Toniolo and Linder, Equation (10+)

Average Accuracy: 49.5% → 85.6%

Time: 32.6s

Precision: binary64

Cost: 46349

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot {\left(\sqrt[3]{k}\right)}^{2}\\ \mathbf{if}\;k \leq -2.7 \cdot 10^{+117}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq -7.8 \cdot 10^{-141} \lor \neg \left(k \leq 1.75 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{t_1}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t_1}\right)\\ \end{array} \]

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 (* t (pow (cbrt k) 2.0))))
   (if (<= k -2.7e+117)
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))
     (if (or (<= k -7.8e-141) (not (<= k 1.75e-152)))
       (*
        (/
         2.0
         (* t (* (/ t l) (* (tan k) (* (sin k) (+ 2.0 (pow (/ k t) 2.0)))))))
        (/ l t))
       (* (/ 1.0 (pow (/ t_1 (cbrt l)) 2.0)) (* (cbrt l) (/ l 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 = t * pow(cbrt(k), 2.0);
	double tmp;
	if (k <= -2.7e+117) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	} else if ((k <= -7.8e-141) || !(k <= 1.75e-152)) {
		tmp = (2.0 / (t * ((t / l) * (tan(k) * (sin(k) * (2.0 + pow((k / t), 2.0))))))) * (l / t);
	} else {
		tmp = (1.0 / pow((t_1 / cbrt(l)), 2.0)) * (cbrt(l) * (l / 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 = t * Math.pow(Math.cbrt(k), 2.0);
	double tmp;
	if (k <= -2.7e+117) {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	} else if ((k <= -7.8e-141) || !(k <= 1.75e-152)) {
		tmp = (2.0 / (t * ((t / l) * (Math.tan(k) * (Math.sin(k) * (2.0 + Math.pow((k / t), 2.0))))))) * (l / t);
	} else {
		tmp = (1.0 / Math.pow((t_1 / Math.cbrt(l)), 2.0)) * (Math.cbrt(l) * (l / 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(t * (cbrt(k) ^ 2.0))
	tmp = 0.0
	if (k <= -2.7e+117)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	elseif ((k <= -7.8e-141) || !(k <= 1.75e-152))
		tmp = Float64(Float64(2.0 / Float64(t * Float64(Float64(t / l) * Float64(tan(k) * Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))))) * Float64(l / t));
	else
		tmp = Float64(Float64(1.0 / (Float64(t_1 / cbrt(l)) ^ 2.0)) * Float64(cbrt(l) * Float64(l / 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[(t * N[Power[N[Power[k, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -2.7e+117], 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], If[Or[LessEqual[k, -7.8e-141], N[Not[LessEqual[k, 1.75e-152]], $MachinePrecision]], N[(N[(2.0 / N[(t * N[(N[(t / l), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Power[N[(t$95$1 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Power[l, 1/3], $MachinePrecision] * N[(l / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -2.7000000000000002e117

   1. Initial program 47.9%

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

   2. Simplified 47.9%

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

      [Start] 47.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 47.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 47.9	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 47.9	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 47.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 47.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 47.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 47.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 66.3%

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

   4. Simplified 91.0%

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

      [Start] 66.3	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 66.3	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 65.7	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 65.7	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 65.7	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 91.0	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 91.0	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if -2.7000000000000002e117 < k < -7.7999999999999994e-141 or 1.7500000000000001e-152 < k

   1. Initial program 52.8%

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

   2. Simplified 53.0%

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

      [Start] 52.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 52.8	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 52.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 52.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 52.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 53.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 53.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 53.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 53.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 64.5%

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

      [Start] 53.0	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow3 [=>] 53.0	\[ \frac{2}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 64.5	\[ \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. Applied egg-rr 81.7%

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

      [Start] 64.5	\[ \frac{2}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l* [=>] 68.3	\[ \frac{2}{\color{blue}{\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}} \]
      associate-/l* [=>] 78.8	\[ \frac{2}{\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \]
      associate-*l/ [=>] 81.5	\[ \frac{2}{\color{blue}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)}{\frac{\ell}{t}}}} \]
      associate-/r/ [=>] 81.7	\[ \color{blue}{\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \cdot \frac{\ell}{t}} \]
      *-commutative [=>] 81.7	\[ \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k\right)}\right)} \cdot \frac{\ell}{t} \]
      associate-*l* [=>] 81.7	\[ \frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \color{blue}{\left(\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)\right)}\right)} \cdot \frac{\ell}{t} \]

   if -7.7999999999999994e-141 < k < 1.7500000000000001e-152

   1. Initial program 37.7%

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

   2. Simplified 6.0%

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

      [Start] 37.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 37.7	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 37.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 37.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 37.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 6.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 6.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 6.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 6.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in k around 0 6.1%

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

   4. Simplified 42.1%

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

      [Start] 6.1	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 6.1	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 6.7	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 6.7	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      associate-*l* [=>] 42.1	\[ \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   5. Applied egg-rr 95.0%

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

      [Start] 42.1	\[ \frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}} \]
      *-un-lft-identity [=>] 42.1	\[ \frac{\color{blue}{1 \cdot \ell}}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}} \]
      add-cube-cbrt [=>] 42.0	\[ \frac{1 \cdot \ell}{\color{blue}{\left(\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}} \cdot \sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\right) \cdot \sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}}} \]
      times-frac [=>] 42.0	\[ \color{blue}{\frac{1}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}} \cdot \sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \cdot \frac{\ell}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}}} \]
      pow2 [=>] 42.0	\[ \frac{1}{\color{blue}{{\left(\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\right)}^{2}}} \cdot \frac{\ell}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]
      cbrt-div [=>] 42.0	\[ \frac{1}{{\color{blue}{\left(\frac{\sqrt[3]{k \cdot \left(k \cdot {t}^{3}\right)}}{\sqrt[3]{\ell}}\right)}}^{2}} \cdot \frac{\ell}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]
      associate-*r* [=>] 6.7	\[ \frac{1}{{\left(\frac{\sqrt[3]{\color{blue}{\left(k \cdot k\right) \cdot {t}^{3}}}}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \frac{\ell}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]
      cbrt-prod [=>] 6.7	\[ \frac{1}{{\left(\frac{\color{blue}{\sqrt[3]{k \cdot k} \cdot \sqrt[3]{{t}^{3}}}}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \frac{\ell}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]
      cbrt-unprod [<=] 41.9	\[ \frac{1}{{\left(\frac{\color{blue}{\left(\sqrt[3]{k} \cdot \sqrt[3]{k}\right)} \cdot \sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \frac{\ell}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]
      pow2 [=>] 41.9	\[ \frac{1}{{\left(\frac{\color{blue}{{\left(\sqrt[3]{k}\right)}^{2}} \cdot \sqrt[3]{{t}^{3}}}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \frac{\ell}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]
      rem-cbrt-cube [=>] 41.9	\[ \frac{1}{{\left(\frac{{\left(\sqrt[3]{k}\right)}^{2} \cdot \color{blue}{t}}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \frac{\ell}{\sqrt[3]{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}} \]
      cbrt-div [=>] 41.9	\[ \frac{1}{{\left(\frac{{\left(\sqrt[3]{k}\right)}^{2} \cdot t}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \frac{\ell}{\color{blue}{\frac{\sqrt[3]{k \cdot \left(k \cdot {t}^{3}\right)}}{\sqrt[3]{\ell}}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{+117}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq -7.8 \cdot 10^{-141} \lor \neg \left(k \leq 1.75 \cdot 10^{-152}\right):\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{2}} \cdot \left(\sqrt[3]{\ell} \cdot \frac{\ell}{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.3%
Cost	26572

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -2.7 \cdot 10^{+117}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot t_1\right)\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-195}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot {\left(\sqrt[3]{k}\right)}^{2}}{\sqrt[3]{\ell}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\left(t \cdot \left(t_1 \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \tan k\right)\right)\right)\right) \cdot 0.5}\\ \end{array} \]

Alternative 2
Accuracy	85.4%
Cost	21005

\[\begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{+117}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq -9 \cdot 10^{-141} \lor \neg \left(k \leq 5 \cdot 10^{-151}\right):\\ \;\;\;\;\frac{2}{t \cdot \left(\frac{t}{\ell} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 3
Accuracy	85.3%
Cost	21004

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -1450:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_2}\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{+100}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(\ell \cdot \frac{\frac{2}{t}}{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(t \cdot \tan k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_2}{t_1}}\\ \end{array} \]

Alternative 4
Accuracy	84.4%
Cost	20620

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -780:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{-189}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{0.5 \cdot \left(t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{k \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	69.9%
Cost	19912

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(t_1 \cdot \frac{1}{k \cdot t}\right)\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-269}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{{\left(\frac{\ell}{k}\right)}^{2}}}{t}\right)}^{3}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{0.5 \cdot \left(t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{k \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 6
Accuracy	71.4%
Cost	14473

\[\begin{array}{l} \mathbf{if}\;k \leq -2.45 \cdot 10^{-160} \lor \neg \left(k \leq 5.4 \cdot 10^{-189}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{0.5 \cdot \left(t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{k \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 7
Accuracy	68.5%
Cost	13833

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-131} \lor \neg \left(t \leq 5.8 \cdot 10^{-87}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\ell \cdot \ell}{k \cdot k}} \cdot \frac{1}{t}\right)}^{3}\\ \end{array} \]

Alternative 8
Accuracy	65.9%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-134} \lor \neg \left(t \leq 3.8 \cdot 10^{-55}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(2 \cdot \frac{k}{\frac{\frac{\ell}{t}}{k}}\right)}\\ \end{array} \]

Alternative 9
Accuracy	66.4%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-139} \lor \neg \left(t \leq 5 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{2}{t \cdot \left(2 \cdot \frac{k \cdot k}{\frac{\ell}{t}}\right)}\\ \end{array} \]

Alternative 10
Accuracy	46.9%
Cost	832

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

Alternative 11
Accuracy	54.9%
Cost	832

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

Alternative 12
Accuracy	62.6%
Cost	832

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

Reproduce

herbie shell --seed 2023138 
(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))))