Toniolo and Linder, Equation (10+)

Average Accuracy: 36.3% → 65.1%

Time: 41.0s

Precision: binary64

Cost: 112009

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ t_3 := \frac{\ell \cdot \sqrt{2}}{t \cdot \left(t_1 \cdot t_2\right)}\\ \mathbf{if}\;k \leq -7.8 \cdot 10^{+157} \lor \neg \left(k \leq 3.6 \cdot 10^{+114}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t \cdot t_2} \cdot \frac{t_3}{t_1}\\ \end{array} \]

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 (cbrt (sin k)))
        (t_2 (cbrt (* (tan k) (+ 2.0 (pow (/ k t) 2.0)))))
        (t_3 (/ (* l (sqrt 2.0)) (* t (* t_1 t_2)))))
   (if (or (<= k -7.8e+157) (not (<= k 3.6e+114)))
     (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t (pow (sin k) 2.0)))))
     (* (/ t_3 (* t t_2)) (/ 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 = cbrt(sin(k));
	double t_2 = cbrt((tan(k) * (2.0 + pow((k / t), 2.0))));
	double t_3 = (l * sqrt(2.0)) / (t * (t_1 * t_2));
	double tmp;
	if ((k <= -7.8e+157) || !(k <= 3.6e+114)) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * pow(sin(k), 2.0))));
	} else {
		tmp = (t_3 / (t * t_2)) * (t_3 / 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 = Math.cbrt(Math.sin(k));
	double t_2 = Math.cbrt((Math.tan(k) * (2.0 + Math.pow((k / t), 2.0))));
	double t_3 = (l * Math.sqrt(2.0)) / (t * (t_1 * t_2));
	double tmp;
	if ((k <= -7.8e+157) || !(k <= 3.6e+114)) {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = (t_3 / (t * t_2)) * (t_3 / 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 = cbrt(sin(k))
	t_2 = cbrt(Float64(tan(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))))
	t_3 = Float64(Float64(l * sqrt(2.0)) / Float64(t * Float64(t_1 * t_2)))
	tmp = 0.0
	if ((k <= -7.8e+157) || !(k <= 3.6e+114))
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(Float64(t_3 / Float64(t * t_2)) * Float64(t_3 / 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[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] / N[(t * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, -7.8e+157], N[Not[LessEqual[k, 3.6e+114]], $MachinePrecision]], 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], N[(N[(t$95$3 / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision] * N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if k < -7.79999999999999941e157 or 3.6000000000000001e114 < k

   1. Initial program 46.9%

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

   2. Simplified 46.9%

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

      [Start] 46.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 46.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 46.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 45.8%

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

      [Start] 46.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      expm1-log1p-u [=>] 46.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\right)\right)} \]
      expm1-udef [=>] 46.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\right)} - 1} \]
      associate-/r* [=>] 45.9	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}\right)} - 1 \]
      associate-*l/ [=>] 45.9	\[ e^{\mathsf{log1p}\left(\frac{\frac{2}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} - 1 \]
      associate-/l* [=>] 45.8	\[ e^{\mathsf{log1p}\left(\frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} - 1 \]
      associate-+r+ [=>] 45.8	\[ e^{\mathsf{log1p}\left(\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}}\right)} - 1 \]
      metadata-eval [=>] 45.8	\[ e^{\mathsf{log1p}\left(\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} - 1 \]

   4. Simplified 51.9%

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

      [Start] 45.8	\[ e^{\mathsf{log1p}\left(\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} - 1 \]
      expm1-def [=>] 46.8	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]
      expm1-log1p [=>] 46.8	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-/l/ [=>] 46.9	\[ \color{blue}{\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}} \]
      unpow2 [<=] 46.9	\[ \frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{{t}^{3}}{\frac{\color{blue}{{\ell}^{2}}}{\sin k}}} \]
      associate-/l* [<=] 46.9	\[ \frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{{\ell}^{2}}}} \]
      associate-*r/ [=>] 46.5	\[ \frac{2}{\color{blue}{\frac{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left({t}^{3} \cdot \sin k\right)}{{\ell}^{2}}}} \]

   5. Taylor expanded in t around 0 61.4%

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

   6. Simplified 87.8%

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

      [Start] 61.4	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 61.4	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 61.4	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 61.4	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 61.4	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 87.8	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 87.8	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if -7.79999999999999941e157 < k < 3.6000000000000001e114

   1. Initial program 32.1%

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

   2. Simplified 32.5%

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

      [Start] 32.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 32.1	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 33.0	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 32.2	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 32.5	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 32.5	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 32.5	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 32.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 32.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 32.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 32.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Applied egg-rr 41.6%

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

      [Start] 32.5	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \]
      associate-*r/ [=>] 32.6	\[ \color{blue}{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}} \]
      add-cube-cbrt [=>] 32.5	\[ \frac{\left(\ell \cdot \ell\right) \cdot 2}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\right) \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]
      associate-/r* [=>] 32.5	\[ \color{blue}{\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)} \cdot \sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}}} \]

   4. Applied egg-rr 56.3%

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

      [Start] 41.6	\[ \frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)} \]
      add-sqr-sqrt [=>] 41.6	\[ \frac{\color{blue}{\sqrt{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)} \]
      associate-*r* [=>] 41.6	\[ \frac{\sqrt{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}} \cdot \sqrt{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}}}{\color{blue}{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot t\right) \cdot \sqrt[3]{\sin k}}} \]
      times-frac [=>] 41.6	\[ \color{blue}{\frac{\sqrt{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot t} \cdot \frac{\sqrt{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}}}{\sqrt[3]{\sin k}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -7.8 \cdot 10^{+157} \lor \neg \left(k \leq 3.6 \cdot 10^{+114}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \sqrt{2}}{t \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}}{t \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\frac{\ell \cdot \sqrt{2}}{t \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}}{\sqrt[3]{\sin k}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	63.9%
Cost	79113

\[\begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{if}\;k \leq -7 \cdot 10^{+157} \lor \neg \left(k \leq 1.35 \cdot 10^{+116}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell \cdot \sqrt{2}}{t \cdot \left(t_1 \cdot t_2\right)}\right)}^{2} \cdot \frac{\frac{1}{t \cdot t_1}}{t_2}\\ \end{array} \]

Alternative 2
Accuracy	63.7%
Cost	78985

\[\begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{if}\;k \leq -6.4 \cdot 10^{+157} \lor \neg \left(k \leq 4.2 \cdot 10^{+114}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\frac{\ell}{t} \cdot \frac{\sqrt{2}}{t_1 \cdot t_2}\right)}^{2}}{t_2 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Accuracy	61.0%
Cost	72713

\[\begin{array}{l} t_1 := t \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{if}\;k \leq -6.4 \cdot 10^{+157} \lor \neg \left(k \leq 2.3 \cdot 10^{+114}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \ell}{t_1} \cdot \frac{\ell}{{t_1}^{2}}\\ \end{array} \]

Alternative 4
Accuracy	62.2%
Cost	46480

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{t}\right)}^{3}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -1.3 \cdot 10^{+158}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.9 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{-119}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	60.1%
Cost	46344

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\sqrt[3]{\ell}}{t}\\ t_3 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_4 := 2 + t_1\\ t_5 := \frac{\ell}{\tan k}\\ t_6 := \frac{2}{\sin k}\\ \mathbf{if}\;k \leq -7 \cdot 10^{+157}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -3.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{+44}:\\ \;\;\;\;\left(\left({t_2}^{2} \cdot \left(t_2 \cdot \frac{1}{t_4}\right)\right) \cdot t_5\right) \cdot t_6\\ \mathbf{elif}\;k \leq 3.6 \cdot 10^{+76}:\\ \;\;\;\;t_6 \cdot \left(t_5 \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{+116}:\\ \;\;\;\;t_6 \cdot \left(t_5 \cdot {\left(\frac{t_2}{\sqrt[3]{t_4}}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 6
Accuracy	59.4%
Cost	40144

\[\begin{array}{l} t_1 := \frac{2}{\sin k} \cdot \left(\frac{\ell}{\tan k} \cdot {\left(\frac{\frac{\sqrt[3]{\ell}}{t}}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{3}\right)\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -6.4 \cdot 10^{+157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.55 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3 \cdot 10^{-68}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Accuracy	56.9%
Cost	21128

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -6.4 \cdot 10^{+157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-292}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right) \cdot \left(\sin k \cdot \frac{t \cdot \frac{t}{\ell}}{\ell \cdot \frac{1}{t}}\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	56.5%
Cost	21000

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -3 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)} \cdot \left(\ell \cdot \left(\frac{1}{t} \cdot \frac{\ell}{t \cdot t}\right)\right)\\ \mathbf{elif}\;k \leq 1.42 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	56.6%
Cost	20872

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -3 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -6 \cdot 10^{-133}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	56.5%
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -5000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -4.9 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \left(\ell \cdot {k}^{-2}\right)}{t}}{t}\\ \mathbf{elif}\;k \leq -2.25 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	56.5%
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{if}\;k \leq -4050000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -4.6 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \left(\ell \cdot {k}^{-2}\right)}{t}}{t}\\ \mathbf{elif}\;k \leq -2.9 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{t_1}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Accuracy	55.7%
Cost	14288

\[\begin{array}{l} t_1 := \frac{2}{\sin k} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \mathbf{if}\;k \leq -4050000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t} \cdot \left(\ell \cdot {k}^{-2}\right)}{t}}{t}\\ \mathbf{elif}\;k \leq -2.9 \cdot 10^{-94}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Accuracy	54.3%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+37} \lor \neg \left(t \leq 1.05 \cdot 10^{+25}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(k \cdot \left(k \cdot \left(t \cdot \sin k\right)\right)\right)}\\ \end{array} \]

Alternative 14
Accuracy	57.9%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+37} \lor \neg \left(t \leq 7.5 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k} \cdot \left(\frac{\ell}{\tan k} \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 15
Accuracy	49.4%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -0.023 \lor \neg \left(t \leq 1.15 \cdot 10^{+18}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{{\left(k \cdot t\right)}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot {\left(\frac{k}{\frac{\ell}{k}}\right)}^{2}}\\ \end{array} \]

Alternative 16
Accuracy	45.9%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -1.36 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{k \cdot t}\\ \mathbf{elif}\;t \leq 10^{-31}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 17
Accuracy	46.5%
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -2.3 \cdot 10^{-75}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t \cdot t}}{k \cdot t}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-31}:\\ \;\;\;\;\frac{2}{t \cdot {\left(\frac{k}{\frac{\ell}{k}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot t\right)}\\ \end{array} \]

Alternative 18
Accuracy	33.9%
Cost	832

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

Alternative 19
Accuracy	34.1%
Cost	832

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

Alternative 20
Accuracy	39.6%
Cost	832

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

Reproduce

herbie shell --seed 2023153 
(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))))