Toniolo and Linder, Equation (10+)

Average Error: 32.8 → 6.0

Time: 59.5s

Precision: binary64

Cost: 59212

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \sqrt[3]{\sin k}\\ t_3 := 2 + t_1\\ \mathbf{if}\;t \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{t_2}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + t_1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\ell}{t_3 \cdot \frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t_2 \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left(\sqrt[3]{\tan k \cdot t_3}\right)}^{3}}\\ \end{array} \]

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 (sin k))) (t_3 (+ 2.0 t_1)))
   (if (<= t -5.5e-5)
     (/
      2.0
      (*
       (pow (* t (/ t_2 (* (cbrt l) (cbrt l)))) 3.0)
       (* (tan k) (+ 1.0 (+ 1.0 t_1)))))
     (if (<= t 9.5e-28)
       (/
        2.0
        (* (/ k (/ l (pow (sin k) 2.0))) (/ (- t) (* l (- (/ (cos k) k))))))
       (if (<= t 8.8e+101)
         (* 2.0 (/ l (* t_3 (/ (* (tan k) (pow t 3.0)) (/ l (sin k))))))
         (/
          2.0
          (*
           (pow (* t_2 (/ t (pow (cbrt l) 2.0))) 3.0)
           (pow (cbrt (* (tan k) t_3)) 3.0))))))))

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(sin(k));
	double t_3 = 2.0 + t_1;
	double tmp;
	if (t <= -5.5e-5) {
		tmp = 2.0 / (pow((t * (t_2 / (cbrt(l) * cbrt(l)))), 3.0) * (tan(k) * (1.0 + (1.0 + t_1))));
	} else if (t <= 9.5e-28) {
		tmp = 2.0 / ((k / (l / pow(sin(k), 2.0))) * (-t / (l * -(cos(k) / k))));
	} else if (t <= 8.8e+101) {
		tmp = 2.0 * (l / (t_3 * ((tan(k) * pow(t, 3.0)) / (l / sin(k)))));
	} else {
		tmp = 2.0 / (pow((t_2 * (t / pow(cbrt(l), 2.0))), 3.0) * pow(cbrt((tan(k) * t_3)), 3.0));
	}
	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.sin(k));
	double t_3 = 2.0 + t_1;
	double tmp;
	if (t <= -5.5e-5) {
		tmp = 2.0 / (Math.pow((t * (t_2 / (Math.cbrt(l) * Math.cbrt(l)))), 3.0) * (Math.tan(k) * (1.0 + (1.0 + t_1))));
	} else if (t <= 9.5e-28) {
		tmp = 2.0 / ((k / (l / Math.pow(Math.sin(k), 2.0))) * (-t / (l * -(Math.cos(k) / k))));
	} else if (t <= 8.8e+101) {
		tmp = 2.0 * (l / (t_3 * ((Math.tan(k) * Math.pow(t, 3.0)) / (l / Math.sin(k)))));
	} else {
		tmp = 2.0 / (Math.pow((t_2 * (t / Math.pow(Math.cbrt(l), 2.0))), 3.0) * Math.pow(Math.cbrt((Math.tan(k) * t_3)), 3.0));
	}
	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(sin(k))
	t_3 = Float64(2.0 + t_1)
	tmp = 0.0
	if (t <= -5.5e-5)
		tmp = Float64(2.0 / Float64((Float64(t * Float64(t_2 / Float64(cbrt(l) * cbrt(l)))) ^ 3.0) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + t_1)))));
	elseif (t <= 9.5e-28)
		tmp = Float64(2.0 / Float64(Float64(k / Float64(l / (sin(k) ^ 2.0))) * Float64(Float64(-t) / Float64(l * Float64(-Float64(cos(k) / k))))));
	elseif (t <= 8.8e+101)
		tmp = Float64(2.0 * Float64(l / Float64(t_3 * Float64(Float64(tan(k) * (t ^ 3.0)) / Float64(l / sin(k))))));
	else
		tmp = Float64(2.0 / Float64((Float64(t_2 * Float64(t / (cbrt(l) ^ 2.0))) ^ 3.0) * (cbrt(Float64(tan(k) * t_3)) ^ 3.0)));
	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[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + t$95$1), $MachinePrecision]}, If[LessEqual[t, -5.5e-5], N[(2.0 / N[(N[Power[N[(t * N[(t$95$2 / N[(N[Power[l, 1/3], $MachinePrecision] * N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e-28], N[(2.0 / N[(N[(k / N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-t) / N[(l * (-N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.8e+101], N[(2.0 * N[(l / N[(t$95$3 * N[(N[(N[Tan[k], $MachinePrecision] * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Power[N[(t$95$2 * N[(t / N[Power[N[Power[l, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[Power[N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -5.5000000000000002e-5

   1. Initial program 22.0

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

   2. Simplified 22.0

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

      [Start] 22.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 22.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 22.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 6.7

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

   4. Applied egg-rr 6.7

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

   5. Simplified 6.7

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

      [Start] 6.7	\[ \frac{2}{{\left(\frac{\frac{\sqrt[3]{\sin k}}{\frac{\sqrt[3]{\ell}}{t}}}{\sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-/l/ [=>] 6.7	\[ \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \frac{\sqrt[3]{\ell}}{t}}\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*r/ [=>] 6.7	\[ \frac{2}{{\left(\frac{\sqrt[3]{\sin k}}{\color{blue}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{t}}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-/r/ [=>] 6.7	\[ \frac{2}{{\color{blue}{\left(\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot t\right)}}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if -5.5000000000000002e-5 < t < 9.50000000000000001e-28

   1. Initial program 51.4

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

   2. Simplified 51.4

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

      [Start] 51.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 51.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 51.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 26.2

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

   4. Simplified 22.1

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

      [Start] 26.2	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 27.6	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 27.6	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 27.6	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 27.6	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 22.1	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 11.4

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

   6. Simplified 5.7

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

      [Start] 11.4	\[ \frac{2}{\frac{k \cdot \left(-t\right)}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}} \]
      times-frac [=>] 5.7	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}}} \]

   if 9.50000000000000001e-28 < t < 8.8000000000000003e101

   1. Initial program 19.7

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

   2. Simplified 13.7

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

      [Start] 19.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 19.7	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 17.8	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 17.1	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 13.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 13.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 13.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 13.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Applied egg-rr 24.3

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

   4. Simplified 7.1

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

      [Start] 24.3	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\frac{\ell}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)} - 1 \]
      expm1-def [=>] 12.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{\frac{\ell}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]
      expm1-log1p [=>] 8.9	\[ \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
      associate-/l/ [=>] 7.1	\[ 2 \cdot \color{blue}{\frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}} \]
      *-commutative [=>] 7.1	\[ 2 \cdot \frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\color{blue}{{t}^{3} \cdot \tan k}}{\frac{\ell}{\sin k}}} \]

   if 8.8000000000000003e101 < t

   1. Initial program 24.5

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

   2. Simplified 24.5

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

      [Start] 24.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 24.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 24.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 4.9

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

   4. Applied egg-rr 4.9

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

4. Final simplification 6.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3} \cdot {\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}^{3}}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.0
Cost	46540

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{{\left(t \cdot \frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}\right)}^{3} \cdot \left(\tan k \cdot \left(1 + \left(1 + t_1\right)\right)\right)}\\ \mathbf{if}\;t \leq -0.00058:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\ell}{\left(2 + t_1\right) \cdot \frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	7.0
Cost	46480

\[\begin{array}{l} t_1 := \frac{2}{{\left(\frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}} \cdot \sqrt[3]{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)}^{3}}\\ t_2 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{t_2}} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;k \leq -1.38 \cdot 10^{-129}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \frac{t \cdot k}{\ell}} \cdot \frac{\ell}{\frac{k}{\cos k}}\\ \end{array} \]

Alternative 3
Error	6.0
Cost	46476

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\left(\tan k \cdot \left(1 + \left(1 + t_1\right)\right)\right) \cdot {\left(\sqrt[3]{\sin k} \cdot \frac{t}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{if}\;t \leq -0.0048:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;2 \cdot \frac{\ell}{\left(2 + t_1\right) \cdot \frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	6.0
Cost	46476

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{2}{\left(\tan k \cdot \left(1 + \left(1 + t_1\right)\right)\right) \cdot {\left(\frac{t \cdot \sqrt[3]{\sin k}}{{\left(\sqrt[3]{\ell}\right)}^{2}}\right)}^{3}}\\ \mathbf{if}\;t \leq -0.028:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{+101}:\\ \;\;\;\;2 \cdot \frac{\ell}{\left(2 + t_1\right) \cdot \frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	8.5
Cost	27212

\[\begin{array}{l} \mathbf{if}\;t \leq -7.2:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+92}:\\ \;\;\;\;2 \cdot \frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{1}{t}}{\frac{{\left(t \cdot k\right)}^{2}}{\ell}}\\ \end{array} \]

Alternative 6
Error	9.5
Cost	20616

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -0.092:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 9.6 \cdot 10^{-16}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \end{array} \]

Alternative 7
Error	9.7
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -4.6 \cdot 10^{+21} \lor \neg \left(k \leq 14200000000000\right):\\ \;\;\;\;\frac{\ell}{\frac{k}{\cos k}} \cdot \left(\frac{2}{t \cdot k} \cdot \left(\ell \cdot {\sin k}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 8
Error	9.6
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -2.45 \cdot 10^{+20} \lor \neg \left(k \leq 12200000000000\right):\\ \;\;\;\;\frac{2}{{\sin k}^{2} \cdot \frac{t \cdot k}{\ell}} \cdot \frac{\ell}{\frac{k}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 9
Error	10.8
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -14:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-35}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{k} \cdot \frac{\ell}{\frac{t \cdot k}{\cos k}}}{{\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \end{array} \]

Alternative 10
Error	11.7
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{+20} \lor \neg \left(k \leq 1.15 \cdot 10^{+20}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{0.5 - \frac{\cos \left(k + k\right)}{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 11
Error	18.1
Cost	14408

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-48}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-48}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{1 - \cos \left(k + k\right)}{\ell \cdot \frac{2 \cdot \ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \end{array} \]

Alternative 12
Error	18.8
Cost	8008

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-44}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\left(\ell \cdot \frac{\cos k}{k}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \end{array} \]

Alternative 13
Error	18.9
Cost	7752

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 4.6 \cdot 10^{-38}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \end{array} \]

Alternative 14
Error	19.5
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -1.04 \cdot 10^{-53}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 7.3 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \end{array} \]

Alternative 15
Error	20.1
Cost	1480

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 2.35 \cdot 10^{-40}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \frac{\ell}{k \cdot k}\right) \cdot \frac{-0.16666666666666666 + \frac{1}{k \cdot k}}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \end{array} \]

Alternative 16
Error	20.9
Cost	1224

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{t_1 \cdot \frac{\ell}{t}}{t \cdot k}\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-51}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{1}{\frac{t \cdot k}{t_1}}\\ \end{array} \]

Alternative 17
Error	30.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -8 \cdot 10^{+28} \lor \neg \left(k \leq 5 \cdot 10^{+23}\right):\\ \;\;\;\;2 \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{t \cdot t}\\ \end{array} \]

Alternative 18
Error	25.7
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-35} \lor \neg \left(t \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{\ell}{\left(t \cdot t\right) \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]

Alternative 19
Error	24.6
Cost	1097

\[\begin{array}{l} \mathbf{if}\;t \leq -2.9 \cdot 10^{-34} \lor \neg \left(t \leq 1.7 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{\ell}{\frac{k \cdot \left(t \cdot t\right)}{\frac{\ell}{t \cdot k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]

Alternative 20
Error	21.2
Cost	1097

\[\begin{array}{l} \mathbf{if}\;k \leq -5 \cdot 10^{+30} \lor \neg \left(k \leq 9.5 \cdot 10^{+23}\right):\\ \;\;\;\;2 \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot k} \cdot \frac{\ell}{t}}{t \cdot k}\\ \end{array} \]

Alternative 21
Error	35.4
Cost	1096

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8.5 \cdot 10^{-140}:\\ \;\;\;\;2 \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \frac{t}{\ell \cdot -0.16666666666666666}}\\ \mathbf{elif}\;\ell \leq 1.65 \cdot 10^{+115}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k \cdot k}{\ell}} \cdot \frac{-0.16666666666666666}{t}\right)\\ \end{array} \]

Alternative 22
Error	35.5
Cost	1092

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+235}:\\ \;\;\;\;2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{k \cdot k}{\ell}} \cdot \frac{-0.16666666666666666}{t}\right)\\ \end{array} \]

Alternative 23
Error	36.2
Cost	832

\[2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{k \cdot \left(t \cdot k\right)}\right) \]

Reproduce

herbie shell --seed 2023038 
(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))))