Toniolo and Linder, Equation (10+)

Average Accuracy: 49.2% → 85.8%

Time: 52.3s

Precision: binary64

Cost: 59012

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \sqrt[3]{\sin k}\\ \mathbf{if}\;t \leq -7.5:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_4\right)\right)}{t_1}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + t_3\right)\right)\right)}\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_2 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_2\right)}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot t_4}{t_1}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + t_3\right)\right)}\\ \end{array} \]

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 (pow (cbrt l) 2.0))
        (t_2 (pow (sin k) 2.0))
        (t_3 (pow (/ k t) 2.0))
        (t_4 (cbrt (sin k))))
   (if (<= t -7.5)
     (/
      2.0
      (*
       (pow (/ (* t (log1p (expm1 t_4))) t_1) 3.0)
       (* (tan k) (+ 1.0 (+ 1.0 t_3)))))
     (if (<= t -2.5e-160)
       (* 2.0 (* (/ l k) (/ l (* t_2 (* k (/ t (cos k)))))))
       (if (<= t 4.9e-219)
         (/ (* -2.0 (* (cos k) (/ l (* t (/ k l))))) (* k (- t_2)))
         (if (<= t 9.2e-55)
           (* 2.0 (* (* (/ l k) (/ l k)) (/ (cos k) (* t t_2))))
           (/
            2.0
            (* (pow (/ (* t t_4) t_1) 3.0) (* (tan k) (+ 2.0 t_3))))))))))

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 = pow(cbrt(l), 2.0);
	double t_2 = pow(sin(k), 2.0);
	double t_3 = pow((k / t), 2.0);
	double t_4 = cbrt(sin(k));
	double tmp;
	if (t <= -7.5) {
		tmp = 2.0 / (pow(((t * log1p(expm1(t_4))) / t_1), 3.0) * (tan(k) * (1.0 + (1.0 + t_3))));
	} else if (t <= -2.5e-160) {
		tmp = 2.0 * ((l / k) * (l / (t_2 * (k * (t / cos(k))))));
	} else if (t <= 4.9e-219) {
		tmp = (-2.0 * (cos(k) * (l / (t * (k / l))))) / (k * -t_2);
	} else if (t <= 9.2e-55) {
		tmp = 2.0 * (((l / k) * (l / k)) * (cos(k) / (t * t_2)));
	} else {
		tmp = 2.0 / (pow(((t * t_4) / t_1), 3.0) * (tan(k) * (2.0 + t_3)));
	}
	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.pow(Math.cbrt(l), 2.0);
	double t_2 = Math.pow(Math.sin(k), 2.0);
	double t_3 = Math.pow((k / t), 2.0);
	double t_4 = Math.cbrt(Math.sin(k));
	double tmp;
	if (t <= -7.5) {
		tmp = 2.0 / (Math.pow(((t * Math.log1p(Math.expm1(t_4))) / t_1), 3.0) * (Math.tan(k) * (1.0 + (1.0 + t_3))));
	} else if (t <= -2.5e-160) {
		tmp = 2.0 * ((l / k) * (l / (t_2 * (k * (t / Math.cos(k))))));
	} else if (t <= 4.9e-219) {
		tmp = (-2.0 * (Math.cos(k) * (l / (t * (k / l))))) / (k * -t_2);
	} else if (t <= 9.2e-55) {
		tmp = 2.0 * (((l / k) * (l / k)) * (Math.cos(k) / (t * t_2)));
	} else {
		tmp = 2.0 / (Math.pow(((t * t_4) / t_1), 3.0) * (Math.tan(k) * (2.0 + t_3)));
	}
	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(l) ^ 2.0
	t_2 = sin(k) ^ 2.0
	t_3 = Float64(k / t) ^ 2.0
	t_4 = cbrt(sin(k))
	tmp = 0.0
	if (t <= -7.5)
		tmp = Float64(2.0 / Float64((Float64(Float64(t * log1p(expm1(t_4))) / t_1) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_3)))));
	elseif (t <= -2.5e-160)
		tmp = Float64(2.0 * Float64(Float64(l / k) * Float64(l / Float64(t_2 * Float64(k * Float64(t / cos(k)))))));
	elseif (t <= 4.9e-219)
		tmp = Float64(Float64(-2.0 * Float64(cos(k) * Float64(l / Float64(t * Float64(k / l))))) / Float64(k * Float64(-t_2)));
	elseif (t <= 9.2e-55)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * Float64(l / k)) * Float64(cos(k) / Float64(t * t_2))));
	else
		tmp = Float64(2.0 / Float64((Float64(Float64(t * t_4) / t_1) ^ 3.0) * Float64(tan(k) * Float64(2.0 + t_3))));
	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[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[t, -7.5], N[(2.0 / N[(N[Power[N[(N[(t * N[Log[1 + N[(Exp[t$95$4] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -2.5e-160], N[(2.0 * N[(N[(l / k), $MachinePrecision] * N[(l / N[(t$95$2 * N[(k * N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 4.9e-219], N[(N[(-2.0 * N[(N[Cos[k], $MachinePrecision] * N[(l / N[(t * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * (-t$95$2)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e-55], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(t * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(N[(t * t$95$4), $MachinePrecision] / t$95$1), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -7.5

   1. Initial program 64.7%

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

   2. Simplified 64.7%

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

      [Start] 64.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 64.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 64.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 89.9%

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

   4. Simplified 89.9%

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

      [Start] 89.9	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*r/ [=>] 89.9	\[ \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Applied egg-rr 89.9%

      \[\leadsto \frac{2}{{\left(\frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right)} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if -7.5 < t < -2.49999999999999997e-160

   1. Initial program 40.8%

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

   2. Simplified 40.9%

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

      [Start] 40.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 40.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 40.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 66.6%

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

   4. Simplified 66.6%

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

      [Start] 66.6	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*r/ [=>] 66.6	\[ \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Taylor expanded in k around inf 58.3%

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

   6. Simplified 60.2%

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

      [Start] 58.3	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      associate-*r* [=>] 58.3	\[ \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot {\sin k}^{2}\right) \cdot t}}{\cos k \cdot {\ell}^{2}}} \]
      associate-/l* [=>] 55.7	\[ \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {\sin k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{t}}}} \]
      unpow2 [=>] 55.7	\[ \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {\sin k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{t}}} \]
      associate-*l* [=>] 55.7	\[ \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot {\sin k}^{2}\right)}}{\frac{\cos k \cdot {\ell}^{2}}{t}}} \]
      associate-/l* [=>] 60.2	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\frac{\cos k \cdot {\ell}^{2}}{t}}{k \cdot {\sin k}^{2}}}}} \]
      *-commutative [=>] 60.2	\[ \frac{2}{\frac{k}{\frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{t}}{k \cdot {\sin k}^{2}}}} \]
      associate-/l* [=>] 60.2	\[ \frac{2}{\frac{k}{\frac{\color{blue}{\frac{{\ell}^{2}}{\frac{t}{\cos k}}}}{k \cdot {\sin k}^{2}}}} \]
      unpow2 [=>] 60.2	\[ \frac{2}{\frac{k}{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{t}{\cos k}}}{k \cdot {\sin k}^{2}}}} \]

   7. Taylor expanded in k around inf 58.5%

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

   8. Simplified 78.2%

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

      [Start] 58.5	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 58.5	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 58.5	\[ 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 58.5	\[ 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
      unpow2 [=>] 58.5	\[ 2 \cdot \frac{\ell \cdot \ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}} \]
      associate-*l* [=>] 61.3	\[ 2 \cdot \frac{\ell \cdot \ell}{\frac{\color{blue}{k \cdot \left(k \cdot \left({\sin k}^{2} \cdot t\right)\right)}}{\cos k}} \]
      associate-*r/ [<=] 61.3	\[ 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot \frac{k \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}} \]
      associate-*r* [=>] 61.3	\[ 2 \cdot \frac{\ell \cdot \ell}{k \cdot \frac{\color{blue}{\left(k \cdot {\sin k}^{2}\right) \cdot t}}{\cos k}} \]
      associate-*r/ [<=] 61.3	\[ 2 \cdot \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(\left(k \cdot {\sin k}^{2}\right) \cdot \frac{t}{\cos k}\right)}} \]
      *-commutative [<=] 61.3	\[ 2 \cdot \frac{\ell \cdot \ell}{k \cdot \color{blue}{\left(\frac{t}{\cos k} \cdot \left(k \cdot {\sin k}^{2}\right)\right)}} \]
      times-frac [=>] 78.1	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{\frac{t}{\cos k} \cdot \left(k \cdot {\sin k}^{2}\right)}\right)} \]
      *-commutative [=>] 78.1	\[ 2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{\color{blue}{\left(k \cdot {\sin k}^{2}\right) \cdot \frac{t}{\cos k}}}\right) \]

   if -2.49999999999999997e-160 < t < 4.8999999999999999e-219

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 0.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 19.6%

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

   4. Simplified 19.6%

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

      [Start] 19.6	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*r/ [=>] 19.6	\[ \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Taylor expanded in k around inf 56.2%

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

   6. Simplified 58.0%

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

      [Start] 56.2	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      associate-*r* [=>] 56.2	\[ \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot {\sin k}^{2}\right) \cdot t}}{\cos k \cdot {\ell}^{2}}} \]
      associate-/l* [=>] 52.6	\[ \frac{2}{\color{blue}{\frac{{k}^{2} \cdot {\sin k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{t}}}} \]
      unpow2 [=>] 52.6	\[ \frac{2}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {\sin k}^{2}}{\frac{\cos k \cdot {\ell}^{2}}{t}}} \]
      associate-*l* [=>] 52.6	\[ \frac{2}{\frac{\color{blue}{k \cdot \left(k \cdot {\sin k}^{2}\right)}}{\frac{\cos k \cdot {\ell}^{2}}{t}}} \]
      associate-/l* [=>] 58.0	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\frac{\cos k \cdot {\ell}^{2}}{t}}{k \cdot {\sin k}^{2}}}}} \]
      *-commutative [=>] 58.0	\[ \frac{2}{\frac{k}{\frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{t}}{k \cdot {\sin k}^{2}}}} \]
      associate-/l* [=>] 58.0	\[ \frac{2}{\frac{k}{\frac{\color{blue}{\frac{{\ell}^{2}}{\frac{t}{\cos k}}}}{k \cdot {\sin k}^{2}}}} \]
      unpow2 [=>] 58.0	\[ \frac{2}{\frac{k}{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{t}{\cos k}}}{k \cdot {\sin k}^{2}}}} \]

   7. Applied egg-rr 67.9%

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

   8. Simplified 82.1%

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

      [Start] 67.9	\[ -2 \cdot \frac{1}{\frac{k}{\cos k \cdot \frac{\ell}{\frac{t}{\ell}}} \cdot \left(k \cdot \left(-{\sin k}^{2}\right)\right)} \]
      associate-/r* [=>] 68.7	\[ -2 \cdot \color{blue}{\frac{\frac{1}{\frac{k}{\cos k \cdot \frac{\ell}{\frac{t}{\ell}}}}}{k \cdot \left(-{\sin k}^{2}\right)}} \]
      associate-*r/ [=>] 68.7	\[ \color{blue}{\frac{-2 \cdot \frac{1}{\frac{k}{\cos k \cdot \frac{\ell}{\frac{t}{\ell}}}}}{k \cdot \left(-{\sin k}^{2}\right)}} \]
      associate-/l* [<=] 68.7	\[ \frac{-2 \cdot \color{blue}{\frac{1 \cdot \left(\cos k \cdot \frac{\ell}{\frac{t}{\ell}}\right)}{k}}}{k \cdot \left(-{\sin k}^{2}\right)} \]
      *-lft-identity [=>] 68.7	\[ \frac{-2 \cdot \frac{\color{blue}{\cos k \cdot \frac{\ell}{\frac{t}{\ell}}}}{k}}{k \cdot \left(-{\sin k}^{2}\right)} \]
      *-commutative [=>] 68.7	\[ \frac{-2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}} \cdot \cos k}}{k}}{k \cdot \left(-{\sin k}^{2}\right)} \]
      associate-*l/ [<=] 68.7	\[ \frac{-2 \cdot \color{blue}{\left(\frac{\frac{\ell}{\frac{t}{\ell}}}{k} \cdot \cos k\right)}}{k \cdot \left(-{\sin k}^{2}\right)} \]
      *-commutative [=>] 68.7	\[ \frac{-2 \cdot \color{blue}{\left(\cos k \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{k}\right)}}{k \cdot \left(-{\sin k}^{2}\right)} \]
      associate-/l/ [=>] 74.3	\[ \frac{-2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell}{k \cdot \frac{t}{\ell}}}\right)}{k \cdot \left(-{\sin k}^{2}\right)} \]
      associate-*r/ [=>] 79.0	\[ \frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{\color{blue}{\frac{k \cdot t}{\ell}}}\right)}{k \cdot \left(-{\sin k}^{2}\right)} \]
      associate-/l* [<=] 69.7	\[ \frac{-2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell \cdot \ell}{k \cdot t}}\right)}{k \cdot \left(-{\sin k}^{2}\right)} \]
      associate-/r* [=>] 67.1	\[ \frac{-2 \cdot \left(\cos k \cdot \color{blue}{\frac{\frac{\ell \cdot \ell}{k}}{t}}\right)}{k \cdot \left(-{\sin k}^{2}\right)} \]
      associate-/l* [=>] 69.1	\[ \frac{-2 \cdot \left(\cos k \cdot \frac{\color{blue}{\frac{\ell}{\frac{k}{\ell}}}}{t}\right)}{k \cdot \left(-{\sin k}^{2}\right)} \]
      associate-/l/ [=>] 82.1	\[ \frac{-2 \cdot \left(\cos k \cdot \color{blue}{\frac{\ell}{t \cdot \frac{k}{\ell}}}\right)}{k \cdot \left(-{\sin k}^{2}\right)} \]
      neg-mul-1 [=>] 82.1	\[ \frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \color{blue}{\left(-1 \cdot {\sin k}^{2}\right)}} \]

   if 4.8999999999999999e-219 < t < 9.20000000000000046e-55

   1. Initial program 19.6%

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

   2. Simplified 18.5%

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

      [Start] 19.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 19.6	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 19.6	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 18.2	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 18.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 18.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 18.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 18.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 59.9%

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

   4. Simplified 77.7%

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

      [Start] 59.9	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 59.9	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 57.9	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 57.9	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 57.9	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 77.7	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 77.7	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if 9.20000000000000046e-55 < t

   1. Initial program 64.9%

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

   2. Simplified 64.9%

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

      [Start] 64.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 64.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 64.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 88.3%

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

   4. Simplified 88.3%

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

      [Start] 88.3	\[ \frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*r/ [=>] 88.3	\[ \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Applied egg-rr 88.3%

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

   6. Simplified 88.3%

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

      [Start] 88.3	\[ \frac{2}{{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k + \tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      *-commutative [=>] 88.3	\[ \frac{2}{{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k + \color{blue}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}\right)} \]
      *-lft-identity [<=] 88.3	\[ \frac{2}{{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\color{blue}{1 \cdot \tan k} + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k\right)} \]
      distribute-rgt-out [=>] 88.3	\[ \frac{2}{{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \color{blue}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      associate-+r+ [=>] 88.3	\[ \frac{2}{{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      metadata-eval [=>] 88.3	\[ \frac{2}{{\left(\frac{\sqrt[3]{\sin k} \cdot t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt[3]{\sin k}\right)\right)}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq -2.5 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{{\sin k}^{2} \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-{\sin k}^{2}\right)}\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.9%
Cost	46480

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \mathbf{if}\;t \leq -9.5:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_1 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_1\right)}\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-46}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Accuracy	82.2%
Cost	46420

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -4:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + t_1\right)\right)\right) \cdot \left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_2 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_2\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-44}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_2}\right)\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+149}:\\ \;\;\;\;\frac{2}{\left(2 + t_1\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot \frac{\sin k}{\cos k}\right)}\\ \end{array} \]

Alternative 3
Accuracy	81.1%
Cost	40276

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -9.5:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot t_3\right) \cdot \left(\sin k \cdot {\left(\frac{t}{t_2}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq -1.35 \cdot 10^{-159}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_1 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_1\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-49}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+154}:\\ \;\;\;\;\frac{2}{t_3 \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{t_2}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 4
Accuracy	81.2%
Cost	40276

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -42:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + t_1\right)\right)\right) \cdot \left(\sin k \cdot {\left(t \cdot {\left(\sqrt[3]{\ell}\right)}^{-2}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq -5.6 \cdot 10^{-161}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_2 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_2\right)}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_2}\right)\\ \mathbf{elif}\;t \leq 1.04 \cdot 10^{+152}:\\ \;\;\;\;\frac{2}{\left(2 + t_1\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 5
Accuracy	80.9%
Cost	39684

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_3 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -6:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\sin k \cdot {\left(\frac{t}{t_2}\right)}^{3}\right)}\\ \mathbf{elif}\;t \leq -5.2 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_1 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_1\right)}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{elif}\;t \leq 10^{+191}:\\ \;\;\;\;t_3 \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{t_2}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ \end{array} \]

Alternative 6
Accuracy	78.7%
Cost	33492

\[\begin{array}{l} t_1 := \frac{2}{{\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot \left(2 \cdot k\right)}\\ t_2 := \frac{\ell}{k \cdot {t}^{1.5}}\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -42:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.8 \cdot 10^{-161}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_3 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 1.35 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_3\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-44}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_3}\right)\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{+191}:\\ \;\;\;\;t_2 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	77.8%
Cost	20868

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -10:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq -3 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_1 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{-219}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_1\right)}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \]

Alternative 8
Accuracy	74.2%
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{t}{\cos k}\\ t_3 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -4:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -9.5 \cdot 10^{-161}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_1 \cdot \left(k \cdot t_2\right)}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-220}:\\ \;\;\;\;\frac{2}{k} \cdot \left(\frac{\ell}{k \cdot t_1} \cdot \frac{\ell}{t_2}\right)\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-43}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot t_3\\ \end{array} \]

Alternative 9
Accuracy	74.4%
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -5.6:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -6.2 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_1 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-219}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(k \cdot \frac{\frac{k}{\ell} \cdot \frac{t}{\ell}}{\cos k}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-44}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_2\\ \end{array} \]

Alternative 10
Accuracy	75.3%
Cost	20752

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;t \leq -32:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq -2 \cdot 10^{-160}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t_2 \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{-2 \cdot \left(\cos k \cdot \frac{\ell}{t \cdot \frac{k}{\ell}}\right)}{k \cdot \left(-t_2\right)}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-43}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 11
Accuracy	73.1%
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -5.2:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 12
Accuracy	77.2%
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -15:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-50}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{{\sin k}^{2} \cdot \left(k \cdot \frac{t}{\cos k}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 13
Accuracy	64.8%
Cost	13896

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot {t}^{1.5}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-52}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_1\\ \end{array} \]

Alternative 14
Accuracy	63.9%
Cost	13772

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-48}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{elif}\;t \leq 7.2 \cdot 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{\left(k \cdot {t}^{1.5}\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 15
Accuracy	62.2%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{-16} \lor \neg \left(t \leq 3.15 \cdot 10^{-52}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \end{array} \]

Alternative 16
Accuracy	55.7%
Cost	1360

\[\begin{array}{l} t_1 := \frac{\ell}{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{if}\;k \leq -2 \cdot 10^{+62}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq -5.9 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k}\right)}{t \cdot k}\\ \end{array} \]

Alternative 17
Accuracy	55.7%
Cost	1360

\[\begin{array}{l} \mathbf{if}\;k \leq -3.6 \cdot 10^{+62}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq -5.9 \cdot 10^{-161}:\\ \;\;\;\;\frac{\ell}{t \cdot \frac{k \cdot k}{\frac{\ell}{t \cdot t}}}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \frac{k \cdot k}{\frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k}\right)}{t \cdot k}\\ \end{array} \]

Alternative 18
Accuracy	55.7%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -3.9 \lor \neg \left(t \leq 7.8 \cdot 10^{-58}\right):\\ \;\;\;\;\frac{\ell}{t \cdot \frac{k \cdot k}{\frac{\ell}{t \cdot t}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \end{array} \]

Alternative 19
Accuracy	41.8%
Cost	704

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

Alternative 20
Accuracy	43.0%
Cost	704

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

Alternative 21
Accuracy	46.8%
Cost	704

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

Reproduce

herbie shell --seed 2023125 
(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))))