Toniolo and Linder, Equation (10+)

Average Error: 33.0 → 7.3

Time: 45.5s

Precision: binary64

Cost: 46800

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\tan k}\\ t_2 := \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ t_3 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+104}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\frac{\sin k}{\ell} \cdot t_3}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{t \cdot k}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(\frac{{t_1}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t_1}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (* t (cbrt (tan k))))
        (t_2 (* (/ 1.0 (/ (* t k) (/ l (* t k)))) (/ l t)))
        (t_3 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -5.6e+104)
     t_2
     (if (<= t -9e-50)
       (/ (* l (* 2.0 (/ (pow t -3.0) (tan k)))) (* (/ (sin k) l) t_3))
       (if (<= t 1.8e-60)
         (* 2.0 (/ (* l (/ (cos k) (* t k))) (* (/ k l) (pow (sin k) 2.0))))
         (if (<= t 5.8e+105)
           (/ 2.0 (* t_3 (* (/ (pow t_1 2.0) (/ l (sin k))) (/ t_1 l))))
           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 = t * cbrt(tan(k));
	double t_2 = (1.0 / ((t * k) / (l / (t * k)))) * (l / t);
	double t_3 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -5.6e+104) {
		tmp = t_2;
	} else if (t <= -9e-50) {
		tmp = (l * (2.0 * (pow(t, -3.0) / tan(k)))) / ((sin(k) / l) * t_3);
	} else if (t <= 1.8e-60) {
		tmp = 2.0 * ((l * (cos(k) / (t * k))) / ((k / l) * pow(sin(k), 2.0)));
	} else if (t <= 5.8e+105) {
		tmp = 2.0 / (t_3 * ((pow(t_1, 2.0) / (l / sin(k))) * (t_1 / l)));
	} 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 = t * Math.cbrt(Math.tan(k));
	double t_2 = (1.0 / ((t * k) / (l / (t * k)))) * (l / t);
	double t_3 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -5.6e+104) {
		tmp = t_2;
	} else if (t <= -9e-50) {
		tmp = (l * (2.0 * (Math.pow(t, -3.0) / Math.tan(k)))) / ((Math.sin(k) / l) * t_3);
	} else if (t <= 1.8e-60) {
		tmp = 2.0 * ((l * (Math.cos(k) / (t * k))) / ((k / l) * Math.pow(Math.sin(k), 2.0)));
	} else if (t <= 5.8e+105) {
		tmp = 2.0 / (t_3 * ((Math.pow(t_1, 2.0) / (l / Math.sin(k))) * (t_1 / l)));
	} 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(t * cbrt(tan(k)))
	t_2 = Float64(Float64(1.0 / Float64(Float64(t * k) / Float64(l / Float64(t * k)))) * Float64(l / t))
	t_3 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -5.6e+104)
		tmp = t_2;
	elseif (t <= -9e-50)
		tmp = Float64(Float64(l * Float64(2.0 * Float64((t ^ -3.0) / tan(k)))) / Float64(Float64(sin(k) / l) * t_3));
	elseif (t <= 1.8e-60)
		tmp = Float64(2.0 * Float64(Float64(l * Float64(cos(k) / Float64(t * k))) / Float64(Float64(k / l) * (sin(k) ^ 2.0))));
	elseif (t <= 5.8e+105)
		tmp = Float64(2.0 / Float64(t_3 * Float64(Float64((t_1 ^ 2.0) / Float64(l / sin(k))) * Float64(t_1 / l))));
	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[(t * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / N[(N[(t * k), $MachinePrecision] / N[(l / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -5.6e+104], t$95$2, If[LessEqual[t, -9e-50], N[(N[(l * N[(2.0 * N[(N[Power[t, -3.0], $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.8e-60], N[(2.0 * N[(N[(l * N[(N[Cos[k], $MachinePrecision] / N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(k / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e+105], N[(2.0 / N[(t$95$3 * N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.6e104 or 5.8000000000000002e105 < t

   1. Initial program 24.2

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

   2. Simplified 24.6

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

      [Start] 24.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 24.2	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 24.6	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 24.6	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 24.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 24.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 24.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 24.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around 0 30.3

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Simplified 27.0

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

      [Start] 30.3	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 30.3	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 27.0	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 27.0	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]

   5. Applied egg-rr 25.2

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

   6. Applied egg-rr 7.8

      \[\leadsto \color{blue}{\frac{1}{\frac{k \cdot t}{\frac{\ell}{k \cdot t}}} \cdot \frac{\ell}{t}} \]

   if -5.6e104 < t < -8.99999999999999924e-50

   1. Initial program 21.5

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

   2. Simplified 15.4

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

      [Start] 21.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 21.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 21.7	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 19.8	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 19.7	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 18.5	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 15.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 15.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 15.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 15.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Applied egg-rr 7.0

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

   if -8.99999999999999924e-50 < t < 1.8e-60

   1. Initial program 55.8

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

   2. Simplified 56.5

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

      [Start] 55.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 55.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 55.8	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 56.5	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 56.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 56.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 56.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 56.5	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 26.5

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

   4. Simplified 16.7

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

      [Start] 26.5	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 26.5	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 28.2	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 28.2	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 28.2	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 16.7	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 16.7	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 3.6

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

   6. Applied egg-rr 5.4

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

   7. Applied egg-rr 5.7

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

   if 1.8e-60 < t < 5.8000000000000002e105

   1. Initial program 21.4

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

   2. Simplified 15.6

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

      [Start] 21.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 21.4	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 19.5	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 19.3	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 15.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 15.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 15.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 15.6	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 9.1

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\ell}\right)} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 7.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{elif}\;t \leq -9 \cdot 10^{-50}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{t \cdot k}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{\left(t \cdot \sqrt[3]{\tan k}\right)}^{2}}{\frac{\ell}{\sin k}} \cdot \frac{t \cdot \sqrt[3]{\tan k}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.5
Cost	27476

\[\begin{array}{l} t_1 := \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + t_2}\\ \mathbf{if}\;t \leq -7.9 \cdot 10^{+96}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.4 \cdot 10^{-21}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-57}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + t_2\right)\right)\right)}\\ \mathbf{elif}\;t \leq 22000000:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+101}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	7.4
Cost	27408

\[\begin{array}{l} t_1 := \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ t_2 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-40}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\frac{\sin k}{\ell} \cdot \left(2 + t_2\right)}\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{t \cdot k}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\tan k}{\ell} \cdot \frac{{t}^{3} \cdot \left(-2 - t_2\right)}{\frac{-\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	8.1
Cost	27344

\[\begin{array}{l} t_1 := \frac{2}{\frac{\frac{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot {t}^{3}\right)}{\ell}}{\frac{\ell}{\sin k}}}\\ t_2 := \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{if}\;t \leq -5 \cdot 10^{+82}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.25 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{t \cdot k}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	7.1
Cost	27344

\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\\ t_2 := \frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{if}\;t \leq -2.35 \cdot 10^{+105}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.45 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{t \cdot k}}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	8.9
Cost	20488

\[\begin{array}{l} t_1 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -265000000000:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{t_1}}}{\frac{k}{\ell} \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\\ \mathbf{elif}\;k \leq 7.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot t_1\right) \cdot \frac{\frac{\ell}{k}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 6
Error	8.8
Cost	20488

\[\begin{array}{l} t_1 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -0.19:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{t_1}}}{\frac{k}{\ell} \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_1}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 7
Error	8.8
Cost	20488

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -7.5 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{t \cdot k}}{t_1}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{t_1}\\ \end{array} \]

Alternative 8
Error	8.8
Cost	20488

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot {\sin k}^{2}\\ t_2 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -8 \cdot 10^{-14}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{t_2}}}{t_1}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_1}\\ \end{array} \]

Alternative 9
Error	19.2
Cost	14416

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := \frac{1}{\frac{t \cdot k}{t_1}} \cdot \frac{\ell}{t}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-60}:\\ \;\;\;\;2 \cdot \frac{t_1}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \mathbf{elif}\;t \leq 2.75 \cdot 10^{+45}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{\tan k}}{2 \cdot \left(t \cdot \sin k\right)}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Error	11.5
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -1100000000 \lor \neg \left(k \leq 3.5 \cdot 10^{-8}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \end{array} \]

Alternative 11
Error	8.9
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -0.185 \lor \neg \left(k \leq 3.7 \cdot 10^{-8}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \end{array} \]

Alternative 12
Error	9.0
Cost	14408

\[\begin{array}{l} t_1 := \frac{k}{\ell} \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)\\ t_2 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -0.2:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{k}{t_2}}}{t_1}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{-8}:\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_1}\\ \end{array} \]

Alternative 13
Error	19.1
Cost	13961

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-41} \lor \neg \left(t \leq 2.1 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{t_1}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_1}{\frac{k}{\ell} \cdot {\sin k}^{2}}\\ \end{array} \]

Alternative 14
Error	19.1
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{-53} \lor \neg \left(t \leq 1.18 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \end{array} \]

Alternative 15
Error	21.2
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-53} \lor \neg \left(t \leq 6.1 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{1}{\frac{t \cdot k}{\frac{\ell}{t \cdot k}}} \cdot \frac{\ell}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k}\right)}{t \cdot k}\\ \end{array} \]

Alternative 16
Error	27.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -5.7 \cdot 10^{-42} \lor \neg \left(t \leq 1.5 \cdot 10^{-60}\right):\\ \;\;\;\;\ell \cdot \frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k}\right)}{t \cdot k}\\ \end{array} \]

Alternative 17
Error	26.0
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{-117} \lor \neg \left(t \leq 7 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k}\right)}{t \cdot k}\\ \end{array} \]

Alternative 18
Error	24.9
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-49} \lor \neg \left(t \leq 7 \cdot 10^{-61}\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k}\right)}{t \cdot k}\\ \end{array} \]

Alternative 19
Error	26.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-60}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot \frac{-0.3333333333333333}{k}\right)}{t \cdot k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]

Alternative 20
Error	35.9
Cost	968

\[\begin{array}{l} \mathbf{if}\;\ell \leq -7.6 \cdot 10^{+93}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot -0.3333333333333333}}\\ \mathbf{elif}\;\ell \leq 6.8 \cdot 10^{-154}:\\ \;\;\;\;\frac{\ell \cdot \left(\ell \cdot -0.3333333333333333\right)}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 21
Error	36.0
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+296}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{t} \cdot \frac{\ell}{k \cdot k}\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 22
Error	35.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot -0.3333333333333333}}\\ \end{array} \]

Alternative 23
Error	34.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+28}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot \frac{-0.3333333333333333}{t \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot -0.3333333333333333}}\\ \end{array} \]

Alternative 24
Error	34.7
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+34}:\\ \;\;\;\;\frac{-0.3333333333333333 \cdot \frac{\ell \cdot \ell}{t \cdot k}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot \left(k \cdot k\right)}{\ell \cdot -0.3333333333333333}}\\ \end{array} \]

Alternative 25
Error	37.6
Cost	704

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

Reproduce

herbie shell --seed 2023028 
(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))))