Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 8.4

Time: 42.2s

Precision: binary64

Cost: 85772

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ t_3 := \sqrt[3]{t_1 \cdot \left(\sin k \cdot \tan k\right)}\\ t_4 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ t_5 := \frac{t}{t_4}\\ \mathbf{if}\;k \leq -6 \cdot 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\frac{t_4}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot t_1}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{\frac{2}{t_5}}{t_3}}{{\left(t_5 \cdot t_3\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2
         (* 2.0 (* (/ (* (/ l k) (/ l k)) (pow (sin k) 2.0)) (/ (cos k) t))))
        (t_3 (cbrt (* t_1 (* (sin k) (tan k)))))
        (t_4 (pow (cbrt l) 2.0))
        (t_5 (/ t t_4)))
   (if (<= k -6e+119)
     t_2
     (if (<= k 1.3e-154)
       (/
        2.0
        (* (pow (/ (* t (cbrt (tan k))) (/ t_4 (cbrt (sin k)))) 3.0) t_1))
       (if (<= k 5.6e+181)
         (/ (/ (/ 2.0 t_5) t_3) (pow (* t_5 t_3) 2.0))
         t_2)))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

↓

double code(double t, double l, double k) {
	double t_1 = 2.0 + pow((k / t), 2.0);
	double t_2 = 2.0 * ((((l / k) * (l / k)) / pow(sin(k), 2.0)) * (cos(k) / t));
	double t_3 = cbrt((t_1 * (sin(k) * tan(k))));
	double t_4 = pow(cbrt(l), 2.0);
	double t_5 = t / t_4;
	double tmp;
	if (k <= -6e+119) {
		tmp = t_2;
	} else if (k <= 1.3e-154) {
		tmp = 2.0 / (pow(((t * cbrt(tan(k))) / (t_4 / cbrt(sin(k)))), 3.0) * t_1);
	} else if (k <= 5.6e+181) {
		tmp = ((2.0 / t_5) / t_3) / pow((t_5 * t_3), 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = 2.0 * ((((l / k) * (l / k)) / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / t));
	double t_3 = Math.cbrt((t_1 * (Math.sin(k) * Math.tan(k))));
	double t_4 = Math.pow(Math.cbrt(l), 2.0);
	double t_5 = t / t_4;
	double tmp;
	if (k <= -6e+119) {
		tmp = t_2;
	} else if (k <= 1.3e-154) {
		tmp = 2.0 / (Math.pow(((t * Math.cbrt(Math.tan(k))) / (t_4 / Math.cbrt(Math.sin(k)))), 3.0) * t_1);
	} else if (k <= 5.6e+181) {
		tmp = ((2.0 / t_5) / t_3) / Math.pow((t_5 * t_3), 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

↓

function code(t, l, k)
	t_1 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(l / k)) / (sin(k) ^ 2.0)) * Float64(cos(k) / t)))
	t_3 = cbrt(Float64(t_1 * Float64(sin(k) * tan(k))))
	t_4 = cbrt(l) ^ 2.0
	t_5 = Float64(t / t_4)
	tmp = 0.0
	if (k <= -6e+119)
		tmp = t_2;
	elseif (k <= 1.3e-154)
		tmp = Float64(2.0 / Float64((Float64(Float64(t * cbrt(tan(k))) / Float64(t_4 / cbrt(sin(k)))) ^ 3.0) * t_1));
	elseif (k <= 5.6e+181)
		tmp = Float64(Float64(Float64(2.0 / t_5) / t_3) / (Float64(t_5 * t_3) ^ 2.0));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(t$95$1 * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(t / t$95$4), $MachinePrecision]}, If[LessEqual[k, -6e+119], t$95$2, If[LessEqual[k, 1.3e-154], N[(2.0 / N[(N[Power[N[(N[(t * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(t$95$4 / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 5.6e+181], N[(N[(N[(2.0 / t$95$5), $MachinePrecision] / t$95$3), $MachinePrecision] / N[Power[N[(t$95$5 * t$95$3), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if k < -6.00000000000000002e119 or 5.59999999999999968e181 < k

   1. Initial program 34.1

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

   2. Simplified 34.1

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

      [Start] 34.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 34.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 34.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 23.1

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

   4. Simplified 23.3

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

      [Start] 23.1	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/r* [=>] 23.3	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]
      *-commutative [=>] 23.3	\[ 2 \cdot \frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{\color{blue}{t \cdot {\sin k}^{2}}} \]
      associate-/r* [=>] 23.3	\[ 2 \cdot \color{blue}{\frac{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{t}}{{\sin k}^{2}}} \]
      *-commutative [=>] 23.3	\[ 2 \cdot \frac{\frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{t}}{{\sin k}^{2}} \]
      associate-/l* [=>] 23.3	\[ 2 \cdot \frac{\frac{\color{blue}{\frac{{\ell}^{2}}{\frac{{k}^{2}}{\cos k}}}}{t}}{{\sin k}^{2}} \]
      unpow2 [=>] 23.3	\[ 2 \cdot \frac{\frac{\frac{\color{blue}{\ell \cdot \ell}}{\frac{{k}^{2}}{\cos k}}}{t}}{{\sin k}^{2}} \]
      unpow2 [=>] 23.3	\[ 2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\frac{\color{blue}{k \cdot k}}{\cos k}}}{t}}{{\sin k}^{2}} \]

   5. Applied egg-rr 6.2

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

   if -6.00000000000000002e119 < k < 1.3e-154

   1. Initial program 33.9

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

   2. Simplified 30.5

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

      [Start] 33.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 33.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 33.0	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 32.9	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 30.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 30.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 30.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 30.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 22.1

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

   4. Simplified 22.1

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

      [Start] 22.1	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{2}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      unpow2 [=>] 22.1	\[ \frac{2}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot \color{blue}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      cube-mult [<=] 22.1	\[ \frac{2}{\color{blue}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell \cdot \frac{\ell}{\sin k}}}\right)}^{3}} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied egg-rr 8.9

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

   if 1.3e-154 < k < 5.59999999999999968e181

   1. Initial program 29.1

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

   2. Simplified 29.0

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

      [Start] 29.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 29.1	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 29.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 29.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 29.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 29.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 29.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 29.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 29.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 10.5

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

   4. Simplified 10.5

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

      [Start] 10.5	\[ \frac{1}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}} \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \]
      associate-*l/ [=>] 10.5	\[ \color{blue}{\frac{1 \cdot \frac{2}{\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{{\left(\sqrt[3]{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{2}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 8.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -6 \cdot 10^{+119}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-154}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\frac{{\left(\sqrt[3]{\ell}\right)}^{2}}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+181}:\\ \;\;\;\;\frac{\frac{\frac{2}{\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}}}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	8.5
Cost	52616

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ t_3 := {\left(\sqrt[3]{\ell}\right)}^{2}\\ \mathbf{if}\;k \leq -2.9 \cdot 10^{+121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{2}{{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\frac{t_3}{\sqrt[3]{\sin k}}}\right)}^{3} \cdot t_1}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+181}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{t_3} \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot t_1\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	9.4
Cost	46612

\[\begin{array}{l} t_1 := t \cdot \sqrt{k}\\ t_2 := \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ t_3 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ t_4 := t \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -8.8 \cdot 10^{+120}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -3.2 \cdot 10^{-128}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.1 \cdot 10^{-299}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot t_4}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-158}:\\ \;\;\;\;\frac{\frac{\ell}{t_1}}{t_4 \cdot t_1}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+181}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 3
Error	10.3
Cost	46408

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;k \leq -1.5 \cdot 10^{+121}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;\frac{2}{t_2 \cdot {\left(\frac{t \cdot \sqrt[3]{\tan k}}{\sqrt[3]{\ell} \cdot \frac{1}{\sqrt[3]{\frac{\sin k}{\ell}}}}\right)}^{3}}\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+181}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot t_2\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	11.4
Cost	40344

\[\begin{array}{l} t_1 := \frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) \cdot {\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_3 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -3.1 \cdot 10^{+121}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{t_3} \cdot \frac{\cos k}{t}\right)\\ \mathbf{elif}\;k \leq -4.8 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-150}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6 \cdot 10^{+136}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot t}\right)}{t_3}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{+181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{+240}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\cos k}{k} \cdot \frac{\frac{\ell}{k}}{\frac{t}{\ell}}\right) \cdot {\sin k}^{-2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{\cos k}{t_3 \cdot t}\right)\\ \end{array} \]

Alternative 5
Error	12.1
Cost	39752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t_1} \cdot \frac{\cos k}{t}\right)\\ \mathbf{if}\;k \leq -4 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{\cos k}{t_1 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}} \cdot \left(\frac{\ell}{t} \cdot \frac{\sqrt[3]{\ell}}{t}\right)\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{-78}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	12.4
Cost	21268

\[\begin{array}{l} t_1 := \frac{2}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ t_2 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ t_3 := \sqrt[3]{\frac{\ell}{t}}\\ \mathbf{if}\;k \leq -2.3 \cdot 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.2 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3 \cdot 10^{-121}:\\ \;\;\;\;\left(\ell \cdot \frac{{t_3}^{2}}{t \cdot \left(k \cdot k\right)}\right) \cdot \frac{t_3}{t}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-78}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	12.5
Cost	21268

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{\sin k}^{2}} \cdot \frac{\cos k}{t}\right)\\ t_3 := \frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\\ \mathbf{if}\;k \leq -4.8 \cdot 10^{+119}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -3.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot t_3\right) \cdot \left(\tan k \cdot \left(1 + \left(t_1 + 1\right)\right)\right)}\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{-251}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-79}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+77}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + t_1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Error	13.8
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t_1 \cdot t}\right)\\ \mathbf{if}\;k \leq -1.75 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 0.0066:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t_1}\\ \mathbf{elif}\;k \leq 1.04 \cdot 10^{+77}:\\ \;\;\;\;\frac{{\sin k}^{-2} \cdot \left(\ell \cdot \cos k\right)}{t} \cdot \frac{\ell}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	13.8
Cost	20752

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t_1} \cdot \frac{\cos k}{t}\right)\\ \mathbf{if}\;k \leq -1.6 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 0.009:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{t_1}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{+77}:\\ \;\;\;\;\frac{{\sin k}^{-2} \cdot \left(\ell \cdot \cos k\right)}{t} \cdot \frac{\ell}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	21.3
Cost	20624

\[\begin{array}{l} \mathbf{if}\;k \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell \cdot \ell}{\frac{k \cdot k}{\cos k}}}{t}}{0.5 - 0.5 \cdot \cos \left(k + k\right)}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 0.007:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{{\sin k}^{-2} \cdot \left(\ell \cdot \cos k\right)}{t} \cdot \frac{\ell}{t \cdot t}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\ell \cdot \left(\ell \cdot {k}^{-2}\right)}}{t}\right)}^{3}\\ \end{array} \]

Alternative 11
Error	12.3
Cost	20620

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t_1} \cdot \frac{\cos k}{t}\right)\\ \mathbf{if}\;k \leq -2.4 \cdot 10^{+16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+199}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k \cdot t}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	20.3
Cost	19912

\[\begin{array}{l} \mathbf{if}\;t \leq -6.9 \cdot 10^{-31}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-74}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{\left(\sqrt[3]{\frac{\ell}{k}}\right)}^{2}}{t}\right)}^{3}\\ \end{array} \]

Alternative 13
Error	20.2
Cost	13961

\[\begin{array}{l} \mathbf{if}\;t \leq -6.9 \cdot 10^{-31} \lor \neg \left(t \leq 7 \cdot 10^{-74}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}}{{\sin k}^{2}}\\ \end{array} \]

Alternative 14
Error	22.3
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-28} \lor \neg \left(t \leq 3.7 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \ell}{t \cdot {k}^{4}}\\ \end{array} \]

Alternative 15
Error	21.4
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -6.9 \cdot 10^{-31} \lor \neg \left(t \leq 3.7 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{\frac{\ell}{k \cdot t}}{t \cdot \left(t \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\ell}{t}}{{k}^{4}}\\ \end{array} \]

Alternative 16
Error	29.0
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -2.2 \cdot 10^{+86} \lor \neg \left(t \leq 1.66 \cdot 10^{-12}\right):\\ \;\;\;\;\ell \cdot \frac{\frac{\ell}{t}}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{t \cdot t}}{k}}{t}\\ \end{array} \]

Alternative 17
Error	29.9
Cost	832

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

Alternative 18
Error	29.7
Cost	832

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

Alternative 19
Error	28.8
Cost	832

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

Alternative 20
Error	23.9
Cost	832

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

Reproduce

herbie shell --seed 2023187 
(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))))