Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 4.9

Time: 43.0s

Precision: binary64

Cost: 53257

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := t \cdot t_1\\ \mathbf{if}\;t \leq -1.62 \cdot 10^{-21} \lor \neg \left(t \leq 9.5 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot t_1\right) \cdot \frac{t_2}{\frac{1}{\sin k}}\right) \cdot \frac{t_2}{\ell}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \end{array} \]

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 (cbrt (tan k))) (t_2 (* t t_1)))
   (if (or (<= t -1.62e-21) (not (<= t 9.5e-22)))
     (/
      2.0
      (*
       (* (* (* (/ t l) t_1) (/ t_2 (/ 1.0 (sin k)))) (/ t_2 l))
       (+ 2.0 (pow (/ k t) 2.0))))
     (* (* (/ 2.0 k) (/ l (pow (sin k) 2.0))) (/ (* l (cos k)) (* t 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 t_1 = cbrt(tan(k));
	double t_2 = t * t_1;
	double tmp;
	if ((t <= -1.62e-21) || !(t <= 9.5e-22)) {
		tmp = 2.0 / (((((t / l) * t_1) * (t_2 / (1.0 / sin(k)))) * (t_2 / l)) * (2.0 + pow((k / t), 2.0)));
	} else {
		tmp = ((2.0 / k) * (l / pow(sin(k), 2.0))) * ((l * cos(k)) / (t * 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 t_1 = Math.cbrt(Math.tan(k));
	double t_2 = t * t_1;
	double tmp;
	if ((t <= -1.62e-21) || !(t <= 9.5e-22)) {
		tmp = 2.0 / (((((t / l) * t_1) * (t_2 / (1.0 / Math.sin(k)))) * (t_2 / l)) * (2.0 + Math.pow((k / t), 2.0)));
	} else {
		tmp = ((2.0 / k) * (l / Math.pow(Math.sin(k), 2.0))) * ((l * Math.cos(k)) / (t * 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)
	t_1 = cbrt(tan(k))
	t_2 = Float64(t * t_1)
	tmp = 0.0
	if ((t <= -1.62e-21) || !(t <= 9.5e-22))
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(Float64(t / l) * t_1) * Float64(t_2 / Float64(1.0 / sin(k)))) * Float64(t_2 / l)) * Float64(2.0 + (Float64(k / t) ^ 2.0))));
	else
		tmp = Float64(Float64(Float64(2.0 / k) * Float64(l / (sin(k) ^ 2.0))) * Float64(Float64(l * cos(k)) / Float64(t * 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_] := Block[{t$95$1 = N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(t * t$95$1), $MachinePrecision]}, If[Or[LessEqual[t, -1.62e-21], N[Not[LessEqual[t, 9.5e-22]], $MachinePrecision]], N[(2.0 / N[(N[(N[(N[(N[(t / l), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(t$95$2 / N[(1.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / k), $MachinePrecision] * N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.62000000000000003e-21 or 9.4999999999999994e-22 < t

   1. Initial program 22.4

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

   2. Simplified 20.5

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

      [Start] 22.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.4	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 21.9	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 21.6	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 20.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 20.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 20.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 20.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 10.5

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

   4. Applied egg-rr 4.1

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

   if -1.62000000000000003e-21 < t < 9.4999999999999994e-22

   1. Initial program 52.1

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

   2. Simplified 52.2

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

      [Start] 52.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 52.1	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 52.1	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 52.6	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 52.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 52.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 52.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 52.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 26.6

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

   4. Simplified 22.5

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

      [Start] 26.6	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 28.0	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 28.0	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 28.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 28.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 22.5	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 11.3

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

   6. Simplified 12.5

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

      [Start] 11.3	\[ \frac{2}{\frac{k \cdot \left(-t\right)}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}} \]
      times-frac [=>] 5.5	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}}} \]
      associate-/r/ [=>] 5.5	\[ \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right)} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}} \]
      associate-*l/ [=>] 5.5	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{-t}{\color{blue}{\frac{\cos k \cdot \left(-\ell\right)}{k}}}} \]
      distribute-rgt-neg-out [=>] 5.5	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{-t}{\frac{\color{blue}{-\cos k \cdot \ell}}{k}}} \]
      distribute-lft-neg-out [<=] 5.5	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{-t}{\frac{\color{blue}{\left(-\cos k\right) \cdot \ell}}{k}}} \]
      associate-/r/ [=>] 12.5	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \color{blue}{\left(\frac{-t}{\left(-\cos k\right) \cdot \ell} \cdot k\right)}} \]
      *-commutative [=>] 12.5	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(\frac{-t}{\color{blue}{\ell \cdot \left(-\cos k\right)}} \cdot k\right)} \]

   7. Applied egg-rr 6.5

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

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.62 \cdot 10^{-21} \lor \neg \left(t \leq 9.5 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{2}{\left(\left(\left(\frac{t}{\ell} \cdot \sqrt[3]{\tan k}\right) \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\frac{1}{\sin k}}\right) \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\ell}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.1
Cost	53261

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+237}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-21} \lor \neg \left(t \leq 6 \cdot 10^{-25}\right):\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t_1}{\ell} \cdot \left(t_1 \cdot \left(t_1 \cdot \frac{\sin k}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \end{array} \]

Alternative 2
Error	9.3
Cost	46536

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := t \cdot t_1\\ \mathbf{if}\;t \leq -7.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t_1 \cdot \sin k\right)\right) \cdot \left(t_2 \cdot {\left(\frac{1}{\frac{\frac{\ell}{t}}{t_1}}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{t_3}{\ell} \cdot \frac{{t_3}^{2}}{\frac{\ell}{\sin k}}\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-21}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_2 \cdot \frac{\sin k}{\ell}}\\ \end{array} \]

Alternative 3
Error	9.4
Cost	46404

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\tan k}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t_2 \cdot \sin k\right)\right) \cdot \left(t_1 \cdot {\left(\frac{1}{\frac{\frac{\ell}{t}}{t_2}}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{-22} \lor \neg \left(t \leq 8.2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_1 \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \end{array} \]

Alternative 4
Error	9.8
Cost	46276

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{2}{t \cdot \left(t_2 \cdot \left(\left(t_1 \cdot \sin k\right) \cdot {\left(t \cdot \frac{t_1}{\ell}\right)}^{2}\right)\right)}\\ \mathbf{elif}\;t \leq -4 \cdot 10^{-27} \lor \neg \left(t \leq 5.8 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_2 \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \end{array} \]

Alternative 5
Error	9.4
Cost	46276

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\tan k}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+103}:\\ \;\;\;\;\frac{2}{\left(t \cdot \left(t_2 \cdot \sin k\right)\right) \cdot \left(t_1 \cdot {\left(\frac{t}{\ell} \cdot t_2\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-23} \lor \neg \left(t \leq 1.95 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_1 \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{{\sin k}^{2}}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \end{array} \]

Alternative 6
Error	10.4
Cost	27477

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+236}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{+163}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t}{\frac{\ell}{t}}}}{t \cdot \frac{t_1}{\cos k}}\\ \mathbf{elif}\;t \leq -4.4 \cdot 10^{+110}:\\ \;\;\;\;\frac{1}{\frac{\left(t \cdot t\right) \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{\tan k}}{t}}}\\ \mathbf{elif}\;t \leq -7.4 \cdot 10^{-25} \lor \neg \left(t \leq 1.4 \cdot 10^{-21}\right):\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{t_1}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \end{array} \]

Alternative 7
Error	10.2
Cost	27476

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{if}\;t \leq -8.4 \cdot 10^{+236}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8.6 \cdot 10^{+165}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t}{\frac{\ell}{t}}}}{t \cdot \frac{t_1}{\cos k}}\\ \mathbf{elif}\;t \leq -6.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{\frac{\left(t \cdot t\right) \cdot \frac{k}{\ell}}{\frac{\frac{\ell}{\tan k}}{t}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\left(\frac{2}{k} \cdot \frac{\ell}{t_1}\right) \cdot \frac{\ell \cdot \cos k}{t \cdot k}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+81}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	8.9
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -1.4 \cdot 10^{-5} \lor \neg \left(k \leq 0.0034\right):\\ \;\;\;\;\cos k \cdot \frac{\frac{\ell \cdot \frac{2}{k}}{{\sin k}^{2}}}{\frac{t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}}\\ \end{array} \]

Alternative 9
Error	12.0
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -4.4 \cdot 10^{-5} \lor \neg \left(k \leq 1.36 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{1 - \cos \left(k + k\right)}{\ell}}{\left(2 \cdot \frac{\ell}{t}\right) \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}}\\ \end{array} \]

Alternative 10
Error	18.8
Cost	14281

\[\begin{array}{l} \mathbf{if}\;k \leq -8.2 \cdot 10^{-5} \lor \neg \left(k \leq 3.3 \cdot 10^{+14}\right):\\ \;\;\;\;4 \cdot \frac{\frac{\ell \cdot \ell}{\frac{k}{\frac{\cos k}{k}}}}{t \cdot \left(1 - \cos \left(k + k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}}\\ \end{array} \]

Alternative 11
Error	14.5
Cost	14281

\[\begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{-5} \lor \neg \left(k \leq 2.9 \cdot 10^{+16}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{4}{k \cdot \left(\frac{1 - \cos \left(k + k\right)}{\ell} \cdot \frac{k}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}}\\ \end{array} \]

Alternative 12
Error	19.0
Cost	13960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot \left(\frac{t}{\ell} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 13
Error	18.7
Cost	13960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{t}{\frac{\ell}{k}} \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 14
Error	19.1
Cost	13960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\left({\sin k}^{2} \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 15
Error	19.5
Cost	13896

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 16
Error	18.9
Cost	7752

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-21}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 17
Error	20.4
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -2.8 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 18
Error	19.6
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-87}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 19
Error	22.4
Cost	1356

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := \frac{\frac{\ell}{t}}{t \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{if}\;k \leq -1.9 \cdot 10^{-107}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.7 \cdot 10^{-239}:\\ \;\;\;\;\frac{t_1}{k \cdot \left(t \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 20
Error	21.7
Cost	1352

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -0.003:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \mathbf{elif}\;t \leq -1.56 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(t \cdot t\right) \cdot \left(\frac{k}{\ell} \cdot \left(k \cdot \frac{2}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 21
Error	21.7
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -1.65 \cdot 10^{-157} \lor \neg \left(k \leq 2 \cdot 10^{+54}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(\frac{t}{\ell} \cdot \left(k \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 22
Error	28.2
Cost	1096

\[\begin{array}{l} t_1 := k \cdot \left(t \cdot t\right)\\ t_2 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{t_2}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot t_2}{t_1}\\ \end{array} \]

Alternative 23
Error	25.1
Cost	1092

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-289}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(t \cdot t\right)}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 24
Error	28.6
Cost	964

\[\begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-96}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \frac{k}{\frac{\ell}{t}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]

Alternative 25
Error	27.9
Cost	964

\[\begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-96}:\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \end{array} \]

Alternative 26
Error	33.8
Cost	832

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

Alternative 27
Error	29.4
Cost	832

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

Reproduce

herbie shell --seed 2023053 
(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))))