cdot \frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]

Alternative 22
Error	22.6
Cost	1096

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{-101}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{k \cdot t}{t_1}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{k \cdot t}\\ \end{array} \]

Alternative 23
Error	35.0
Cost	832

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

Alternative 24
Error	30.1
Cost	832

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

Reproduce

herbie shell --seed 2023237 
(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))))