Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 10.3

Time: 1.1min

Precision: binary64

Cost: 53128

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\tan k}\\ t_3 := t \cdot t_2\\ t_4 := t_3 \cdot \frac{t_3}{\ell}\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(t_4 \cdot \frac{t_3 \cdot \sin k}{-\ell}\right) \cdot \left(-2 - t_1\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-97}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + t_1\right) \cdot \left(t_4 \cdot \left(t_2 \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}\\ \end{array} \]

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 (/ k t) 2.0))
        (t_2 (cbrt (tan k)))
        (t_3 (* t t_2))
        (t_4 (* t_3 (/ t_3 l))))
   (if (<= t -8.2e-20)
     (/ 2.0 (* (* t_4 (/ (* t_3 (sin k)) (- l))) (- -2.0 t_1)))
     (if (<= t 8e-97)
       (/ 2.0 (* (/ k (/ (cos k) k)) (* (/ t l) (/ (pow (sin k) 2.0) l))))
       (/ 2.0 (* (+ 2.0 t_1) (* t_4 (* t_2 (/ (sin k) (/ l t))))))))))

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((k / t), 2.0);
	double t_2 = cbrt(tan(k));
	double t_3 = t * t_2;
	double t_4 = t_3 * (t_3 / l);
	double tmp;
	if (t <= -8.2e-20) {
		tmp = 2.0 / ((t_4 * ((t_3 * sin(k)) / -l)) * (-2.0 - t_1));
	} else if (t <= 8e-97) {
		tmp = 2.0 / ((k / (cos(k) / k)) * ((t / l) * (pow(sin(k), 2.0) / l)));
	} else {
		tmp = 2.0 / ((2.0 + t_1) * (t_4 * (t_2 * (sin(k) / (l / t)))));
	}
	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((k / t), 2.0);
	double t_2 = Math.cbrt(Math.tan(k));
	double t_3 = t * t_2;
	double t_4 = t_3 * (t_3 / l);
	double tmp;
	if (t <= -8.2e-20) {
		tmp = 2.0 / ((t_4 * ((t_3 * Math.sin(k)) / -l)) * (-2.0 - t_1));
	} else if (t <= 8e-97) {
		tmp = 2.0 / ((k / (Math.cos(k) / k)) * ((t / l) * (Math.pow(Math.sin(k), 2.0) / l)));
	} else {
		tmp = 2.0 / ((2.0 + t_1) * (t_4 * (t_2 * (Math.sin(k) / (l / t)))));
	}
	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(k / t) ^ 2.0
	t_2 = cbrt(tan(k))
	t_3 = Float64(t * t_2)
	t_4 = Float64(t_3 * Float64(t_3 / l))
	tmp = 0.0
	if (t <= -8.2e-20)
		tmp = Float64(2.0 / Float64(Float64(t_4 * Float64(Float64(t_3 * sin(k)) / Float64(-l))) * Float64(-2.0 - t_1)));
	elseif (t <= 8e-97)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(cos(k) / k)) * Float64(Float64(t / l) * Float64((sin(k) ^ 2.0) / l))));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 + t_1) * Float64(t_4 * Float64(t_2 * Float64(sin(k) / Float64(l / t))))));
	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[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(t * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[(t$95$3 / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8.2e-20], N[(2.0 / N[(N[(t$95$4 * N[(N[(t$95$3 * N[Sin[k], $MachinePrecision]), $MachinePrecision] / (-l)), $MachinePrecision]), $MachinePrecision] * N[(-2.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-97], N[(2.0 / N[(N[(k / N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] * N[(N[(t / l), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 + t$95$1), $MachinePrecision] * N[(t$95$4 * N[(t$95$2 * N[(N[Sin[k], $MachinePrecision] / N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.2000000000000002e-20

   1. Initial program 22.7

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

   2. Simplified 20.8

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

      [Start] 22.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.7	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 22.3	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 22.1	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 20.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 20.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 20.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 20.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 11.0

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

   4. Simplified 9.4

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

      [Start] 11.0	\[ \frac{2}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\ell} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\frac{\ell}{\sin k}}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      associate-/r/ [=>] 9.4	\[ \frac{2}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\ell} \cdot \color{blue}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\ell} \cdot \sin k\right)}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied egg-rr 3.8

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

   6. Applied egg-rr 4.3

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

   if -8.2000000000000002e-20 < t < 8.00000000000000029e-97

   1. Initial program 55.9

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

   2. Simplified 56.1

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

      [Start] 55.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 55.9	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 55.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 55.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 55.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 56.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 56.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 56.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 56.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in t around 0 26.9

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

   4. Simplified 22.1

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

      [Start] 26.9	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 28.8	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 28.8	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 28.8	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 28.8	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 28.8	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 22.1	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   if 8.00000000000000029e-97 < t

   1. Initial program 22.7

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

   2. Simplified 21.5

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

      [Start] 22.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 22.7	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 22.7	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 22.8	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 21.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 21.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 21.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 21.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 12.6

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

   4. Simplified 11.2

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

      [Start] 12.6	\[ \frac{2}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\ell} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\frac{\ell}{\sin k}}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
      associate-/r/ [=>] 11.2	\[ \frac{2}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\ell} \cdot \color{blue}{\left(\frac{t \cdot \sqrt[3]{\tan k}}{\ell} \cdot \sin k\right)}\right) \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   5. Applied egg-rr 6.4

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

   6. Applied egg-rr 6.6

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

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(\left(\left(t \cdot \sqrt[3]{\tan k}\right) \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\ell}\right) \cdot \frac{\left(t \cdot \sqrt[3]{\tan k}\right) \cdot \sin k}{-\ell}\right) \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-97}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\left(t \cdot \sqrt[3]{\tan k}\right) \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\ell}\right) \cdot \left(\sqrt[3]{\tan k} \cdot \frac{\sin k}{\frac{\ell}{t}}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	53129

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := t \cdot t_1\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{-20} \lor \neg \left(t \leq 2.5 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{t_2}{\ell} \cdot \sin k\right) \cdot \left(t_2 \cdot \left(t \cdot \frac{t_1}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	10.1
Cost	53129

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ t_2 := \frac{t_1}{\ell}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{-20} \lor \neg \left(t \leq 2.55 \cdot 10^{-95}\right):\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t_1 \cdot t_2\right) \cdot \left(t_2 \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \end{array} \]

Alternative 3
Error	10.2
Cost	53128

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := t \cdot \sqrt[3]{\tan k}\\ t_3 := \frac{t_2}{\ell}\\ t_4 := t_2 \cdot t_3\\ \mathbf{if}\;t \leq -9.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\left(t_4 \cdot \frac{t_2 \cdot \sin k}{-\ell}\right) \cdot \left(-2 - t_1\right)}\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(2 + t_1\right) \cdot \left(t_4 \cdot \left(t_3 \cdot \sin k\right)\right)}\\ \end{array} \]

Alternative 4
Error	12.9
Cost	46860

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := t \cdot t_1\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \frac{2}{t_3 \cdot \left(\frac{{t_2}^{2}}{\ell} \cdot \frac{t \cdot \sin k}{\frac{\ell}{t_1}}\right)}\\ t_5 := \frac{t_2}{\ell}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{-20}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+167}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\left(t_5 \cdot \sin k\right) \cdot \left(t_5 \cdot \left(t \cdot \sqrt[3]{k}\right)\right)\right)}\\ \end{array} \]

Alternative 5
Error	13.8
Cost	46668

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{2}{t_2 \cdot \left(\left(\frac{t_1}{\ell} \cdot \sin k\right) \cdot \frac{{t_1}^{2}}{\ell}\right)}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-19}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{+246}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+278}:\\ \;\;\;\;\frac{2}{{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot t_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \end{array} \]

Alternative 6
Error	13.9
Cost	46668

\[\begin{array}{l} t_1 := \sqrt[3]{\tan k}\\ t_2 := t \cdot t_1\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_4 := \frac{{t_2}^{2}}{\ell}\\ \mathbf{if}\;t \leq -6.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(t_4 \cdot \frac{t \cdot \sin k}{\frac{\ell}{t_1}}\right)}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+246}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\left(\frac{t_2}{\ell} \cdot \sin k\right) \cdot t_4\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+278}:\\ \;\;\;\;\frac{2}{{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot t_3\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \end{array} \]

Alternative 7
Error	17.4
Cost	40268

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+101}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(t_2 \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot t_2\right)\right) \cdot {\left({t}^{0.75} \cdot \left({t}^{0.75} \cdot \frac{1}{\ell}\right)\right)}^{2}}\\ \end{array} \]

Alternative 8
Error	17.4
Cost	40140

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 2 + t_1\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(t_2 \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left({t}^{0.75} \cdot \frac{{t}^{0.75}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot t_2\right)\right)}\\ \end{array} \]

Alternative 9
Error	16.8
Cost	33676

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 1 + \left(1 + t_1\right)\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot t_2}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+75}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right)}\\ \end{array} \]

Alternative 10
Error	16.9
Cost	27600

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(2 + t_1\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+195}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{1}{\frac{t}{k} \cdot \frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+245}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \end{array} \]

Alternative 11
Error	17.2
Cost	27472

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -8.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+117}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(2 + t_1\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+197}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\ell}\right)}^{2} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{+245}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{k}{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell}}}{k}}}\\ \end{array} \]

Alternative 12
Error	16.4
Cost	27212

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 3.3 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2 \cdot \ell}{\left(\left(2 + t_1\right) \cdot \frac{\sin k}{\ell}\right) \cdot \left(\tan k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 13
Error	16.8
Cost	27212

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 1.12 \cdot 10^{-97}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+99}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 14
Error	18.2
Cost	27089

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.95 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 0.108 \lor \neg \left(t \leq 8.2 \cdot 10^{+148}\right):\\ \;\;\;\;\frac{2}{{\left(\mathsf{hypot}\left(1, \mathsf{hypot}\left(1, \frac{k}{t}\right)\right) \cdot \left(k \cdot \frac{{t}^{1.5}}{\ell}\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{2 + t_1}\\ \end{array} \]

Alternative 15
Error	18.8
Cost	21528

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.3 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -2.55 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot t_1\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 0.1:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{-\ell}{-k}\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{t \cdot \left(\frac{\ell}{k} \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)\right)}{\frac{k}{t} \cdot \frac{\sin k}{\ell}}\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{+103}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 16
Error	18.3
Cost	21396

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right) \cdot \left(1 + \left(1 + t_1\right)\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{elif}\;t \leq 0.108:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{-\ell}{-k}\\ \mathbf{elif}\;t \leq 3.7 \cdot 10^{+24}:\\ \;\;\;\;\frac{t \cdot \left(\frac{\ell}{k} \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)\right)}{\frac{k}{t} \cdot \frac{\sin k}{\ell}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+102}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{\frac{\frac{2}{\sin k}}{t \cdot t} \cdot \frac{\frac{1}{\tan k}}{t}}{2 + t_1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 17
Error	18.6
Cost	20872

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq -0.00155:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 18
Error	22.0
Cost	20488

\[\begin{array}{l} \mathbf{if}\;t \leq -2.5 \cdot 10^{+67}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 19
Error	21.4
Cost	20488

\[\begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+65}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 20
Error	21.8
Cost	20488

\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+65}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{\cos k}{k \cdot k} \cdot \left(2 \cdot \frac{\frac{\ell \cdot \ell}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 21
Error	21.7
Cost	20488

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+66}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\cos k}\right) \cdot \left({\sin k}^{2} \cdot \frac{t}{\ell \cdot \ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 22
Error	20.0
Cost	20488

\[\begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+65}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 2.55 \cdot 10^{-95}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 23
Error	22.9
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq -3.1 \cdot 10^{-7}:\\ \;\;\;\;\frac{\ell \cdot \ell}{{\left(t \cdot {\left(\sqrt[3]{k}\right)}^{2}\right)}^{3}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 24
Error	23.2
Cost	13512

\[\begin{array}{l} \mathbf{if}\;t \leq -7.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\frac{2}{\ell} \cdot \frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 25
Error	24.5
Cost	7432

\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{-\ell}{-k}\\ \end{array} \]

Alternative 26
Error	25.0
Cost	7432

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\frac{2}{\ell} \cdot \frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-107}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{-\ell}{-k}\\ \end{array} \]

Alternative 27
Error	24.5
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-20} \lor \neg \left(t \leq 4.3 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{k \cdot {t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \end{array} \]

Alternative 28
Error	28.2
Cost	2256

\[\begin{array}{l} t_1 := \frac{\frac{-0.3333333333333333 + \frac{2}{k \cdot k}}{\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}}}{2 + \frac{k \cdot k}{t \cdot t}}\\ \mathbf{if}\;k \leq -1450000000000:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}\\ \mathbf{elif}\;k \leq -1.05 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-154}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 4800:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\ell}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 29
Error	38.0
Cost	704

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

Alternative 30
Error	37.3
Cost	704

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

Alternative 31
Error	34.9
Cost	704

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

Reproduce

herbie shell --seed 2023067 
(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))))