Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 7.9

Time: 43.2s

Precision: binary64

Cost: 65744

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{2}{{\left({\left(\sqrt[3]{\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ t_3 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-75}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-254}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 10^{-260}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{t_1}\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-8}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

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

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0))
        (t_2
         (/
          2.0
          (pow
           (*
            (pow (cbrt (/ (* t (cbrt (sin k))) (pow (cbrt l) 2.0))) 3.0)
            (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
           3.0)))
        (t_3 (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_1))))))
   (if (<= t -1.2e-75)
     t_2
     (if (<= t -2.4e-254)
       t_3
       (if (<= t 1e-260)
         (* 2.0 (* (/ (cos k) (* k k)) (/ (/ l (/ t l)) t_1)))
         (if (<= t 6.8e-8) t_3 t_2))))))

C

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

↓

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = 2.0 / pow((pow(cbrt(((t * cbrt(sin(k))) / pow(cbrt(l), 2.0))), 3.0) * cbrt((tan(k) * (2.0 + pow((k / t), 2.0))))), 3.0);
	double t_3 = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_1)));
	double tmp;
	if (t <= -1.2e-75) {
		tmp = t_2;
	} else if (t <= -2.4e-254) {
		tmp = t_3;
	} else if (t <= 1e-260) {
		tmp = 2.0 * ((cos(k) / (k * k)) * ((l / (t / l)) / t_1));
	} else if (t <= 6.8e-8) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

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

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double t_2 = 2.0 / Math.pow((Math.pow(Math.cbrt(((t * Math.cbrt(Math.sin(k))) / Math.pow(Math.cbrt(l), 2.0))), 3.0) * Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))))), 3.0);
	double t_3 = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_1)));
	double tmp;
	if (t <= -1.2e-75) {
		tmp = t_2;
	} else if (t <= -2.4e-254) {
		tmp = t_3;
	} else if (t <= 1e-260) {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * ((l / (t / l)) / t_1));
	} else if (t <= 6.8e-8) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

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

↓

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(2.0 / (Float64((cbrt(Float64(Float64(t * cbrt(sin(k))) / (cbrt(l) ^ 2.0))) ^ 3.0) * cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))) ^ 3.0))
	t_3 = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * t_1))))
	tmp = 0.0
	if (t <= -1.2e-75)
		tmp = t_2;
	elseif (t <= -2.4e-254)
		tmp = t_3;
	elseif (t <= 1e-260)
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(Float64(l / Float64(t / l)) / t_1)));
	elseif (t <= 6.8e-8)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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

↓

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[Power[N[(N[Power[N[Power[N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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, -1.2e-75], t$95$2, If[LessEqual[t, -2.4e-254], t$95$3, If[LessEqual[t, 1e-260], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[(t / l), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.8e-8], t$95$3, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2000000000000001e-75 or 6.8e-8 < t

   1. Initial program 22.7

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

   2. Simplified 22.6

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

      [Start] 22.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 22.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 22.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 7.4

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

   4. Applied egg-rr 7.4

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

   5. Simplified 7.4

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

      [Start] 7.4	\[ \frac{2}{{\left(\frac{\frac{\sqrt[3]{\sin k}}{\frac{\sqrt[3]{\ell}}{t}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-/l/ [=>] 7.4	\[ \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   6. Applied egg-rr 6.6

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

   7. Applied egg-rr 3.6

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

   if -1.2000000000000001e-75 < t < -2.40000000000000002e-254 or 9.99999999999999961e-261 < t < 6.8e-8

   1. Initial program 51.9

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

   2. Simplified 51.8

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

      [Start] 51.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 51.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 51.9	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 52.4	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 51.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 51.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 51.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 51.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 26.4

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

   4. Simplified 16.2

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

      [Start] 26.4	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 26.4	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 27.4	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 27.4	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 27.4	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 16.2	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 16.2	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if -2.40000000000000002e-254 < t < 9.99999999999999961e-261

   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

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

      [Start] 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	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 64.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 64.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 64.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 64.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 64.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 64.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 64.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 64.0

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

   4. Simplified 64.0

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

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

   5. Taylor expanded in t around 0 30.5

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

   6. Simplified 24.4

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

      [Start] 30.5	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 34.3	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 34.3	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 34.3	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{{\ell}^{2}}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]
      associate-/r* [=>] 32.6	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{\frac{{\ell}^{2}}{t}}{{\sin k}^{2}}}\right) \]
      unpow2 [=>] 32.6	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\color{blue}{\ell \cdot \ell}}{t}}{{\sin k}^{2}}\right) \]
      associate-/l* [=>] 24.4	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{{\sin k}^{2}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-254}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 10^{-260}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{{\sin k}^{2}}\right)\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-8}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({\left(\sqrt[3]{\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}}\right)}^{3} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	9.2
Cost	46348

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;k \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_2}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_2}\right)\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	9.1
Cost	46348

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;k \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.55 \cdot 10^{-139}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_2}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_2}\right)\\ \mathbf{elif}\;k \leq 5.1 \cdot 10^{+103}:\\ \;\;\;\;{\left(\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\tan k \cdot \left(\sin k \cdot 0.5\right)\right)}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	10.1
Cost	46216

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;k \leq -5 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-107}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_2}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_2}\right)\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	9.8
Cost	40140

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -1.6 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 8.3 \cdot 10^{-109}:\\ \;\;\;\;{\left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.65 \cdot 10^{+101}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	9.7
Cost	39948

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -6 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{-61}:\\ \;\;\;\;{\left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	10.2
Cost	32840

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.3 \cdot 10^{-20}:\\ \;\;\;\;{\left(\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)}^{3}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	12.0
Cost	26440

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -1.1 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	12.0
Cost	26440

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -7.2 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{{\left(\sqrt[3]{k}\right)}^{2}}{\frac{\sqrt[3]{\ell}}{t}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 2.02 \cdot 10^{+46}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.2
Cost	20884

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+75}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-254}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-261}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{t_1}\right)\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 10
Error	16.2
Cost	20884

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+74}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t \leq -1.1 \cdot 10^{-252}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{-261}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{t_1}\right)\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{+33}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 11
Error	16.3
Cost	20620

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 12
Error	20.3
Cost	19908

\[\begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+74}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -1.4 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-102}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 13
Error	20.3
Cost	19908

\[\begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}{\ell}}\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-98}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-102}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 14
Error	20.6
Cost	14600

\[\begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{1}{\frac{k}{\ell} \cdot \left(t \cdot \left(t \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-97}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\frac{k}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-102}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 15
Error	21.2
Cost	13896

\[\begin{array}{l} t_1 := \ell \cdot {k}^{-2}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-94}:\\ \;\;\;\;\frac{1}{\frac{k}{\ell} \cdot \left(t \cdot \left(t \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-103}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 16
Error	21.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{\frac{k}{\ell} \cdot \left(t \cdot \left(t \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-102}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{1}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 17
Error	22.6
Cost	1352

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{1}{\frac{k}{\ell} \cdot \left(t \cdot \left(t \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{1}{t}}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot k}\right)\\ \end{array} \]

Alternative 18
Error	24.1
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-94} \lor \neg \left(t \leq 1.55 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 19
Error	23.5
Cost	1225

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-94} \lor \neg \left(t \leq 1.55 \cdot 10^{-102}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot t_1}{t}\\ \end{array} \]

Alternative 20
Error	23.4
Cost	1224

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-94}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-104}:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot k}\right)\\ \end{array} \]

Alternative 21
Error	22.8
Cost	1224

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-94}:\\ \;\;\;\;\frac{1}{\frac{k}{\ell} \cdot \left(t \cdot \left(t \cdot k\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-107}:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t \cdot k}\right)\\ \end{array} \]

Alternative 22
Error	28.8
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-95} \lor \neg \left(t \leq 6.2 \cdot 10^{-105}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternative 23
Error	28.8
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{-95} \lor \neg \left(t \leq 3.9 \cdot 10^{-103}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array} \]

Alternative 24
Error	29.3
Cost	832

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

Reproduce

herbie shell --seed 2023056 
(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))))