Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 9.4

Time: 51.4s

Precision: binary64

Cost: 40340

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := 2 + t_2\\ t_4 := \left(t_1 \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\frac{2}{\tan k}}{t_3}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-76}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+76}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{t_3 \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot t_1}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left(t_2 + 1\right)\right)\right)}\\ \end{array} \]

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 (sin k)))
        (t_2 (pow (/ k t) 2.0))
        (t_3 (+ 2.0 t_2))
        (t_4 (* (* t_1 (/ l (pow t 3.0))) (/ (/ 2.0 (tan k)) t_3))))
   (if (<= t -5.6e+102)
     (* l (/ l (pow (* t (pow (cbrt k) 2.0)) 3.0)))
     (if (<= t -7.8e-76)
       t_4
       (if (<= t 3.4e-67)
         (* 2.0 (/ (* (/ l k) (/ (cos k) t)) (* (/ k l) (pow (sin k) 2.0))))
         (if (<= t 1.1e+76)
           t_4
           (if (<= t 9.2e+150)
             (/
              2.0
              (* t_3 (pow (/ (* t (cbrt (tan k))) (cbrt (* l t_1))) 3.0)))
             (/
              2.0
              (*
               (* (sin k) (pow (/ t (pow (cbrt l) 2.0)) 3.0))
               (* (tan k) (+ 1.0 (+ t_2 1.0))))))))))))

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 / sin(k);
	double t_2 = pow((k / t), 2.0);
	double t_3 = 2.0 + t_2;
	double t_4 = (t_1 * (l / pow(t, 3.0))) * ((2.0 / tan(k)) / t_3);
	double tmp;
	if (t <= -5.6e+102) {
		tmp = l * (l / pow((t * pow(cbrt(k), 2.0)), 3.0));
	} else if (t <= -7.8e-76) {
		tmp = t_4;
	} else if (t <= 3.4e-67) {
		tmp = 2.0 * (((l / k) * (cos(k) / t)) / ((k / l) * pow(sin(k), 2.0)));
	} else if (t <= 1.1e+76) {
		tmp = t_4;
	} else if (t <= 9.2e+150) {
		tmp = 2.0 / (t_3 * pow(((t * cbrt(tan(k))) / cbrt((l * t_1))), 3.0));
	} else {
		tmp = 2.0 / ((sin(k) * pow((t / pow(cbrt(l), 2.0)), 3.0)) * (tan(k) * (1.0 + (t_2 + 1.0))));
	}
	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 / Math.sin(k);
	double t_2 = Math.pow((k / t), 2.0);
	double t_3 = 2.0 + t_2;
	double t_4 = (t_1 * (l / Math.pow(t, 3.0))) * ((2.0 / Math.tan(k)) / t_3);
	double tmp;
	if (t <= -5.6e+102) {
		tmp = l * (l / Math.pow((t * Math.pow(Math.cbrt(k), 2.0)), 3.0));
	} else if (t <= -7.8e-76) {
		tmp = t_4;
	} else if (t <= 3.4e-67) {
		tmp = 2.0 * (((l / k) * (Math.cos(k) / t)) / ((k / l) * Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 1.1e+76) {
		tmp = t_4;
	} else if (t <= 9.2e+150) {
		tmp = 2.0 / (t_3 * Math.pow(((t * Math.cbrt(Math.tan(k))) / Math.cbrt((l * t_1))), 3.0));
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((t / Math.pow(Math.cbrt(l), 2.0)), 3.0)) * (Math.tan(k) * (1.0 + (t_2 + 1.0))));
	}
	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(l / sin(k))
	t_2 = Float64(k / t) ^ 2.0
	t_3 = Float64(2.0 + t_2)
	t_4 = Float64(Float64(t_1 * Float64(l / (t ^ 3.0))) * Float64(Float64(2.0 / tan(k)) / t_3))
	tmp = 0.0
	if (t <= -5.6e+102)
		tmp = Float64(l * Float64(l / (Float64(t * (cbrt(k) ^ 2.0)) ^ 3.0)));
	elseif (t <= -7.8e-76)
		tmp = t_4;
	elseif (t <= 3.4e-67)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(cos(k) / t)) / Float64(Float64(k / l) * (sin(k) ^ 2.0))));
	elseif (t <= 1.1e+76)
		tmp = t_4;
	elseif (t <= 9.2e+150)
		tmp = Float64(2.0 / Float64(t_3 * (Float64(Float64(t * cbrt(tan(k))) / cbrt(Float64(l * t_1))) ^ 3.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64(t / (cbrt(l) ^ 2.0)) ^ 3.0)) * Float64(tan(k) * Float64(1.0 + Float64(t_2 + 1.0)))));
	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[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$1 * N[(l / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e+102], N[(l * N[(l / N[Power[N[(t * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -7.8e-76], t$95$4, If[LessEqual[t, 3.4e-67], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.1e+76], t$95$4, If[LessEqual[t, 9.2e+150], N[(2.0 / N[(t$95$3 * N[Power[N[(N[(t * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l * t$95$1), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -5.60000000000000037e102

   1. Initial program 24.0

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

   2. Simplified 24.4

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

      [Start] 24.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 24.0	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 24.4	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 24.4	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 24.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 24.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 24.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 24.4	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around 0 30.3

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

   4. Simplified 27.3

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

      [Start] 30.3	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 30.3	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 27.3	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 27.3	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

   5. Applied egg-rr 20.9

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

   6. Applied egg-rr 12.8

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

   if -5.60000000000000037e102 < t < -7.8000000000000005e-76 or 3.4000000000000001e-67 < t < 1.1e76

   1. Initial program 20.8

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

   2. Simplified 20.8

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

      [Start] 20.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 20.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 20.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 32.6

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} - 1} \]

   4. Simplified 12.8

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

      [Start] 32.6	\[ e^{\mathsf{log1p}\left(\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} - 1 \]
      expm1-def [=>] 24.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]
      expm1-log1p [=>] 19.3	\[ \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k} \cdot \left(\ell \cdot \ell\right)}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-/l* [=>] 19.7	\[ \color{blue}{\frac{\frac{2}{{t}^{3} \cdot \sin k}}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \ell}}} \]
      associate-/l/ [=>] 19.7	\[ \color{blue}{\frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell \cdot \ell} \cdot \left({t}^{3} \cdot \sin k\right)}} \]
      unpow2 [<=] 19.7	\[ \frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\color{blue}{{\ell}^{2}}} \cdot \left({t}^{3} \cdot \sin k\right)} \]
      associate-/r/ [<=] 19.3	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{{\ell}^{2}}{{t}^{3} \cdot \sin k}}}} \]
      unpow2 [=>] 19.3	\[ \frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\color{blue}{\ell \cdot \ell}}{{t}^{3} \cdot \sin k}}} \]
      associate-/r* [=>] 20.8	\[ \frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\color{blue}{\frac{\frac{\ell \cdot \ell}{{t}^{3}}}{\sin k}}}} \]
      unpow2 [<=] 20.8	\[ \frac{2}{\frac{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\frac{\frac{\color{blue}{{\ell}^{2}}}{{t}^{3}}}{\sin k}}} \]
      associate-/l* [<=] 23.7	\[ \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k}{\frac{{\ell}^{2}}{{t}^{3}}}}} \]

   if -7.8000000000000005e-76 < t < 3.4000000000000001e-67

   1. Initial program 58.5

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

   2. Simplified 58.9

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

      [Start] 58.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 58.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 58.5	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 58.9	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 58.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 58.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 58.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 58.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 27.3

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

   4. Simplified 16.9

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

      [Start] 27.3	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 27.3	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 29.4	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 29.4	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 29.4	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 16.9	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 16.9	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 2.4

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

   if 1.1e76 < t < 9.20000000000000004e150

   1. Initial program 26.8

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

   2. Simplified 23.3

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

      [Start] 26.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 26.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 24.9	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 24.3	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 23.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 23.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 23.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 23.3	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 15.4

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

   4. Simplified 15.4

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

      [Start] 15.4	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{2}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      unpow2 [=>] 15.4	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot \color{blue}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      cube-mult [<=] 15.4	\[ \frac{2}{\color{blue}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   if 9.20000000000000004e150 < t

   1. Initial program 22.6

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

   2. Simplified 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] 22.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      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 8.6

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

   4. Simplified 8.6

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

      [Start] 8.6	\[ \frac{2}{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow2 [=>] 8.6	\[ \frac{2}{\left(\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \color{blue}{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      cube-mult [<=] 8.6	\[ \frac{2}{\left(\color{blue}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -7.8 \cdot 10^{-76}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-67}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+76}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	7.8
Cost	85904

\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\ell}{k}}\\ t_2 := \frac{\cos k}{t}\\ t_3 := {\sin k}^{2}\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_5 := \frac{t}{t_4}\\ t_6 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_7 := \sqrt[3]{t_6 \cdot \left(\sin k \cdot \tan k\right)}\\ t_8 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -0.00041:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{\frac{k}{\ell} \cdot t_3}\\ \mathbf{elif}\;k \leq -1.15 \cdot 10^{-128}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot t_6\right)} \cdot t_5\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{-145}:\\ \;\;\;\;\frac{t_1 \cdot \left(\ell \cdot {\left(\frac{t_1}{t_8}\right)}^{2}\right)}{t_8}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\frac{2}{t_5}}{t_7}}{{\left(\frac{t}{\frac{t_4}{t_7}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{t_2}{t_3}\right)\right) \cdot -2\\ \end{array} \]

Alternative 2
Error	7.9
Cost	85904

\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\ell}{k}}\\ t_2 := \frac{\cos k}{t}\\ t_3 := {\sin k}^{2}\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_5 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_6 := \frac{t_4}{\sqrt[3]{t_5 \cdot \left(\sin k \cdot \tan k\right)}}\\ t_7 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -0.00037:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{\frac{k}{\ell} \cdot t_3}\\ \mathbf{elif}\;k \leq -1.15 \cdot 10^{-128}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot t_5\right)} \cdot \frac{t}{t_4}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{-145}:\\ \;\;\;\;\frac{t_1 \cdot \left(\ell \cdot {\left(\frac{t_1}{t_7}\right)}^{2}\right)}{t_7}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+90}:\\ \;\;\;\;\frac{t_6 \cdot \frac{2}{t}}{{\left(\frac{t}{t_6}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{t_2}{t_3}\right)\right) \cdot -2\\ \end{array} \]

Alternative 3
Error	7.9
Cost	46480

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ t_3 := \sqrt[3]{\frac{\ell}{k}}\\ t_4 := \frac{\cos k}{t}\\ t_5 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -0.00041:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_4}{\frac{k}{\ell} \cdot t_1}\\ \mathbf{elif}\;k \leq -1.15 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-142}:\\ \;\;\;\;\frac{t_3 \cdot \left(\ell \cdot {\left(\frac{t_3}{t_5}\right)}^{2}\right)}{t_5}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+88}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{t_4}{t_1}\right)\right) \cdot -2\\ \end{array} \]

Alternative 4
Error	10.3
Cost	40276

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := 2 + t_2\\ t_4 := \left(t_1 \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\frac{2}{\tan k}}{t_3}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+102}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-84}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{+76}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+151}:\\ \;\;\;\;\frac{2}{t_3 \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot t_1}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_2 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]

Alternative 5
Error	10.5
Cost	40212

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ t_3 := \sqrt[3]{\frac{\ell}{k}}\\ t_4 := t \cdot \sqrt[3]{k}\\ t_5 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_6 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -1.75 \cdot 10^{-6}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_6}{\frac{k}{\ell} \cdot t_2}\\ \mathbf{elif}\;k \leq -2 \cdot 10^{-240}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\sqrt[3]{\ell}} \cdot \sqrt[3]{\tan k \cdot \left(\frac{k}{\ell} \cdot t_1\right)}\right)}^{3}}\\ \mathbf{elif}\;k \leq -2.45 \cdot 10^{-288}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{t_5}}{{t_5}^{2}}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{t_3 \cdot \left(\ell \cdot {\left(\frac{t_3}{t_4}\right)}^{2}\right)}{t_4}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+90}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot t_1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{t_6}{t_2}\right)\right) \cdot -2\\ \end{array} \]

Alternative 6
Error	10.4
Cost	33676

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+102}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-77}:\\ \;\;\;\;\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\frac{2}{\tan k}}{2 + t_1}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{-48}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right) \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)}\\ \end{array} \]

Alternative 7
Error	10.4
Cost	27344

\[\begin{array}{l} t_1 := \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{{t}^{3}}\right) \cdot \frac{\frac{2}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ t_2 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;t \leq -1 \cdot 10^{+108}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-66}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t_2}^{2}}}{t_2}\\ \end{array} \]

Alternative 8
Error	11.7
Cost	21264

\[\begin{array}{l} t_1 := \frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ t_2 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{+54}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-110}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t_2}^{2}}}{t_2}\\ \end{array} \]

Alternative 9
Error	12.9
Cost	21136

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;t \leq -2.2 \cdot 10^{+91}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}{-1} \cdot \left(\tan k \cdot \left(-1 + \left(-1 - t_1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-48}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{+128}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + t_1\right)\right)\right) \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t_2}^{2}}}{t_2}\\ \end{array} \]

Alternative 10
Error	13.0
Cost	20756

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+91}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -1.02 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}{-1} \cdot \left(\tan k \cdot \left(-1 + \left(-1 - {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\frac{k}{\frac{\ell}{{t}^{3}}}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+53}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\ell}{k \cdot {t_1}^{2}}}{t_1}\\ \end{array} \]

Alternative 11
Error	14.6
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -0.00038:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-242}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}{-1} \cdot \left(\tan k \cdot \left(-1 + \left(-1 - {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	12.1
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell}{\frac{k}{\ell} \cdot \left(k \cdot \frac{{\sin k}^{2}}{\frac{\cos k}{t}}\right)}\\ \mathbf{if}\;k \leq -6.8 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-241}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 62:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}{-1} \cdot \left(\tan k \cdot \left(-1 + \left(-1 - {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	14.7
Cost	20040

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{if}\;k \leq -0.00038:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-242}:\\ \;\;\;\;\ell \cdot \frac{\ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.75 \cdot 10^{+81}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}{-1} \cdot \left(\tan k \cdot \left(-1 + \left(-1 - {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	18.3
Cost	14804

\[\begin{array}{l} t_1 := k \cdot {t}^{1.5}\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_3 := 2 \cdot \left(t_2 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ t_4 := \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{t_2}}\\ \mathbf{if}\;t \leq -2.32 \cdot 10^{+55}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot {\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq -3.6 \cdot 10^{-91}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-73}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 7 \cdot 10^{+15}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 2.85 \cdot 10^{+51}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{t_1}{\ell}}\\ \end{array} \]

Alternative 15
Error	14.7
Cost	14729

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00041 \lor \neg \left(k \leq 1.3 \cdot 10^{+78}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \frac{\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)}{\frac{1}{t}}}\\ \end{array} \]

Alternative 16
Error	14.8
Cost	14729

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00041 \lor \neg \left(k \leq 1.3 \cdot 10^{+79}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{t}{\ell}\right)\right)}{-1} \cdot \left(\tan k \cdot \left(-1 + \left(-1 - {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]

Alternative 17
Error	15.9
Cost	14601

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00041 \lor \neg \left(k \leq 2.7 \cdot 10^{+77}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(t \cdot \frac{\frac{k}{\ell} \cdot \left(t \cdot t\right)}{\ell}\right)}\\ \end{array} \]

Alternative 18
Error	21.9
Cost	14280

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+55}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot {\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq -4.6 \cdot 10^{-89}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-49}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 19
Error	22.1
Cost	13644

\[\begin{array}{l} \mathbf{if}\;t \leq -9.8 \cdot 10^{+54}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot {\left(t \cdot \sqrt[3]{k}\right)}^{3}}\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-84}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-49}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 20
Error	22.9
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-48}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 21
Error	24.7
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-44} \lor \neg \left(t \leq 1.1 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 22
Error	24.5
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{-77} \lor \neg \left(t \leq 1.3 \cdot 10^{-48}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 23
Error	28.0
Cost	7244

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{if}\;k \leq -13600000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3 \cdot 10^{-158}:\\ \;\;\;\;\frac{\ell}{t \cdot \frac{k \cdot k}{\frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\left|\ell \cdot \left(\ell \cdot \frac{-0.11666666666666667}{t}\right)\right|\\ \mathbf{elif}\;k \leq 1.7 \cdot 10^{+42}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 24
Error	28.0
Cost	7244

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{if}\;k \leq -115000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3 \cdot 10^{-158}:\\ \;\;\;\;\frac{\ell}{t \cdot \frac{k \cdot k}{\frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;-0.11666666666666667 \cdot \left|\ell \cdot \frac{\ell}{t}\right|\\ \mathbf{elif}\;k \leq 4.9 \cdot 10^{+42}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 25
Error	28.0
Cost	1616

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{if}\;k \leq -225000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3 \cdot 10^{-158}:\\ \;\;\;\;\frac{\ell}{t \cdot \frac{k \cdot k}{\frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 5.7 \cdot 10^{+41}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 26
Error	30.7
Cost	1097

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-161} \lor \neg \left(\ell \leq 5.8 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.11666666666666667}{\frac{t}{\ell \cdot \ell}}\\ \end{array} \]

Alternative 27
Error	30.7
Cost	1096

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{-159}:\\ \;\;\;\;\frac{\ell}{t \cdot \frac{k \cdot k}{\frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;\ell \leq 6.5 \cdot 10^{-146}:\\ \;\;\;\;\frac{-0.11666666666666667}{\frac{t}{\ell \cdot \ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \end{array} \]

Alternative 28
Error	43.4
Cost	576

\[\frac{1}{\frac{t}{-0.11666666666666667 \cdot \left(\ell \cdot \ell\right)}} \]

Alternative 29
Error	45.6
Cost	448

\[-0.11666666666666667 \cdot \left(\ell \cdot \frac{\ell}{t}\right) \]

Alternative 30
Error	43.6
Cost	448

\[-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

Alternative 31
Error	43.4
Cost	448

\[\frac{-0.11666666666666667}{\frac{t}{\ell \cdot \ell}} \]

Reproduce

herbie shell --seed 2023018 
(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))))