Toniolo and Linder, Equation (10+)

Average Error: 32.9 → 7.2

Time: 36.8s

Precision: binary64

Cost: 46348

Math

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

↓

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -3 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot \left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot t_3\right)\right) \cdot 0.5}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_2}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_2}\right)\\ \mathbf{elif}\;k \leq 1.06 \cdot 10^{+92}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot t_3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0))))))
        (t_2 (* t (pow (cbrt k) 2.0)))
        (t_3 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= k -3e+75)
     t_1
     (if (<= k -8.5e-165)
       (/ (/ l t) (* (* t (* (* (sin k) (* (/ t l) (tan k))) t_3)) 0.5))
       (if (<= k 3.6e-155)
         (* (/ 1.0 (pow (/ t_2 (cbrt l)) 2.0)) (* (cbrt l) (/ l t_2)))
         (if (<= k 1.06e+92)
           (* (/ l t) (/ 2.0 (* t (* (/ t l) (* (tan k) (* (sin k) t_3))))))
           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) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	double t_2 = t * pow(cbrt(k), 2.0);
	double t_3 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (k <= -3e+75) {
		tmp = t_1;
	} else if (k <= -8.5e-165) {
		tmp = (l / t) / ((t * ((sin(k) * ((t / l) * tan(k))) * t_3)) * 0.5);
	} else if (k <= 3.6e-155) {
		tmp = (1.0 / pow((t_2 / cbrt(l)), 2.0)) * (cbrt(l) * (l / t_2));
	} else if (k <= 1.06e+92) {
		tmp = (l / t) * (2.0 / (t * ((t / l) * (tan(k) * (sin(k) * t_3)))));
	} 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) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	double t_2 = t * Math.pow(Math.cbrt(k), 2.0);
	double t_3 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (k <= -3e+75) {
		tmp = t_1;
	} else if (k <= -8.5e-165) {
		tmp = (l / t) / ((t * ((Math.sin(k) * ((t / l) * Math.tan(k))) * t_3)) * 0.5);
	} else if (k <= 3.6e-155) {
		tmp = (1.0 / Math.pow((t_2 / Math.cbrt(l)), 2.0)) * (Math.cbrt(l) * (l / t_2));
	} else if (k <= 1.06e+92) {
		tmp = (l / t) * (2.0 / (t * ((t / l) * (Math.tan(k) * (Math.sin(k) * t_3)))));
	} 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) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))))
	t_2 = Float64(t * (cbrt(k) ^ 2.0))
	t_3 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (k <= -3e+75)
		tmp = t_1;
	elseif (k <= -8.5e-165)
		tmp = Float64(Float64(l / t) / Float64(Float64(t * Float64(Float64(sin(k) * Float64(Float64(t / l) * tan(k))) * t_3)) * 0.5));
	elseif (k <= 3.6e-155)
		tmp = Float64(Float64(1.0 / (Float64(t_2 / cbrt(l)) ^ 2.0)) * Float64(cbrt(l) * Float64(l / t_2)));
	elseif (k <= 1.06e+92)
		tmp = Float64(Float64(l / t) * Float64(2.0 / Float64(t * Float64(Float64(t / l) * Float64(tan(k) * Float64(sin(k) * t_3))))));
	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[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -3e+75], t$95$1, If[LessEqual[k, -8.5e-165], N[(N[(l / t), $MachinePrecision] / N[(N[(t * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 3.6e-155], N[(N[(1.0 / N[Power[N[(t$95$2 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] * N[(l / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.06e+92], N[(N[(l / t), $MachinePrecision] * N[(2.0 / N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -3e75 or 1.05999999999999999e92 < k

   1. Initial program 34.2

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

   2. Simplified 34.2

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

      [Start] 34.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 34.2	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 34.2	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 34.2	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 34.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 34.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 34.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 34.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 21.7

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

   4. Simplified 8.0

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

      [Start] 21.7	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 21.7	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 23.0	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 23.0	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 23.0	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 8.0	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 8.0	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if -3e75 < k < -8.5e-165

   1. Initial program 29.9

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

   2. Simplified 30.1

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

      [Start] 29.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 29.9	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 29.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 29.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 29.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 30.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 30.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 30.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 30.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 20.3

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

   4. Applied egg-rr 8.2

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

   5. Applied egg-rr 7.6

      \[\leadsto \color{blue}{\frac{\frac{\ell}{t}}{\left(t \cdot \left(\left(\left(\frac{t}{\ell} \cdot \tan k\right) \cdot \sin k\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot 0.5}} \]

   if -8.5e-165 < k < 3.59999999999999989e-155

   1. Initial program 38.2

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

   2. Simplified 62.8

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

      [Start] 38.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 38.2	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 38.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 38.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 38.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 61.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 61.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 61.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 61.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in k around 0 62.8

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

   4. Simplified 36.8

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

      [Start] 62.8	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 62.8	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 62.6	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 62.6	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      associate-*l* [=>] 36.8	\[ \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   5. Applied egg-rr 2.9

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

   if 3.59999999999999989e-155 < k < 1.05999999999999999e92

   1. Initial program 29.0

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

   2. Simplified 28.8

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

      [Start] 29.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 29.0	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 29.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 29.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 28.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 28.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 20.3

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

   4. Applied egg-rr 8.2

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

4. Final simplification 7.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3 \cdot 10^{+75}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot \left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot 0.5}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{1}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)\\ \mathbf{elif}\;k \leq 1.06 \cdot 10^{+92}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.3
Cost	33292

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -9 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.3 \cdot 10^{-223}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot \left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot t_3\right)\right) \cdot 0.5}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \frac{\ell}{{t_2}^{2}}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{+92}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot t_3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	8.3
Cost	26572

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -1 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.35 \cdot 10^{-237}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot \left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot t_2\right)\right) \cdot 0.5}\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 6.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot t_2\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	9.2
Cost	21136

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_3 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -1.02 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{t_3 \cdot \frac{k}{\frac{\ell}{{t_3}^{2}}}}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	9.2
Cost	21136

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -3.4 \cdot 10^{+77}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -7.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot t_3\right)}}{\frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{+92}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot t_3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	9.1
Cost	21136

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := t \cdot \sqrt[3]{k}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -3.2 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot \left(\left(\sin k \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right) \cdot t_3\right)\right) \cdot 0.5}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{elif}\;k \leq 9.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot t_3\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	11.1
Cost	20752

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{0.5 \cdot \left(t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{k \cdot t}{\ell}\right)\right)\right)}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_3 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -1.4 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.5 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-174}:\\ \;\;\;\;\frac{\ell}{t_3 \cdot \frac{k}{\frac{\ell}{{t_3}^{2}}}}\\ \mathbf{elif}\;k \leq 0.00024:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	20.3
Cost	20361

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -3.6 \cdot 10^{-224} \lor \neg \left(k \leq 3.65 \cdot 10^{-168}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{0.5 \cdot \left(t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{k \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \frac{k}{\frac{\ell}{{t_1}^{2}}}}\\ \end{array} \]

Alternative 8
Error	21.0
Cost	14473

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+109} \lor \neg \left(t \leq -3.9 \cdot 10^{-23}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{0.5 \cdot \left(t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{k \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{-k} \cdot \frac{\ell}{k \cdot \left(-{t}^{3}\right)}\\ \end{array} \]

Alternative 9
Error	21.9
Cost	7820

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{2}{t \cdot \left(2 \cdot \frac{k}{\frac{\frac{\ell}{t}}{k}}\right)}\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.15 \cdot 10^{-103}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-127}:\\ \;\;\;\;\frac{\ell}{\frac{t}{\ell}} \cdot \left(\frac{2}{{k}^{4}} - \frac{0.3333333333333333}{k \cdot k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	21.7
Cost	7436

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{2}{t \cdot \left(2 \cdot \frac{k}{\frac{\frac{\ell}{t}}{k}}\right)}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-84}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{-127}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	22.1
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-84} \lor \neg \left(t \leq 2.75 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(2 \cdot \frac{k}{\frac{\frac{\ell}{t}}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 12
Error	28.7
Cost	1360

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}\\ \mathbf{if}\;k \leq -5.5 \cdot 10^{+44}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-174}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{t} \cdot \frac{1}{k}\right)\right)\\ \mathbf{elif}\;k \leq 16000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 13
Error	28.2
Cost	1360

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\frac{\ell}{t \cdot t}}{k \cdot k}\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{+45}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq -1.55 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-174}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\frac{\ell}{k} \cdot \left(\frac{\ell}{t} \cdot \frac{1}{k}\right)\right)\\ \mathbf{elif}\;k \leq 1250:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 14
Error	28.0
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-85} \lor \neg \left(t \leq 2.75 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{2 \cdot \frac{t \cdot t}{\frac{\frac{\ell}{k}}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 + \frac{2}{k \cdot k}}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 15
Error	22.9
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-84} \lor \neg \left(t \leq 2.75 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(2 \cdot \frac{k}{\frac{\frac{\ell}{t}}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333 + \frac{2}{k \cdot k}}{k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 16
Error	22.8
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{-103} \lor \neg \left(t \leq 2.75 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(2 \cdot \frac{k}{\frac{\frac{\ell}{t}}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333 + \frac{2}{k \cdot k}}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t}\\ \end{array} \]

Alternative 17
Error	36.6
Cost	704

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

Alternative 18
Error	35.1
Cost	704

\[-0.3333333333333333 \cdot \frac{\ell \cdot \frac{\ell}{k \cdot t}}{k} \]

Reproduce

herbie shell --seed 2023034 
(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))))