Toniolo and Linder, Equation (10+)

Average Error: 50.41% → 8.9%

Time: 46.8s

Precision: binary64

Cost: 52617

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-47} \lor \neg \left(t \leq 6.8 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\sin k}}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \end{array} \]

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
 (if (or (<= t -1.05e-47) (not (<= t 6.8e-106)))
   (/
    2.0
    (pow
     (/
      (cbrt (sin k))
      (/
       (/ (pow (cbrt l) 2.0) t)
       (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0))))))
     3.0))
   (/ 2.0 (/ (* t k) (* (/ l (pow (sin k) 2.0)) (* l (/ (cos k) k)))))))

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 tmp;
	if ((t <= -1.05e-47) || !(t <= 6.8e-106)) {
		tmp = 2.0 / pow((cbrt(sin(k)) / ((pow(cbrt(l), 2.0) / t) / cbrt((tan(k) * (2.0 + pow((k / t), 2.0)))))), 3.0);
	} else {
		tmp = 2.0 / ((t * k) / ((l / pow(sin(k), 2.0)) * (l * (cos(k) / k))));
	}
	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 tmp;
	if ((t <= -1.05e-47) || !(t <= 6.8e-106)) {
		tmp = 2.0 / Math.pow((Math.cbrt(Math.sin(k)) / ((Math.pow(Math.cbrt(l), 2.0) / t) / Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0)))))), 3.0);
	} else {
		tmp = 2.0 / ((t * k) / ((l / Math.pow(Math.sin(k), 2.0)) * (l * (Math.cos(k) / k))));
	}
	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)
	tmp = 0.0
	if ((t <= -1.05e-47) || !(t <= 6.8e-106))
		tmp = Float64(2.0 / (Float64(cbrt(sin(k)) / Float64(Float64((cbrt(l) ^ 2.0) / t) / cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(t * k) / Float64(Float64(l / (sin(k) ^ 2.0)) * Float64(l * Float64(cos(k) / k)))));
	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_] := If[Or[LessEqual[t, -1.05e-47], N[Not[LessEqual[t, 6.8e-106]], $MachinePrecision]], N[(2.0 / N[Power[N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] / N[(N[(N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision] / t), $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]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(t * k), $MachinePrecision] / N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e-47 or 6.79999999999999965e-106 < t

   1. Initial program 36.03

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

   2. Simplified 43.01

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

      [Start] 36.03	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 36.03	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 36.03	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 36.03	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 36.02	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 42.81	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 42.81	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 42.81	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 42.81	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 38.69

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

   4. Simplified 38.7

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

      [Start] 38.69	\[ \frac{2}{{\left(\sqrt[3]{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3}} \]
      associate-*r* [=>] 38.7	\[ \frac{2}{{\left(\sqrt[3]{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{t}{\sqrt[3]{\ell \cdot \ell}}\right)}^{3}} \]

   5. Applied egg-rr 6.68

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

   if -1.05e-47 < t < 6.79999999999999965e-106

   1. Initial program 91.05

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

   2. Simplified 91.22

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

      [Start] 91.05	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 91.05	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 91.05	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 91.05	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 91.01	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 91.19	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 91.19	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 91.19	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 91.19	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in t around 0 41.75

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

   4. Simplified 32.61

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

      [Start] 41.75	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 43.88	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 43.88	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 43.89	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 43.89	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 43.89	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 32.61	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 15.16

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

4. Final simplification 8.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-47} \lor \neg \left(t \leq 6.8 \cdot 10^{-106}\right):\\ \;\;\;\;\frac{2}{{\left(\frac{\sqrt[3]{\sin k}}{\frac{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.7%
Cost	46212

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+186}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\frac{\ell}{2 + t_1}}}\right)}^{3} \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	12.89%
Cost	40144

\[\begin{array}{l} t_1 := \frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\frac{\ell}{2 + {\left(\frac{k}{t}\right)}^{2}}}}\right)}^{3} \cdot \frac{\sin k}{\ell}}\\ t_2 := \frac{\frac{\ell}{t \cdot k}}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{+168}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -7.6 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	14.07%
Cost	40144

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq -2.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}} \cdot \frac{-2 - t_1}{-\ell}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+188}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\frac{\ell}{2 + t_1}}}\right)}^{3} \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 4
Error	13.96%
Cost	40144

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \frac{\frac{\ell}{t \cdot k}}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+207}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_3 + 1\right)\right)\right) \cdot {\left(t \cdot \frac{\sqrt[3]{t_1}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\frac{\ell}{2 + t_3}}}\right)}^{3} \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	15.13%
Cost	33808

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t \cdot k}}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}} \cdot \frac{-2 - t_2}{-\ell}}\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+191}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.43%
Cost	27608

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{\sin k}\\ t_3 := \frac{\ell}{t \cdot k}\\ t_4 := 2 + t_1\\ t_5 := \frac{2}{\frac{\tan k \cdot t_4}{t_2 \cdot \frac{\ell}{{t}^{3}}}}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+165}:\\ \;\;\;\;\frac{t_3}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq -3.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{\frac{t_4 \cdot \left(\tan k \cdot {t}^{3}\right)}{\ell}}{t_2}}\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-48}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;t_5\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \frac{t_3}{t}\\ \end{array} \]

Alternative 7
Error	15.86%
Cost	27344

\[\begin{array}{l} t_1 := \frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\ell}{\sin k} \cdot \frac{\ell}{{t}^{3}}}}\\ t_2 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{t_2}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-49}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \end{array} \]

Alternative 8
Error	15.62%
Cost	27344

\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}\right)}\\ t_2 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{t_2}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \end{array} \]

Alternative 9
Error	14.56%
Cost	27344

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq -8.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}} \cdot \frac{-2 - t_2}{-\ell}}\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-100}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k \cdot \left(\left(2 + t_2\right) \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \end{array} \]

Alternative 10
Error	16.84%
Cost	27212

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{2}{\left(\tan k \cdot \left(1 + \left(t_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+165}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq -4.1 \cdot 10^{+57}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\left(2 + t_2\right) \cdot \left({t}^{3} \cdot \left(\frac{\sin k}{\ell} \cdot \frac{\tan k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+103}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \end{array} \]

Alternative 11
Error	17.62%
Cost	21396

\[\begin{array}{l} t_1 := \frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)}\\ t_2 := \frac{\ell}{t \cdot k}\\ t_3 := \frac{t_2}{t}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+165}:\\ \;\;\;\;\frac{t_2}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-50}:\\ \;\;\;\;\frac{t_3}{t} \cdot \frac{\ell}{k}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_3\\ \end{array} \]

Alternative 12
Error	16.84%
Cost	20752

\[\begin{array}{l} t_1 := \ell \cdot \frac{\cos k}{k}\\ t_2 := \frac{\ell}{t \cdot k}\\ t_3 := {\sin k}^{2}\\ t_4 := \frac{2}{\frac{t \cdot k}{\frac{\ell}{t_3} \cdot t_1}}\\ \mathbf{if}\;k \leq -5.3 \cdot 10^{+166}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -1.6 \cdot 10^{+79}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t_3 \cdot \frac{t}{\ell}\right)}{t_1}}\\ \mathbf{elif}\;k \leq -5.3 \cdot 10^{-40}:\\ \;\;\;\;\frac{2}{\left(-k\right) \cdot \frac{t_3 \cdot \frac{\frac{t \cdot k}{\ell}}{\ell}}{-\cos k}}\\ \mathbf{elif}\;k \leq 2.55 \cdot 10^{-34}:\\ \;\;\;\;\frac{t_2}{\frac{t}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 13
Error	16.98%
Cost	20620

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := \frac{\cos k}{k}\\ t_3 := \frac{\ell}{{\sin k}^{2}}\\ t_4 := \frac{2}{\frac{t \cdot k}{t_3 \cdot \left(\ell \cdot t_2\right)}}\\ \mathbf{if}\;k \leq -2.8 \cdot 10^{+173}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -1.45 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{t}{\ell}}{t_3 \cdot t_2}}\\ \mathbf{elif}\;k \leq 2.55 \cdot 10^{-34}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 14
Error	17.28%
Cost	20489

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -5.3 \cdot 10^{-40} \lor \neg \left(k \leq 2.55 \cdot 10^{-34}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 15
Error	17.94%
Cost	14409

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -5.2 \cdot 10^{+39} \lor \neg \left(k \leq 0.00017\right):\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(\left(k \cdot \frac{t}{\ell}\right) \cdot \frac{1 - \cos \left(k + k\right)}{2 \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 16
Error	27.49%
Cost	14408

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -1.45 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \frac{t \cdot {k}^{2}}{\ell}}{\ell \cdot \cos k}}{\frac{1}{k}}}\\ \mathbf{elif}\;k \leq 0.00017:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(k \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\frac{\ell}{\frac{t}{\ell}}}\right)}\\ \end{array} \]

Alternative 17
Error	27.6%
Cost	7752

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-67}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-68}:\\ \;\;\;\;\frac{2}{\frac{k}{\cos k} \cdot \left(k \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 18
Error	27.61%
Cost	7752

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -1.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 19
Error	28.55%
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -1.18 \cdot 10^{-67}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 20
Error	33.01%
Cost	7176

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-59}:\\ \;\;\;\;\frac{{\left(\frac{\frac{\ell}{k}}{t}\right)}^{2}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 21
Error	36.2%
Cost	1224

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{-48}:\\ \;\;\;\;\frac{t_1}{t \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{t_1 \cdot \frac{-\ell}{k}}{t \cdot \left(-t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{t_1}}\\ \end{array} \]

Alternative 22
Error	45.56%
Cost	964

\[\begin{array}{l} \mathbf{if}\;t \leq 6200000:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k}}{k} \cdot \frac{\ell}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot k\right) \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \end{array} \]

Alternative 23
Error	47.26%
Cost	832

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

Alternative 24
Error	36.58%
Cost	832

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_1 \cdot \frac{t_1}{t} \end{array} \]

Alternative 25
Error	36.58%
Cost	832

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \frac{t_1}{\frac{t}{t_1}} \end{array} \]

Reproduce

herbie shell --seed 2023089 
(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))))