Toniolo and Linder, Equation (10+)

Average Error: 50.45% → 8.65%

Time: 48.8s

Precision: binary64

Cost: 98768

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \frac{t}{t_2}\\ t_4 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{if}\;k \leq -9.5 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -7.8 \cdot 10^{-118}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{2 \cdot \frac{t_2}{t \cdot t_4}}\right)}^{3}}{{\left(\frac{t_4}{\frac{t_2}{t}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{t_4}\right)}^{3} \cdot t_3}}{{\left(t_4 \cdot t_3\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (* 2.0 (/ (* (/ l k) (cos k)) (* (/ k l) (* t (pow (sin k) 2.0))))))
        (t_2 (pow (cbrt l) 2.0))
        (t_3 (/ t t_2))
        (t_4 (cbrt (* (+ 2.0 (pow (/ k t) 2.0)) (* (sin k) (tan k))))))
   (if (<= k -9.5e+85)
     t_1
     (if (<= k -7.8e-118)
       (/
        (pow (cbrt (* 2.0 (/ t_2 (* t t_4)))) 3.0)
        (pow (/ t_4 (/ t_2 t)) 2.0))
       (if (<= k 2.8e-141)
         (* (/ (/ l t) (* k t)) (/ l (* k t)))
         (if (<= k 4.1e+98)
           (/ (/ 2.0 (* (pow (cbrt t_4) 3.0) t_3)) (pow (* t_4 t_3) 2.0))
           t_1))))))

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 = 2.0 * (((l / k) * cos(k)) / ((k / l) * (t * pow(sin(k), 2.0))));
	double t_2 = pow(cbrt(l), 2.0);
	double t_3 = t / t_2;
	double t_4 = cbrt(((2.0 + pow((k / t), 2.0)) * (sin(k) * tan(k))));
	double tmp;
	if (k <= -9.5e+85) {
		tmp = t_1;
	} else if (k <= -7.8e-118) {
		tmp = pow(cbrt((2.0 * (t_2 / (t * t_4)))), 3.0) / pow((t_4 / (t_2 / t)), 2.0);
	} else if (k <= 2.8e-141) {
		tmp = ((l / t) / (k * t)) * (l / (k * t));
	} else if (k <= 4.1e+98) {
		tmp = (2.0 / (pow(cbrt(t_4), 3.0) * t_3)) / pow((t_4 * t_3), 2.0);
	} else {
		tmp = t_1;
	}
	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 = 2.0 * (((l / k) * Math.cos(k)) / ((k / l) * (t * Math.pow(Math.sin(k), 2.0))));
	double t_2 = Math.pow(Math.cbrt(l), 2.0);
	double t_3 = t / t_2;
	double t_4 = Math.cbrt(((2.0 + Math.pow((k / t), 2.0)) * (Math.sin(k) * Math.tan(k))));
	double tmp;
	if (k <= -9.5e+85) {
		tmp = t_1;
	} else if (k <= -7.8e-118) {
		tmp = Math.pow(Math.cbrt((2.0 * (t_2 / (t * t_4)))), 3.0) / Math.pow((t_4 / (t_2 / t)), 2.0);
	} else if (k <= 2.8e-141) {
		tmp = ((l / t) / (k * t)) * (l / (k * t));
	} else if (k <= 4.1e+98) {
		tmp = (2.0 / (Math.pow(Math.cbrt(t_4), 3.0) * t_3)) / Math.pow((t_4 * t_3), 2.0);
	} else {
		tmp = t_1;
	}
	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(2.0 * Float64(Float64(Float64(l / k) * cos(k)) / Float64(Float64(k / l) * Float64(t * (sin(k) ^ 2.0)))))
	t_2 = cbrt(l) ^ 2.0
	t_3 = Float64(t / t_2)
	t_4 = cbrt(Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(sin(k) * tan(k))))
	tmp = 0.0
	if (k <= -9.5e+85)
		tmp = t_1;
	elseif (k <= -7.8e-118)
		tmp = Float64((cbrt(Float64(2.0 * Float64(t_2 / Float64(t * t_4)))) ^ 3.0) / (Float64(t_4 / Float64(t_2 / t)) ^ 2.0));
	elseif (k <= 2.8e-141)
		tmp = Float64(Float64(Float64(l / t) / Float64(k * t)) * Float64(l / Float64(k * t)));
	elseif (k <= 4.1e+98)
		tmp = Float64(Float64(2.0 / Float64((cbrt(t_4) ^ 3.0) * t_3)) / (Float64(t_4 * t_3) ^ 2.0));
	else
		tmp = t_1;
	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[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[k, -9.5e+85], t$95$1, If[LessEqual[k, -7.8e-118], N[(N[Power[N[Power[N[(2.0 * N[(t$95$2 / N[(t * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / N[Power[N[(t$95$4 / N[(t$95$2 / t), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.8e-141], N[(N[(N[(l / t), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] * N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.1e+98], N[(N[(2.0 / N[(N[Power[N[Power[t$95$4, 1/3], $MachinePrecision], 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] / N[Power[N[(t$95$4 * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -9.49999999999999945e85 or 4.1e98 < k

   1. Initial program 52.56

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

   2. Simplified 52.57

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

      [Start] 52.56	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 52.56	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 52.56	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 52.56	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 52.57	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 52.57	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 52.57	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 52.57	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 32.9

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

   4. Simplified 12.29

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

      [Start] 32.9	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 32.9	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 34.95	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 34.95	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 34.95	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 12.29	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 12.29	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 5.56

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

   if -9.49999999999999945e85 < k < -7.80000000000000002e-118

   1. Initial program 42.95

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

   2. Simplified 42.94

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

      [Start] 42.95	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 42.95	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 42.95	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 42.95	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 42.95	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 42.94	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 42.94	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 42.94	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 42.94	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 14.23

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

   4. Simplified 14.23

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

      [Start] 14.23	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 14.23	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]
      associate-*r/ [=>] 14.23	\[ \frac{\color{blue}{\frac{1 \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      metadata-eval [=>] 14.23	\[ \frac{\frac{\color{blue}{2}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-commutative [=>] 14.23	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*l* [=>] 14.22	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   5. Applied egg-rr 14.22

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

   6. Applied egg-rr 14.29

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

   if -7.80000000000000002e-118 < k < 2.80000000000000012e-141

   1. Initial program 59.31

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

   2. Simplified 48.16

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

      [Start] 59.31	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 59.31	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 56.45	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 56.62	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 48.16	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 48.16	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 48.16	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 48.16	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around 0 89.36

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

   4. Simplified 88.83

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

      [Start] 89.36	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 89.36	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 88.83	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 88.83	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

   5. Applied egg-rr 85.95

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

   6. Taylor expanded in k around 0 88.83

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

   7. Simplified 27.18

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

      [Start] 88.83	\[ \frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}} \]
      unpow2 [=>] 88.83	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      unpow3 [=>] 88.83	\[ \frac{\ell}{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(\left(t \cdot t\right) \cdot t\right)}}{\ell}} \]
      associate-*r* [=>] 87.02	\[ \frac{\ell}{\frac{\color{blue}{\left(\left(k \cdot k\right) \cdot \left(t \cdot t\right)\right) \cdot t}}{\ell}} \]
      swap-sqr [<=] 33.62	\[ \frac{\ell}{\frac{\color{blue}{\left(\left(k \cdot t\right) \cdot \left(k \cdot t\right)\right)} \cdot t}{\ell}} \]
      unpow2 [<=] 33.62	\[ \frac{\ell}{\frac{\color{blue}{{\left(k \cdot t\right)}^{2}} \cdot t}{\ell}} \]
      *-commutative [=>] 33.62	\[ \frac{\ell}{\frac{\color{blue}{t \cdot {\left(k \cdot t\right)}^{2}}}{\ell}} \]
      *-commutative [<=] 33.62	\[ \frac{\ell}{\frac{\color{blue}{{\left(k \cdot t\right)}^{2} \cdot t}}{\ell}} \]
      associate-*l/ [<=] 27.18	\[ \frac{\ell}{\color{blue}{\frac{{\left(k \cdot t\right)}^{2}}{\ell} \cdot t}} \]
      *-commutative [=>] 27.18	\[ \frac{\ell}{\color{blue}{t \cdot \frac{{\left(k \cdot t\right)}^{2}}{\ell}}} \]

   8. Applied egg-rr 5.67

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

   if 2.80000000000000012e-141 < k < 4.1e98

   1. Initial program 44.06

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

   2. Simplified 44

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

      [Start] 44.06	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 44.06	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 44.06	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 44.06	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 44	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 44.01	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 44	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 44	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 44	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 13.7

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

   4. Simplified 13.69

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

      [Start] 13.7	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 13.7	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]
      associate-*r/ [=>] 13.7	\[ \frac{\color{blue}{\frac{1 \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      metadata-eval [=>] 13.7	\[ \frac{\frac{\color{blue}{2}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      *-commutative [=>] 13.7	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right) \cdot \tan k}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
      associate-*l* [=>] 13.69	\[ \frac{\frac{2}{\sqrt[3]{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]

   5. Applied egg-rr 13.77

      \[\leadsto \frac{\frac{2}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}\right)}^{3}} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.65

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.5 \cdot 10^{+85}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq -7.8 \cdot 10^{-118}:\\ \;\;\;\;\frac{{\left(\sqrt[3]{2 \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}\right)}^{3}}{{\left(\frac{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}\right)}^{3} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.66%
Cost	98768

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := \frac{t_1}{t}\\ t_3 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ t_4 := 2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ t_5 := \frac{t}{t_1}\\ \mathbf{if}\;k \leq -3.7 \cdot 10^{+83}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq -1.2 \cdot 10^{-114}:\\ \;\;\;\;\frac{\frac{2 \cdot t_2}{t_3}}{{\left(\frac{t_3}{t_2}\right)}^{2}}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+100}:\\ \;\;\;\;\frac{\frac{2}{{\left(\sqrt[3]{t_3}\right)}^{3} \cdot t_5}}{{\left(t_3 \cdot t_5\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 2
Error	8.65%
Cost	85904

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ t_2 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\\ \mathbf{if}\;k \leq -8.1 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{t_2}^{3}}\\ \mathbf{elif}\;k \leq 2.25 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{2}{t_2}}{{t_2}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	8.65%
Cost	85904

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \frac{t_2}{t}\\ t_4 := \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ t_5 := t_4 \cdot \frac{t}{t_2}\\ \mathbf{if}\;k \leq -6.2 \cdot 10^{+81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{2 \cdot t_3}{t_4}}{{\left(\frac{t_4}{t_3}\right)}^{2}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{2}{t_5}}{{t_5}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	8.73%
Cost	46480

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ t_2 := \frac{2}{{\left(\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{if}\;k \leq -2.05 \cdot 10^{+83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1.3 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{+97}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	12.22%
Cost	39948

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1050000:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{t_1}{\frac{\cos k}{t}}\right)}\\ \mathbf{elif}\;k \leq 1.9 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{+97}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\sin k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 6
Error	12.01%
Cost	27276

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{-44}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{-\ell}{\sin k}} \cdot \frac{-2 - {\left(\frac{k}{t}\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	12.22%
Cost	27212

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{+23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-44}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+86}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+99}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	14.13%
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \frac{\ell \cdot \cos k}{\frac{k}{\ell} \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}\\ \mathbf{if}\;k \leq -3400000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{+239}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \end{array} \]

Alternative 9
Error	16.96%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -1500000 \lor \neg \left(k \leq 1.7 \cdot 10^{+14}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 10
Error	16.96%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -4000000 \lor \neg \left(k \leq 1.9 \cdot 10^{+14}\right):\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 11
Error	12.9%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -3400000 \lor \neg \left(k \leq 1.5 \cdot 10^{-35}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 12
Error	12.58%
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -6500000:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \cos k}{\frac{k}{\ell} \cdot \left(t \cdot t_1\right)}\\ \mathbf{elif}\;k \leq 2.45 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot t_1}\\ \end{array} \]

Alternative 13
Error	12.63%
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -950000:\\ \;\;\;\;2 \cdot \frac{1}{\frac{k}{\ell} \cdot \left(\frac{k}{\ell} \cdot \frac{t_1}{t_2}\right)}\\ \mathbf{elif}\;k \leq 9.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{\frac{k}{\ell} \cdot t_1}\\ \end{array} \]

Alternative 14
Error	17.15%
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -8000000 \lor \neg \left(k \leq 1.3 \cdot 10^{+19}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 15
Error	17.12%
Cost	14408

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ \mathbf{if}\;k \leq -560000:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\frac{2 \cdot \cos k}{t}}{1 - t_1}\right)\\ \mathbf{elif}\;k \leq 5.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{t_1}{2}\right)}\right)\\ \end{array} \]

Alternative 16
Error	17.11%
Cost	14408

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ \mathbf{if}\;k \leq -340000:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{2 \cdot \cos k}{t}}{1 - t_1}\right)\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t \cdot \left(0.5 - \frac{t_1}{2}\right)} \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell}}\right)\\ \end{array} \]

Alternative 17
Error	17.23%
Cost	14408

\[\begin{array}{l} t_1 := \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\\ \mathbf{if}\;k \leq -1020000:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{1}{\frac{k}{\ell} \cdot \frac{k}{\ell}}\right)\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell}}\right)\\ \end{array} \]

Alternative 18
Error	28.81%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -2.25 \cdot 10^{-72} \lor \neg \left(t \leq 3.1 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 19
Error	29.45%
Cost	1609

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-73} \lor \neg \left(t \leq 2 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \left(\frac{1}{t \cdot \left(k \cdot k\right)} + \frac{-0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 20
Error	30.01%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{-73} \lor \neg \left(t \leq 2.25 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 21
Error	29.99%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-78} \lor \neg \left(t \leq 1.3 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 22
Error	33.08%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -1.65 \cdot 10^{+159} \lor \neg \left(k \leq 1.12 \cdot 10^{+157}\right):\\ \;\;\;\;\frac{\ell}{t \cdot \frac{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot t} \cdot \frac{\ell}{k \cdot t}\\ \end{array} \]

Alternative 23
Error	52.44%
Cost	832

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

Alternative 24
Error	36.85%
Cost	832

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

Reproduce

herbie shell --seed 2023121 
(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))))