Toniolo and Linder, Equation (10+)

Average Accuracy: 48.8% → 85.1%

Time: 42.6s

Precision: binary64

Cost: 46216.00

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\\ t_3 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -9.2 \cdot 10^{+58}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \mathbf{elif}\;k \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot t_2} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{2}{t_2}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \end{array} \]

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 (* (/ l k) (/ l k)))
        (t_2 (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))
        (t_3 (* t (pow (sin k) 2.0))))
   (if (<= k -9.2e+58)
     (* 2.0 (* t_1 (/ (cos k) t_3)))
     (if (<= k -7.5e-149)
       (/ 2.0 (pow (* (cbrt (* (tan k) t_2)) (/ t (pow (cbrt l) 2.0))) 3.0))
       (if (<= k 1.35e+88)
         (/ (* (* (/ l t) (/ (/ l t) (tan k))) (/ 2.0 t_2)) t)
         (* 2.0 (/ (cos 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 = (l / k) * (l / k);
	double t_2 = sin(k) * (2.0 + pow((k / t), 2.0));
	double t_3 = t * pow(sin(k), 2.0);
	double tmp;
	if (k <= -9.2e+58) {
		tmp = 2.0 * (t_1 * (cos(k) / t_3));
	} else if (k <= -7.5e-149) {
		tmp = 2.0 / pow((cbrt((tan(k) * t_2)) * (t / pow(cbrt(l), 2.0))), 3.0);
	} else if (k <= 1.35e+88) {
		tmp = (((l / t) * ((l / t) / tan(k))) * (2.0 / t_2)) / t;
	} else {
		tmp = 2.0 * (cos(k) / (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 = (l / k) * (l / k);
	double t_2 = Math.sin(k) * (2.0 + Math.pow((k / t), 2.0));
	double t_3 = t * Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= -9.2e+58) {
		tmp = 2.0 * (t_1 * (Math.cos(k) / t_3));
	} else if (k <= -7.5e-149) {
		tmp = 2.0 / Math.pow((Math.cbrt((Math.tan(k) * t_2)) * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0);
	} else if (k <= 1.35e+88) {
		tmp = (((l / t) * ((l / t) / Math.tan(k))) * (2.0 / t_2)) / t;
	} else {
		tmp = 2.0 * (Math.cos(k) / (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 = Float64(Float64(l / k) * Float64(l / k))
	t_2 = Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))
	t_3 = Float64(t * (sin(k) ^ 2.0))
	tmp = 0.0
	if (k <= -9.2e+58)
		tmp = Float64(2.0 * Float64(t_1 * Float64(cos(k) / t_3)));
	elseif (k <= -7.5e-149)
		tmp = Float64(2.0 / (Float64(cbrt(Float64(tan(k) * t_2)) * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0));
	elseif (k <= 1.35e+88)
		tmp = Float64(Float64(Float64(Float64(l / t) * Float64(Float64(l / t) / tan(k))) * Float64(2.0 / t_2)) / t);
	else
		tmp = Float64(2.0 * Float64(cos(k) / 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[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -9.2e+58], N[(2.0 * N[(t$95$1 * N[(N[Cos[k], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -7.5e-149], N[(2.0 / N[Power[N[(N[Power[N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision] * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.35e+88], N[(N[(N[(N[(l / t), $MachinePrecision] * N[(N[(l / t), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / t$95$2), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(t$95$3 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -9.2000000000000001e58

   1. Initial program 47.2%

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

   2. Simplified 47.2%

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

      [Start] 47.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 47.2	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 47.2	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 47.2	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 47.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 47.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 47.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 47.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 67.2%

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

   4. Simplified 87.3%

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

      [Start] 67.2	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 67.2	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 65.4	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 65.4	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 65.4	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 87.3	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 87.3	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if -9.2000000000000001e58 < k < -7.49999999999999995e-149

   1. Initial program 54.5%

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

   2. Simplified 54.5%

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

      [Start] 54.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 54.5	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 54.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 86.8%

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

   if -7.49999999999999995e-149 < k < 1.35000000000000008e88

   1. Initial program 47.5%

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

   2. Simplified 32.5%

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

      [Start] 47.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 47.5	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 47.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 47.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 47.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 33.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 33.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 33.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 33.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 35.1%

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

   4. Applied egg-rr 67.5%

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

   5. Applied egg-rr 80.1%

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

   if 1.35000000000000008e88 < k

   1. Initial program 48.6%

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

   2. Simplified 48.6%

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

      [Start] 48.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 48.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 48.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 69.1%

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

   4. Simplified 88.7%

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

      [Start] 69.1	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 69.1	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      *-commutative [=>] 69.1	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]
      associate-/l* [=>] 66.6	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]
      unpow2 [=>] 66.6	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]
      unpow2 [=>] 66.6	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}} \]
      times-frac [=>] 88.7	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.2 \cdot 10^{+58}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq -7.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+88}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{2}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.3%
Cost	21136.00

\[\begin{array}{l} t_1 := \frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_3 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -3 \cdot 10^{+50}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t_3}\right)\\ \mathbf{elif}\;k \leq -1.6 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 9 \cdot 10^{-140}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{1}{k}}{t}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_2}}\\ \end{array} \]

Alternative 2
Accuracy	84.2%
Cost	21136.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\\ t_3 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.35 \cdot 10^{+59}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \mathbf{elif}\;k \leq -7.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;k \leq 8.6 \cdot 10^{-140}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{1}{k}}{t}\\ \mathbf{elif}\;k \leq 3.35 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{2}{t_2}}{t} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \end{array} \]

Alternative 3
Accuracy	84.5%
Cost	21136.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -3.8 \cdot 10^{+58}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \mathbf{elif}\;k \leq -5.6 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \left(\tan k \cdot \left(\sin k \cdot t_2\right)\right)\right)}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-189}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{1}{k}}{t}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t \cdot \tan k}}{\frac{t}{\frac{\frac{2}{t_2}}{\sin k}}}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \end{array} \]

Alternative 4
Accuracy	83.7%
Cost	21004.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{2}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ t_3 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.65 \cdot 10^{+47}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\frac{\ell}{t}}{\tan k}}{t \cdot \frac{\sin k}{t_2}}}{t}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{+100}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{\ell}{t \cdot \tan k}}{\frac{t}{\frac{t_2}{\sin k}}}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \end{array} \]

Alternative 5
Accuracy	83.9%
Cost	21004.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := t \cdot {\sin k}^{2}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -3.5 \cdot 10^{+51}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_2}\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-154}:\\ \;\;\;\;\frac{\frac{\ell \cdot \frac{\frac{\ell}{t}}{\tan k}}{t \cdot \frac{\sin k}{\frac{2}{t_3}}}}{t}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{+98}:\\ \;\;\;\;\frac{\frac{\frac{2}{\sin k \cdot t_3} \cdot \frac{\ell}{t \cdot \tan k}}{\frac{t}{\ell}}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_2}{t_1}}\\ \end{array} \]

Alternative 6
Accuracy	84.3%
Cost	20872.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -9 \cdot 10^{+44}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_2}\right)\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{2}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_2}{t_1}}\\ \end{array} \]

Alternative 7
Accuracy	82.3%
Cost	20752.00

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := t \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -5.8 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.05 \cdot 10^{-271}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\left(t \cdot t_3\right) \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{1}{k}}{t}\\ \mathbf{elif}\;k \leq 6.5:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(t_2 \cdot t_3\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Accuracy	82.2%
Cost	20752.00

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_3 := t \cdot \frac{k}{\ell}\\ t_4 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -6.8 \cdot 10^{+32}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t_4}\right)\\ \mathbf{elif}\;k \leq -8.5 \cdot 10^{-269}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\left(t \cdot t_3\right) \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;k \leq 1.26 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{1}{k}}{t}\\ \mathbf{elif}\;k \leq 48:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(t_1 \cdot t_3\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_2}}\\ \end{array} \]

Alternative 9
Accuracy	70.8%
Cost	14284.00

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-71}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{-171}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right) \cdot \frac{k}{\frac{\cos k}{\sin k}}}\\ \mathbf{elif}\;t \leq 3.15 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	69.7%
Cost	8336.00

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-71}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \mathbf{elif}\;t \leq 5.1 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\frac{\ell}{k \cdot \frac{t}{\ell}}}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \left(t \cdot \frac{k}{\ell}\right)\right) \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Accuracy	69.7%
Cost	8336.00

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\frac{\ell}{k \cdot \frac{t}{\ell}}}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{t}{\frac{\ell}{k}} \cdot \frac{t \cdot t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	69.8%
Cost	8336.00

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{1}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+119}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Accuracy	67.6%
Cost	7888.00

\[\begin{array}{l} t_1 := \frac{1}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\frac{\ell}{k \cdot \frac{t}{\ell}}}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+69}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{\ell}{t} \cdot \frac{\frac{\ell}{t}}{\tan k}\right) \cdot \frac{1}{k}}{t}\\ \end{array} \]

Alternative 14
Accuracy	67.3%
Cost	7696.00

\[\begin{array}{l} t_1 := \frac{1}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ t_2 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -3.3 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -3.4 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 15
Accuracy	67.6%
Cost	7696.00

\[\begin{array}{l} t_1 := \frac{1}{k} \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)\\ t_2 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{{k}^{3}}{\frac{\ell}{k \cdot \frac{t}{\ell}}}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 16
Accuracy	67.4%
Cost	7568.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+87}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.9 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-109}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 17
Accuracy	68.5%
Cost	7568.00

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.4 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 18
Accuracy	66.1%
Cost	7436.00

\[\begin{array}{l} t_1 := \frac{1}{k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+89}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-79}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell \cdot \ell}{{k}^{4}}}{t}\\ \mathbf{elif}\;t \leq 6 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 19
Accuracy	61.9%
Cost	1488.00

\[\begin{array}{l} t_1 := \frac{1}{k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ t_2 := k \cdot \left(k \cdot t\right)\\ t_3 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot t_2}\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{+87}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.5 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t \cdot t_2}}{t}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 20
Accuracy	63.4%
Cost	1488.00

\[\begin{array}{l} t_1 := \frac{1}{k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -4.3 \cdot 10^{-94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{-t}}{t \cdot \left(\left(k \cdot k\right) \cdot \left(-t\right)\right)}\\ \mathbf{elif}\;t \leq 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 21
Accuracy	63.0%
Cost	1488.00

\[\begin{array}{l} t_1 := \frac{1}{k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -2.3 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{-t}}{t \cdot \left(t \cdot \left(k \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 22
Accuracy	63.2%
Cost	1488.00

\[\begin{array}{l} t_1 := \frac{1}{k} \cdot \frac{\ell}{\frac{t}{\ell} \cdot \left(t \cdot \left(k \cdot t\right)\right)}\\ t_2 := \frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.7 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{-87}:\\ \;\;\;\;\frac{1}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell \cdot \ell}{t} \cdot \frac{1}{t}\right)\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{+118}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 23
Accuracy	59.3%
Cost	1097.00

\[\begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;t \leq -5 \cdot 10^{-221} \lor \neg \left(t \leq 3.8 \cdot 10^{-126}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{\frac{t}{\ell} \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \ell}{t \cdot t_1}}{t}\\ \end{array} \]

Alternative 24
Accuracy	45.9%
Cost	832.00

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

Alternative 25
Accuracy	55.9%
Cost	832.00

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

Alternative 26
Accuracy	57.5%
Cost	832.00

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

Reproduce

herbie shell --seed 2023096 
(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))))