Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 7.9

Time: 48.7s

Precision: binary64

Cost: 52616

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := {\sin k}^{2} \cdot t\\ \mathbf{if}\;k \leq -1.3 \cdot 10^{+187}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_2}{t_1}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\tan k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_2}\right)\\ \end{array} \]

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 (* (/ l k) (/ l k))) (t_2 (* (pow (sin k) 2.0) t)))
   (if (<= k -1.3e+187)
     (* 2.0 (/ (cos k) (/ t_2 t_1)))
     (if (<= k 2.1e+57)
       (/
        2.0
        (pow
         (/
          (cbrt (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))
          (/ (/ (pow (cbrt l) 2.0) t) (cbrt (tan k))))
         3.0))
       (* 2.0 (* t_1 (/ (cos k) t_2)))))))

C

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

↓

double code(double t, double l, double k) {
	double t_1 = (l / k) * (l / k);
	double t_2 = pow(sin(k), 2.0) * t;
	double tmp;
	if (k <= -1.3e+187) {
		tmp = 2.0 * (cos(k) / (t_2 / t_1));
	} else if (k <= 2.1e+57) {
		tmp = 2.0 / pow((cbrt((sin(k) * (2.0 + pow((k / t), 2.0)))) / ((pow(cbrt(l), 2.0) / t) / cbrt(tan(k)))), 3.0);
	} else {
		tmp = 2.0 * (t_1 * (cos(k) / 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 = (l / k) * (l / k);
	double t_2 = Math.pow(Math.sin(k), 2.0) * t;
	double tmp;
	if (k <= -1.3e+187) {
		tmp = 2.0 * (Math.cos(k) / (t_2 / t_1));
	} else if (k <= 2.1e+57) {
		tmp = 2.0 / Math.pow((Math.cbrt((Math.sin(k) * (2.0 + Math.pow((k / t), 2.0)))) / ((Math.pow(Math.cbrt(l), 2.0) / t) / Math.cbrt(Math.tan(k)))), 3.0);
	} else {
		tmp = 2.0 * (t_1 * (Math.cos(k) / 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 = Float64(Float64(l / k) * Float64(l / k))
	t_2 = Float64((sin(k) ^ 2.0) * t)
	tmp = 0.0
	if (k <= -1.3e+187)
		tmp = Float64(2.0 * Float64(cos(k) / Float64(t_2 / t_1)));
	elseif (k <= 2.1e+57)
		tmp = Float64(2.0 / (Float64(cbrt(Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))) / Float64(Float64((cbrt(l) ^ 2.0) / t) / cbrt(tan(k)))) ^ 3.0));
	else
		tmp = Float64(2.0 * Float64(t_1 * Float64(cos(k) / 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[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[k, -1.3e+187], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.1e+57], N[(2.0 / N[Power[N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.2999999999999999e187

   1. Initial program 35.2

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

   2. Simplified 35.2

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

      [Start] 35.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 35.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 35.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 24.4

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

   4. Simplified 5.5

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

      [Start] 24.4	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 24.4	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      *-commutative [=>] 24.4	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]
      associate-/l* [=>] 24.4	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]
      unpow2 [=>] 24.4	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]
      unpow2 [=>] 24.4	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}} \]
      times-frac [=>] 5.5	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]

   if -1.2999999999999999e187 < k < 2.09999999999999991e57

   1. Initial program 31.3

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

   2. Simplified 37.0

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

      [Start] 31.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 31.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 31.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 31.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 31.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 36.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 36.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 36.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 36.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 22.3

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

   4. Applied egg-rr 8.0

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

   if 2.09999999999999991e57 < k

   1. Initial program 33.4

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

   2. Simplified 33.4

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

      [Start] 33.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 33.4	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 33.4	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 33.4	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 33.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 33.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 33.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 33.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 20.9

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

   4. Simplified 9.1

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

      [Start] 20.9	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 20.9	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 22.8	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 22.8	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 22.8	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 9.1	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 9.1	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.3 \cdot 10^{+187}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\tan k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	21268

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := {\sin k}^{2}\\ t_3 := t_2 \cdot t\\ \mathbf{if}\;k \leq -1.4 \cdot 10^{+179}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \mathbf{elif}\;k \leq -7.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(k \cdot \frac{k}{\cos k}\right) \cdot \left(t_2 \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq -7.5 \cdot 10^{-241}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)}\\ \mathbf{elif}\;k \leq 4.4 \cdot 10^{-127}:\\ \;\;\;\;\frac{1}{\left(k \cdot 2\right) \cdot \frac{\left(t \cdot \tan k\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}{2}}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+58}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{2}{t \cdot \frac{t}{\ell}}}{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \end{array} \]

Alternative 2
Error	9.7
Cost	21136

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \left(\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \tan k\right)\right)}{\frac{\ell}{t}}}\\ t_3 := t \cdot \sqrt[3]{k}\\ t_4 := {\sin k}^{2} \cdot t\\ \mathbf{if}\;k \leq -1.4 \cdot 10^{+186}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_1}}\\ \mathbf{elif}\;k \leq -2.6 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{-157}:\\ \;\;\;\;\frac{\ell}{t_3 \cdot \frac{k}{\frac{\ell}{{t_3}^{2}}}}\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{+57}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_4}\right)\\ \end{array} \]

Alternative 3
Error	8.6
Cost	21001

\[\begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-32} \lor \neg \left(t \leq 5 \cdot 10^{-104}\right):\\ \;\;\;\;\frac{\frac{\frac{1}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\frac{t}{\ell}} \cdot \frac{\frac{2}{t \cdot \tan k}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(k \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 4
Error	15.5
Cost	20884

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)\\ t_3 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1 \cdot t}\right)\\ \mathbf{if}\;k \leq -1.14 \cdot 10^{+179}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -6 \cdot 10^{-85}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{t_1}\right)\\ \mathbf{elif}\;k \leq -6.5 \cdot 10^{-244}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_2}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\left(k \cdot 2\right) \cdot \frac{\left(t \cdot \tan k\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}{2}}\\ \mathbf{elif}\;k \leq 7000000000:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 5
Error	15.5
Cost	20884

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)\\ t_3 := {\sin k}^{2}\\ t_4 := t_3 \cdot t\\ \mathbf{if}\;k \leq -1.14 \cdot 10^{+179}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_1}}\\ \mathbf{elif}\;k \leq -5.6 \cdot 10^{-85}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{t_3}\right)\\ \mathbf{elif}\;k \leq -5.5 \cdot 10^{-241}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_2}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\left(k \cdot 2\right) \cdot \frac{\left(t \cdot \tan k\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}{2}}\\ \mathbf{elif}\;k \leq 10500000000:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_4}\right)\\ \end{array} \]

Alternative 6
Error	13.8
Cost	20884

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)\\ t_3 := {\sin k}^{2}\\ t_4 := t_3 \cdot t\\ \mathbf{if}\;k \leq -2 \cdot 10^{+179}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_1}}\\ \mathbf{elif}\;k \leq -8.4 \cdot 10^{-29}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(k \cdot \frac{k}{\cos k}\right) \cdot \left(t_3 \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq -6.7 \cdot 10^{-242}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_2}\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\left(k \cdot 2\right) \cdot \frac{\left(t \cdot \tan k\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}{2}}\\ \mathbf{elif}\;k \leq 90000000000:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_4}\right)\\ \end{array} \]

Alternative 7
Error	13.9
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\\ t_2 := t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)\\ \mathbf{if}\;k \leq -3.6 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.25 \cdot 10^{-243}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_2}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\left(k \cdot 2\right) \cdot \frac{\left(t \cdot \tan k\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}{2}}\\ \mathbf{elif}\;k \leq 200000000000:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.4
Cost	14804

\[\begin{array}{l} t_1 := \frac{\frac{\cos k}{k}}{k} \cdot \left(2 \cdot \frac{\ell \cdot \ell}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ t_2 := t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)\\ \mathbf{if}\;k \leq -1.24 \cdot 10^{+148}:\\ \;\;\;\;-0.3333333333333333 \cdot \left|\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right|\\ \mathbf{elif}\;k \leq -0.0006:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -8.2 \cdot 10^{-244}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_2}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\left(k \cdot 2\right) \cdot \frac{\left(t \cdot \tan k\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}{2}}\\ \mathbf{elif}\;k \leq 4200000000:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	20.4
Cost	14804

\[\begin{array}{l} t_1 := \frac{\frac{\cos k}{k}}{k}\\ t_2 := t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)\\ t_3 := 0.5 - \frac{\cos \left(k + k\right)}{2}\\ \mathbf{if}\;k \leq -5.4 \cdot 10^{+147}:\\ \;\;\;\;-0.3333333333333333 \cdot \left|\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right|\\ \mathbf{elif}\;k \leq -0.0028:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\frac{\ell \cdot \ell}{t}}{t_3}\right)\\ \mathbf{elif}\;k \leq -7.2 \cdot 10^{-244}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_2}\\ \mathbf{elif}\;k \leq 1.12 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\left(k \cdot 2\right) \cdot \frac{\left(t \cdot \tan k\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}{2}}\\ \mathbf{elif}\;k \leq 420000000000:\\ \;\;\;\;\frac{2}{\frac{t_2}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell \cdot \ell}{t \cdot t_3}\right)\\ \end{array} \]

Alternative 10
Error	22.8
Cost	14220

\[\begin{array}{l} t_1 := t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)\\ \mathbf{if}\;k \leq -8.7 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{\cos k}{k}}{k} \cdot \left(2 \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}\right)\\ \mathbf{elif}\;k \leq -7.2 \cdot 10^{-243}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_1}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-185}:\\ \;\;\;\;\frac{1}{\left(k \cdot 2\right) \cdot \frac{\left(t \cdot \tan k\right) \cdot {\left(\frac{t}{\ell}\right)}^{2}}{2}}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \end{array} \]

Alternative 11
Error	22.6
Cost	8016

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)}\\ \mathbf{if}\;k \leq -2.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\cos k}{k}}{k} \cdot \left(2 \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}\right)\\ \mathbf{elif}\;k \leq -1.9 \cdot 10^{-242}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 7.4 \cdot 10^{-196}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \end{array} \]

Alternative 12
Error	22.6
Cost	8016

\[\begin{array}{l} t_1 := t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)\\ \mathbf{if}\;k \leq -9 \cdot 10^{-29}:\\ \;\;\;\;\frac{\frac{\cos k}{k}}{k} \cdot \left(2 \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot t\right)}{\ell}}\right)\\ \mathbf{elif}\;k \leq -7.8 \cdot 10^{-243}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t_1}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{-202}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\frac{t_1}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \end{array} \]

Alternative 13
Error	21.6
Cost	7884

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\left(\sin k \cdot \frac{t}{\ell}\right) \cdot \left(k + k\right)\right)}\\ \mathbf{if}\;t \leq -2.25 \cdot 10^{-20}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-105}:\\ \;\;\;\;\frac{\ell}{\frac{t}{\ell}} \cdot \left(\frac{2}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot k}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	24.4
Cost	7688

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\frac{\ell}{k \cdot {t}^{3}}}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-102}:\\ \;\;\;\;\frac{\ell}{\frac{t}{\ell}} \cdot \left(\frac{2}{{k}^{4}} + \frac{-0.3333333333333333}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 15
Error	25.1
Cost	7305

\[\begin{array}{l} \mathbf{if}\;k \leq -5.2 \cdot 10^{+23} \lor \neg \left(k \leq 6 \cdot 10^{+27}\right):\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \end{array} \]

Alternative 16
Error	25.1
Cost	7304

\[\begin{array}{l} \mathbf{if}\;k \leq -7.2 \cdot 10^{+22}:\\ \;\;\;\;-0.3333333333333333 \cdot \left|\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right|\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \end{array} \]

Alternative 17
Error	28.3
Cost	1616

\[\begin{array}{l} t_1 := \frac{2}{\left(t \cdot t\right) \cdot \left(2 \cdot \frac{k \cdot k}{\ell \cdot \frac{\ell}{t}}\right)}\\ \mathbf{if}\;k \leq -2.7 \cdot 10^{+68}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-168}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 2.95 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \end{array} \]

Alternative 18
Error	28.1
Cost	1616

\[\begin{array}{l} t_1 := \frac{2}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{+68}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-168}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{+27}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \end{array} \]

Alternative 19
Error	27.8
Cost	1616

\[\begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{+68}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq -1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}}{\ell} \cdot \left(t \cdot t\right)}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-168}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{+28}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(2 \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \end{array} \]

Alternative 20
Error	31.6
Cost	1352

\[\begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-168}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{2}{\left(t \cdot t\right) \cdot \left(2 \cdot \frac{t}{\frac{\ell \cdot \ell}{k \cdot k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{k}{\ell} \cdot \frac{k \cdot t}{\ell}}\\ \end{array} \]

Alternative 21
Error	37.1
Cost	704

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

Alternative 22
Error	35.8
Cost	704

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

Alternative 23
Error	33.9
Cost	704

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

Reproduce

herbie shell --seed 2023047 
(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))))