Toniolo and Linder, Equation (10-)

Average Error: 47.8 → 9.3

Time: 1.8min

Precision: binary64

Cost: 73108

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \sqrt[3]{\frac{\sin k \cdot \tan k}{\ell \cdot \ell}} \cdot t\\ t_3 := \frac{\tan k}{\ell}\\ t_4 := \sqrt[3]{t_1}\\ t_5 := \sqrt[3]{t_3}\\ t_6 := t \cdot t_4\\ t_7 := \frac{8}{t_3}\\ t_8 := \frac{\frac{\frac{\sqrt[3]{t_7}}{t}}{t_4}}{\frac{k}{t}}\\ t_9 := \frac{\frac{\sqrt[3]{\frac{t_7}{t_1}}}{t_2 \cdot \frac{k \cdot t}{t}}}{\frac{t_2 \cdot k}{t}}\\ t_10 := \sqrt[3]{\sin k}\\ \mathbf{if}\;\ell \leq -1.15 \cdot 10^{+136}:\\ \;\;\;\;\frac{t_8}{{\left(t_5 \cdot \left(t \cdot \frac{t_10}{\sqrt[3]{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-161}:\\ \;\;\;\;t_9\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{t_3}}}{t_6} \cdot \frac{\frac{\sqrt[3]{2}}{k} \cdot \sqrt[3]{4}}{\frac{1}{t}}}{{\left(t_5 \cdot t_6\right)}^{2} \cdot \frac{k}{t}}\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{+125}:\\ \;\;\;\;t_9\\ \mathbf{else}:\\ \;\;\;\;\frac{t_8}{{\left(t_5 \cdot \left(t \cdot \begin{array}{l} \mathbf{if}\;t_10 \ne 0:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array}\right)\right)}^{2} \cdot \frac{k}{t}}\\ \end{array} \]

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 (/ (sin k) l))
        (t_2 (* (cbrt (/ (* (sin k) (tan k)) (* l l))) t))
        (t_3 (/ (tan k) l))
        (t_4 (cbrt t_1))
        (t_5 (cbrt t_3))
        (t_6 (* t t_4))
        (t_7 (/ 8.0 t_3))
        (t_8 (/ (/ (/ (cbrt t_7) t) t_4) (/ k t)))
        (t_9 (/ (/ (cbrt (/ t_7 t_1)) (* t_2 (/ (* k t) t))) (/ (* t_2 k) t)))
        (t_10 (cbrt (sin k))))
   (if (<= l -1.15e+136)
     (/ t_8 (* (pow (* t_5 (* t (/ t_10 (cbrt l)))) 2.0) (/ k t)))
     (if (<= l -5e-161)
       t_9
       (if (<= l 7e-151)
         (/
          (*
           (/ (cbrt (/ 1.0 t_3)) t_6)
           (/ (* (/ (cbrt 2.0) k) (cbrt 4.0)) (/ 1.0 t)))
          (* (pow (* t_5 t_6) 2.0) (/ k t)))
         (if (<= l 4.7e+125)
           t_9
           (/
            t_8
            (*
             (pow
              (* t_5 (* t (if (!= t_10 0.0) (/ 1.0 (cbrt (/ l (sin k)))) t_4)))
              2.0)
             (/ k 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 = sin(k) / l;
	double t_2 = cbrt(((sin(k) * tan(k)) / (l * l))) * t;
	double t_3 = tan(k) / l;
	double t_4 = cbrt(t_1);
	double t_5 = cbrt(t_3);
	double t_6 = t * t_4;
	double t_7 = 8.0 / t_3;
	double t_8 = ((cbrt(t_7) / t) / t_4) / (k / t);
	double t_9 = (cbrt((t_7 / t_1)) / (t_2 * ((k * t) / t))) / ((t_2 * k) / t);
	double t_10 = cbrt(sin(k));
	double tmp;
	if (l <= -1.15e+136) {
		tmp = t_8 / (pow((t_5 * (t * (t_10 / cbrt(l)))), 2.0) * (k / t));
	} else if (l <= -5e-161) {
		tmp = t_9;
	} else if (l <= 7e-151) {
		tmp = ((cbrt((1.0 / t_3)) / t_6) * (((cbrt(2.0) / k) * cbrt(4.0)) / (1.0 / t))) / (pow((t_5 * t_6), 2.0) * (k / t));
	} else if (l <= 4.7e+125) {
		tmp = t_9;
	} else {
		double tmp_1;
		if (t_10 != 0.0) {
			tmp_1 = 1.0 / cbrt((l / sin(k)));
		} else {
			tmp_1 = t_4;
		}
		tmp = t_8 / (pow((t_5 * (t * tmp_1)), 2.0) * (k / 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.sin(k) / l;
	double t_2 = Math.cbrt(((Math.sin(k) * Math.tan(k)) / (l * l))) * t;
	double t_3 = Math.tan(k) / l;
	double t_4 = Math.cbrt(t_1);
	double t_5 = Math.cbrt(t_3);
	double t_6 = t * t_4;
	double t_7 = 8.0 / t_3;
	double t_8 = ((Math.cbrt(t_7) / t) / t_4) / (k / t);
	double t_9 = (Math.cbrt((t_7 / t_1)) / (t_2 * ((k * t) / t))) / ((t_2 * k) / t);
	double t_10 = Math.cbrt(Math.sin(k));
	double tmp;
	if (l <= -1.15e+136) {
		tmp = t_8 / (Math.pow((t_5 * (t * (t_10 / Math.cbrt(l)))), 2.0) * (k / t));
	} else if (l <= -5e-161) {
		tmp = t_9;
	} else if (l <= 7e-151) {
		tmp = ((Math.cbrt((1.0 / t_3)) / t_6) * (((Math.cbrt(2.0) / k) * Math.cbrt(4.0)) / (1.0 / t))) / (Math.pow((t_5 * t_6), 2.0) * (k / t));
	} else if (l <= 4.7e+125) {
		tmp = t_9;
	} else {
		double tmp_1;
		if (t_10 != 0.0) {
			tmp_1 = 1.0 / Math.cbrt((l / Math.sin(k)));
		} else {
			tmp_1 = t_4;
		}
		tmp = t_8 / (Math.pow((t_5 * (t * tmp_1)), 2.0) * (k / 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(sin(k) / l)
	t_2 = Float64(cbrt(Float64(Float64(sin(k) * tan(k)) / Float64(l * l))) * t)
	t_3 = Float64(tan(k) / l)
	t_4 = cbrt(t_1)
	t_5 = cbrt(t_3)
	t_6 = Float64(t * t_4)
	t_7 = Float64(8.0 / t_3)
	t_8 = Float64(Float64(Float64(cbrt(t_7) / t) / t_4) / Float64(k / t))
	t_9 = Float64(Float64(cbrt(Float64(t_7 / t_1)) / Float64(t_2 * Float64(Float64(k * t) / t))) / Float64(Float64(t_2 * k) / t))
	t_10 = cbrt(sin(k))
	tmp = 0.0
	if (l <= -1.15e+136)
		tmp = Float64(t_8 / Float64((Float64(t_5 * Float64(t * Float64(t_10 / cbrt(l)))) ^ 2.0) * Float64(k / t)));
	elseif (l <= -5e-161)
		tmp = t_9;
	elseif (l <= 7e-151)
		tmp = Float64(Float64(Float64(cbrt(Float64(1.0 / t_3)) / t_6) * Float64(Float64(Float64(cbrt(2.0) / k) * cbrt(4.0)) / Float64(1.0 / t))) / Float64((Float64(t_5 * t_6) ^ 2.0) * Float64(k / t)));
	elseif (l <= 4.7e+125)
		tmp = t_9;
	else
		tmp_1 = 0.0
		if (t_10 != 0.0)
			tmp_1 = Float64(1.0 / cbrt(Float64(l / sin(k))));
		else
			tmp_1 = t_4;
		end
		tmp = Float64(t_8 / Float64((Float64(t_5 * Float64(t * tmp_1)) ^ 2.0) * Float64(k / 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[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$1, 1/3], $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$3, 1/3], $MachinePrecision]}, Block[{t$95$6 = N[(t * t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(8.0 / t$95$3), $MachinePrecision]}, Block[{t$95$8 = N[(N[(N[(N[Power[t$95$7, 1/3], $MachinePrecision] / t), $MachinePrecision] / t$95$4), $MachinePrecision] / N[(k / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[(N[Power[N[(t$95$7 / t$95$1), $MachinePrecision], 1/3], $MachinePrecision] / N[(t$95$2 * N[(N[(k * t), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$2 * k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[l, -1.15e+136], N[(t$95$8 / N[(N[Power[N[(t$95$5 * N[(t * N[(t$95$10 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, -5e-161], t$95$9, If[LessEqual[l, 7e-151], N[(N[(N[(N[Power[N[(1.0 / t$95$3), $MachinePrecision], 1/3], $MachinePrecision] / t$95$6), $MachinePrecision] * N[(N[(N[(N[Power[2.0, 1/3], $MachinePrecision] / k), $MachinePrecision] * N[Power[4.0, 1/3], $MachinePrecision]), $MachinePrecision] / N[(1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[(t$95$5 * t$95$6), $MachinePrecision], 2.0], $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[l, 4.7e+125], t$95$9, N[(t$95$8 / N[(N[Power[N[(t$95$5 * N[(t * If[Unequal[t$95$10, 0.0], N[(1.0 / N[Power[N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], t$95$4]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(k / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if l < -1.15e136

   1. Initial program 62.4

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

   2. Simplified 61.6

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

   3. Applied egg-rr 23.1

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

   4. Applied egg-rr 27.7

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{\frac{k}{t}}}{\sqrt[3]{\frac{\sin k}{\ell}}} \cdot \frac{1}{t}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}} \]

   5. Simplified 23.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{t}}{\sqrt[3]{\frac{\sin k}{\ell}}}}{\frac{k}{t}}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}} \]
      Proof

   6. Applied egg-rr 23.2

      \[\leadsto \frac{\frac{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{t}}{\sqrt[3]{\frac{\sin k}{\ell}}}}{\frac{k}{t}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \color{blue}{\frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}}\right)\right)}^{2} \cdot \frac{k}{t}} \]

   if -1.15e136 < l < -4.9999999999999999e-161 or 6.99999999999999991e-151 < l < 4.69999999999999972e125

   1. Initial program 44.1

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

   2. Simplified 34.8

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

   3. Applied egg-rr 18.4

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

   4. Applied egg-rr 19.4

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{\frac{k}{t}}}{\sqrt[3]{\frac{\sin k}{\ell}}} \cdot \frac{1}{t}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}} \]

   5. Simplified 18.3

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{t}}{\sqrt[3]{\frac{\sin k}{\ell}}}}{\frac{k}{t}}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}} \]
      Proof

   6. Applied egg-rr 10.5

      \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt[3]{\frac{\frac{8}{\frac{\tan k}{\ell}}}{\frac{\sin k}{\ell}}}}{\frac{k}{t} \cdot t}}{\sqrt[3]{\frac{\sin k}{\ell} \cdot \frac{\tan k}{\ell}} \cdot t}}{\frac{\left(\sqrt[3]{\frac{\sin k}{\ell} \cdot \frac{\tan k}{\ell}} \cdot t\right) \cdot k}{t}}} \]

   7. Simplified 3.4

      \[\leadsto \color{blue}{\frac{\frac{\sqrt[3]{\frac{\frac{8}{\frac{\tan k}{\ell}}}{\frac{\sin k}{\ell}}}}{\left(\sqrt[3]{\frac{\sin k \cdot \tan k}{\ell \cdot \ell}} \cdot t\right) \cdot \frac{k \cdot t}{t}}}{\frac{\left(\sqrt[3]{\frac{\sin k \cdot \tan k}{\ell \cdot \ell}} \cdot t\right) \cdot k}{t}}} \]
      Proof

   if -4.9999999999999999e-161 < l < 6.99999999999999991e-151

   1. Initial program 44.9

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

   2. Simplified 36.0

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

   3. Applied egg-rr 9.4

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

   4. Applied egg-rr 9.4

      \[\leadsto \frac{\frac{\sqrt[3]{\frac{1}{\frac{\tan k}{\ell}}}}{t \cdot \sqrt[3]{\frac{\sin k}{\ell}}} \cdot \color{blue}{\frac{\frac{\sqrt[3]{2}}{k} \cdot \sqrt[3]{4}}{\frac{1}{t}}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}} \]

   if 4.69999999999999972e125 < l

   1. Initial program 61.1

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

   2. Simplified 60.1

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

   3. Applied egg-rr 24.5

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

   4. Applied egg-rr 28.9

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{\frac{k}{t}}}{\sqrt[3]{\frac{\sin k}{\ell}}} \cdot \frac{1}{t}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}} \]

   5. Simplified 24.5

      \[\leadsto \frac{\color{blue}{\frac{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{t}}{\sqrt[3]{\frac{\sin k}{\ell}}}}{\frac{k}{t}}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}} \]
      Proof

   6. Applied egg-rr 24.5

      \[\leadsto \frac{\frac{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{t}}{\sqrt[3]{\frac{\sin k}{\ell}}}}{\frac{k}{t}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sqrt[3]{\sin k} \ne 0:\\ \;\;\;\;\frac{1}{\frac{\sqrt[3]{\ell}}{\sqrt[3]{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sin k}{\ell}}\\ } \end{array}}\right)\right)}^{2} \cdot \frac{k}{t}} \]

   7. Simplified 24.4

      \[\leadsto \frac{\frac{\frac{\frac{\sqrt[3]{\frac{8}{\frac{\tan k}{\ell}}}}{t}}{\sqrt[3]{\frac{\sin k}{\ell}}}}{\frac{k}{t}}}{{\left(\sqrt[3]{\frac{\tan k}{\ell}} \cdot \left(t \cdot \color{blue}{\begin{array}{l} \color{blue}{\mathbf{if}\;\sqrt[3]{\sin k} \ne 0:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sin k}{\ell}}\\ } \end{array}}\right)\right)}^{2} \cdot \frac{k}{t}} \]
      Proof
3. Recombined 4 regimes into one program.

Alternatives

Alternative 1
Error	9.4
Cost	73108

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \sqrt[3]{\frac{\sin k \cdot \tan k}{\ell \cdot \ell}} \cdot t\\ t_3 := \frac{\tan k}{\ell}\\ t_4 := \sqrt[3]{t_1}\\ t_5 := \sqrt[3]{t_3}\\ t_6 := t \cdot t_4\\ t_7 := \frac{8}{t_3}\\ t_8 := \frac{\frac{\frac{\sqrt[3]{t_7}}{t}}{t_4}}{\frac{k}{t}}\\ t_9 := \frac{\frac{\sqrt[3]{\frac{t_7}{t_1}}}{t_2 \cdot \frac{k \cdot t}{t}}}{\frac{t_2 \cdot k}{t}}\\ t_10 := \sqrt[3]{\sin k}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{t_8}{{\left(t_5 \cdot \left(t \cdot \frac{t_10}{\sqrt[3]{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-169}:\\ \;\;\;\;t_9\\ \mathbf{elif}\;\ell \leq 7.2 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{t_3}}}{t_6} \cdot \frac{2 \cdot t}{k}}{{\left(t_5 \cdot t_6\right)}^{2} \cdot \frac{k}{t}}\\ \mathbf{elif}\;\ell \leq 4.7 \cdot 10^{+125}:\\ \;\;\;\;t_9\\ \mathbf{else}:\\ \;\;\;\;\frac{t_8}{{\left(t_5 \cdot \left(t \cdot \begin{array}{l} \mathbf{if}\;t_10 \ne 0:\\ \;\;\;\;\frac{1}{\sqrt[3]{\frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array}\right)\right)}^{2} \cdot \frac{k}{t}}\\ \end{array} \]

Alternative 2
Error	9.5
Cost	66052

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \sqrt[3]{\frac{\sin k \cdot \tan k}{\ell \cdot \ell}} \cdot t\\ t_3 := \frac{\tan k}{\ell}\\ t_4 := \sqrt[3]{t_1}\\ t_5 := \sqrt[3]{t_3}\\ t_6 := t \cdot t_4\\ t_7 := \frac{8}{t_3}\\ t_8 := \frac{\frac{\sqrt[3]{\frac{t_7}{t_1}}}{t_2 \cdot \frac{k \cdot t}{t}}}{\frac{t_2 \cdot k}{t}}\\ t_9 := {\left(t_5 \cdot t_6\right)}^{2} \cdot \frac{k}{t}\\ \mathbf{if}\;\ell \leq -2.5 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\sqrt[3]{t_7}}{t}}{t_4}}{\frac{k}{t}}}{{\left(t_5 \cdot \left(t \cdot \frac{\sqrt[3]{\sin k}}{\sqrt[3]{\ell}}\right)\right)}^{2} \cdot \frac{k}{t}}\\ \mathbf{elif}\;\ell \leq -4 \cdot 10^{-169}:\\ \;\;\;\;t_8\\ \mathbf{elif}\;\ell \leq 1.28 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{t_3}}}{t_6} \cdot \frac{2 \cdot t}{k}}{t_9}\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{+129}:\\ \;\;\;\;t_8\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\frac{k}{t}}}{t_6}}{t_5}}{t_9}\\ \end{array} \]

Alternative 3
Error	9.3
Cost	60624

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \sqrt[3]{t_1}\\ t_3 := \sqrt[3]{\frac{\sin k \cdot \tan k}{\ell \cdot \ell}} \cdot t\\ t_4 := t \cdot t_2\\ t_5 := \frac{\tan k}{\ell}\\ t_6 := \sqrt[3]{t_5}\\ t_7 := {\left(t_6 \cdot t_4\right)}^{2} \cdot \frac{k}{t}\\ t_8 := \frac{8}{t_5}\\ t_9 := \frac{\frac{\sqrt[3]{\frac{t_8}{t_1}}}{t_3 \cdot \frac{k \cdot t}{t}}}{\frac{t_3 \cdot k}{t}}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+136}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\sqrt[3]{t_8}}{t}}{t_2}}{\frac{k}{t}}}{t_7}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-161}:\\ \;\;\;\;t_9\\ \mathbf{elif}\;\ell \leq 7 \cdot 10^{-151}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{t_5}}}{t_4} \cdot \frac{2 \cdot t}{k}}{t_7}\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{+129}:\\ \;\;\;\;t_9\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\frac{k}{t}}}{t_4}}{t_6}}{t_7}\\ \end{array} \]

Alternative 4
Error	13.3
Cost	60176

\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\sin k}{\ell}}\\ t_2 := t \cdot t_1\\ t_3 := \frac{\tan k}{\ell}\\ t_4 := \sqrt[3]{t_3}\\ t_5 := {\left(t_4 \cdot t_2\right)}^{2} \cdot \frac{k}{t}\\ t_6 := \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ t_7 := \frac{2}{\frac{k}{t}}\\ \mathbf{if}\;\ell \leq -3.7 \cdot 10^{+78}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\sqrt[3]{\frac{8}{t_3}}}{t}}{t_1}}{\frac{k}{t}}}{t_5}\\ \mathbf{elif}\;\ell \leq -4.3 \cdot 10^{-165}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;\ell \leq 9.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{\frac{\frac{t_7}{t_2}}{t_4}}{t_5}\\ \mathbf{elif}\;\ell \leq 10000000000:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\frac{1}{t_3}}}{t_2} \cdot t_7}{t_5}\\ \end{array} \]

Alternative 5
Error	13.4
Cost	60176

\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\sin k}{\ell}}\\ t_2 := t \cdot t_1\\ t_3 := \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ t_4 := \frac{\tan k}{\ell}\\ t_5 := {\left(\sqrt[3]{t_4} \cdot t_2\right)}^{2} \cdot \frac{k}{t}\\ t_6 := \frac{\sqrt[3]{\frac{1}{t_4}}}{t_2}\\ \mathbf{if}\;\ell \leq -6.2 \cdot 10^{+77}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\sqrt[3]{\frac{8}{t_4}}}{t}}{t_1}}{\frac{k}{t}}}{t_5}\\ \mathbf{elif}\;\ell \leq -3 \cdot 10^{-167}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 1.2 \cdot 10^{-198}:\\ \;\;\;\;\frac{t_6 \cdot \frac{2 \cdot t}{k}}{t_5}\\ \mathbf{elif}\;\ell \leq 10000000000:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_6 \cdot \frac{2}{\frac{k}{t}}}{t_5}\\ \end{array} \]

Alternative 6
Error	14.6
Cost	60048

\[\begin{array}{l} t_1 := t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\\ t_2 := \frac{\tan k}{\ell}\\ t_3 := \frac{\frac{\frac{\sqrt[3]{\frac{8}{t_2}}}{\frac{k}{t}}}{t_1}}{{\left(\sqrt[3]{t_2} \cdot t_1\right)}^{2} \cdot \frac{k}{t}}\\ t_4 := \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+185}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-161}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 1.15 \cdot 10^{-197}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq 10000000000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Error	14.3
Cost	60048

\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\tan k}{\ell}}\\ t_2 := \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ t_3 := t \cdot \sqrt[3]{\frac{\sin k}{\ell}}\\ t_4 := \frac{\frac{\frac{\frac{2}{\frac{k}{t}}}{t_3}}{t_1}}{{\left(t_1 \cdot t_3\right)}^{2} \cdot \frac{k}{t}}\\ \mathbf{if}\;\ell \leq -3.4 \cdot 10^{+185}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq -3.5 \cdot 10^{-167}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\ell \leq 10^{-199}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;\ell \leq 9500000000:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 8
Error	13.3
Cost	60048

\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\sin k}{\ell}}\\ t_2 := t \cdot t_1\\ t_3 := \frac{\tan k}{\ell}\\ t_4 := \sqrt[3]{t_3}\\ t_5 := {\left(t_4 \cdot t_2\right)}^{2} \cdot \frac{k}{t}\\ t_6 := \frac{\frac{\frac{\frac{2}{\frac{k}{t}}}{t_2}}{t_4}}{t_5}\\ t_7 := \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ \mathbf{if}\;\ell \leq -4.8 \cdot 10^{+82}:\\ \;\;\;\;\frac{\frac{\frac{\frac{\sqrt[3]{\frac{8}{t_3}}}{t}}{t_1}}{\frac{k}{t}}}{t_5}\\ \mathbf{elif}\;\ell \leq -1.18 \cdot 10^{-163}:\\ \;\;\;\;t_7\\ \mathbf{elif}\;\ell \leq 6.6 \cdot 10^{-198}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;\ell \leq 8200000000:\\ \;\;\;\;t_7\\ \mathbf{else}:\\ \;\;\;\;t_6\\ \end{array} \]

Alternative 9
Error	16.0
Cost	47512

\[\begin{array}{l} t_1 := \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ t_2 := \sin k \cdot \tan k\\ t_3 := {\left(\frac{k}{t}\right)}^{2}\\ t_4 := t_2 \cdot t_3\\ t_5 := \left(t_2 \cdot \left(t_3 \cdot \frac{t}{\ell}\right)\right) \cdot t\\ \mathbf{if}\;\ell \leq -3 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{t_4}}{t}}{\frac{t}{\ell}}\\ \mathbf{elif}\;\ell \leq -5 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{-192}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{2}{{\left(\frac{k}{t \cdot \ell}\right)}^{2}}}}{\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{\tan k} \cdot t\right)}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 400000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 9.2 \cdot 10^{+177}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{2}{\frac{\frac{\frac{k}{t}}{\ell}}{\ell}}}}{t \cdot \sqrt[3]{t_2 \cdot \frac{k}{t}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\begin{array}{l} \mathbf{if}\;t_5 \ne 0:\\ \;\;\;\;\frac{t}{\frac{\ell}{t_5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_4 \cdot {t}^{3}}{\ell \cdot \ell}\\ \end{array}}\\ \end{array} \]

Alternative 10
Error	15.9
Cost	46156

\[\begin{array}{l} t_1 := \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ t_2 := \sin k \cdot \tan k\\ t_3 := \frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{t_2 \cdot {\left(\frac{k}{t}\right)}^{2}}}{t}}{\frac{t}{\ell}}\\ \mathbf{if}\;\ell \leq -5.8 \cdot 10^{+142}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-166}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 2 \cdot 10^{-192}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{2}{{\left(\frac{k}{t \cdot \ell}\right)}^{2}}}}{\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{\tan k} \cdot t\right)}\right)}^{3}\\ \mathbf{elif}\;\ell \leq 1750:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.95 \cdot 10^{+177}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{2}{\frac{\frac{\frac{k}{t}}{\ell}}{\ell}}}}{t \cdot \sqrt[3]{t_2 \cdot \frac{k}{t}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Error	17.2
Cost	33940

\[\begin{array}{l} t_1 := \frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ t_2 := \sin k \cdot \tan k\\ t_3 := \frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{t_2 \cdot {\left(\frac{k}{t}\right)}^{2}}}{t}}{\frac{t}{\ell}}\\ \mathbf{if}\;\ell \leq -3.3 \cdot 10^{+140}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \leq -2 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 5 \cdot 10^{-240}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \mathbf{elif}\;\ell \leq 5200:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 1.8 \cdot 10^{+177}:\\ \;\;\;\;{\left(\frac{\sqrt[3]{\frac{2}{\frac{\frac{\frac{k}{t}}{\ell}}{\ell}}}}{t \cdot \sqrt[3]{t_2 \cdot \frac{k}{t}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 12
Error	17.7
Cost	27220

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}}{t}}{\frac{t}{\ell}}\\ t_2 := \frac{2}{\frac{\left(\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right) \cdot t}{\cos k}}\\ \mathbf{if}\;k \leq -2 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.3 \cdot 10^{-94}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{\frac{{\left(k \cdot t\right)}^{2}}{{\left(\ell \cdot t\right)}^{2}}}}{k \cdot t}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{-5}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{+79}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	19.5
Cost	26824

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}}{t}}{\frac{t}{\ell}}\\ \mathbf{if}\;k \leq -3.5 \cdot 10^{+85}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.6 \cdot 10^{-105}:\\ \;\;\;\;\frac{2}{\frac{\left(k \cdot k\right) \cdot \left(t \cdot \left({\sin k}^{2} \cdot {\ell}^{-2}\right)\right)}{\cos k}}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{\frac{{\left(k \cdot t\right)}^{2}}{{\left(\ell \cdot t\right)}^{2}}}}{k \cdot t}\\ \mathbf{elif}\;k \leq 10^{-14}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	24.1
Cost	21520

\[\begin{array}{l} t_1 := \frac{2}{\frac{t \cdot \left(t \cdot \left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \frac{k \cdot k}{t}\right)\right)}{\ell \cdot t}}\\ t_2 := {k}^{-4} \cdot \ell\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-300}:\\ \;\;\;\;2 \cdot \begin{array}{l} \mathbf{if}\;t_2 \ne 0:\\ \;\;\;\;\frac{\ell}{\frac{t}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{\ell}{t}\\ \end{array}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+250}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \end{array} \]

Alternative 15
Error	22.6
Cost	21008

\[\begin{array}{l} t_1 := \frac{k}{\ell \cdot t}\\ t_2 := \ell \cdot {k}^{-4}\\ \mathbf{if}\;t \leq -4 \cdot 10^{+226}:\\ \;\;\;\;2 \cdot \left({k}^{-4} \cdot \left(\ell \cdot \left(-\left(-\frac{\ell}{t}\right)\right)\right)\right)\\ \mathbf{elif}\;t \leq -1.2 \cdot 10^{-137}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 2.45 \cdot 10^{-107}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;{k}^{-4} \cdot \left(\ell \cdot \ell\right) \ne 0:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell}}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot t_2\right) \cdot \ell\\ \end{array}\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+73}:\\ \;\;\;\;\frac{\frac{2}{t_1}}{\left(\left(\tan k \cdot {t}^{3}\right) \cdot \sin k\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot t}{\ell} \cdot \frac{\frac{\left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot k\right) \cdot k}{t}}{t}}\\ \end{array} \]

Alternative 16
Error	21.5
Cost	20876

\[\begin{array}{l} t_1 := \frac{\frac{\frac{\frac{2}{\frac{t}{\ell}}}{\left(\sin k \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}}{t}}{\frac{t}{\ell}}\\ \mathbf{if}\;k \leq -2.65 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{\frac{{\left(k \cdot t\right)}^{2}}{{\left(\ell \cdot t\right)}^{2}}}}{k \cdot t}\\ \mathbf{elif}\;k \leq 10^{-14}:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	24.7
Cost	15444

\[\begin{array}{l} t_1 := {k}^{-4} \cdot \ell\\ t_2 := \frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\\ t_3 := \frac{2}{\frac{t \cdot \left(t \cdot \left(t_2 \cdot \frac{k \cdot k}{t}\right)\right)}{\ell \cdot t}}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-300}:\\ \;\;\;\;2 \cdot \begin{array}{l} \mathbf{if}\;t_1 \ne 0:\\ \;\;\;\;\frac{\ell}{\frac{t}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{t}\\ \end{array}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+37}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+250}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\begin{array}{l} \mathbf{if}\;t \ne 0:\\ \;\;\;\;\frac{t}{\frac{\ell}{t}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot t}{\ell}\\ \end{array} \cdot \frac{t_2 \cdot \left(\frac{k}{t} \cdot k\right)}{t}}\\ \end{array} \]

Alternative 18
Error	24.7
Cost	15312

\[\begin{array}{l} t_1 := {k}^{-4} \cdot \ell\\ t_2 := \frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\\ t_3 := \frac{2}{\frac{t \cdot \left(t \cdot \left(t_2 \cdot \frac{k \cdot k}{t}\right)\right)}{\ell \cdot t}}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-300}:\\ \;\;\;\;2 \cdot \begin{array}{l} \mathbf{if}\;t_1 \ne 0:\\ \;\;\;\;\frac{\ell}{\frac{t}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\ell}{t}\\ \end{array}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+37}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \ell\right)}{{k}^{4} \cdot t}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+250}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{t_2 \cdot \left(\frac{k}{t} \cdot k\right)}{t}}\\ \end{array} \]

Alternative 19
Error	21.2
Cost	14668

\[\begin{array}{l} t_1 := \frac{2}{\left(\frac{t}{\ell} \cdot t\right) \cdot \frac{\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \tan k\right)\right) \cdot \left(\frac{k}{t} \cdot k\right)}{t}}\\ \mathbf{if}\;k \leq -9.6 \cdot 10^{-41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 10^{-77}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{\frac{{\left(k \cdot t\right)}^{2}}{{\left(\ell \cdot t\right)}^{2}}}}{k \cdot t}\\ \mathbf{elif}\;k \leq 0.000475:\\ \;\;\;\;\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 20
Error	27.0
Cost	14152

\[\begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \left(-\frac{{k}^{-4}}{-t}\right)\right)\right)\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{\frac{2}{k}}{\frac{{\left(k \cdot t\right)}^{2}}{{\left(\ell \cdot t\right)}^{2}}}}{k \cdot t}\\ \mathbf{elif}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array} \]

Alternative 21
Error	27.5
Cost	7944

\[\begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \left(-\frac{{k}^{-4}}{-t}\right)\right)\right)\\ \mathbf{elif}\;k \leq 10^{-77}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell \cdot \left(\ell \cdot t\right)}}}{{\left(t \cdot k\right)}^{2} \cdot \frac{t \cdot k}{t}}\\ \mathbf{elif}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array} \]

Alternative 22
Error	27.4
Cost	7816

\[\begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \left(-\frac{{k}^{-4}}{-t}\right)\right)\right)\\ \mathbf{elif}\;k \leq 10^{-77}:\\ \;\;\;\;\frac{\frac{2}{\frac{{\left(t \cdot k\right)}^{3}}{t}}}{\frac{1}{\ell \cdot t} \cdot \frac{k}{\ell}}\\ \mathbf{elif}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array} \]

Alternative 23
Error	27.9
Cost	7688

\[\begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \left(-\frac{{k}^{-4}}{-t}\right)\right)\right)\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{k}}{\frac{\frac{{\left(t \cdot k\right)}^{3}}{t}}{\ell \cdot \left(\ell \cdot t\right)}}\\ \mathbf{elif}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array} \]

Alternative 24
Error	28.2
Cost	7564

\[\begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \left(-\frac{{k}^{-4}}{-t}\right)\right)\right)\\ \mathbf{elif}\;k \leq 9 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell \cdot \left(\ell \cdot t\right)}}}{\left(\left(t \cdot k\right) \cdot k\right) \cdot \frac{\left(t \cdot k\right) \cdot t}{t}}\\ \mathbf{elif}\;\ell \cdot \ell \ne 0:\\ \;\;\;\;\frac{2 \cdot {k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \end{array} \]

Alternative 25
Error	30.2
Cost	7304

\[\begin{array}{l} t_1 := 2 \cdot \frac{{k}^{-4} \cdot \left(\ell \cdot \ell\right)}{t}\\ \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell \cdot \left(\ell \cdot t\right)}}}{\left(\left(t \cdot k\right) \cdot k\right) \cdot \frac{\left(t \cdot k\right) \cdot t}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 26
Error	27.9
Cost	7304

\[\begin{array}{l} t_1 := \left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell \cdot \left(\ell \cdot t\right)}}}{\left(\left(t \cdot k\right) \cdot k\right) \cdot \frac{\left(t \cdot k\right) \cdot t}{t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 27
Error	27.9
Cost	7304

\[\begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;\left(\frac{2}{t} \cdot \left(\ell \cdot {k}^{-4}\right)\right) \cdot \ell\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell \cdot \left(\ell \cdot t\right)}}}{\left(\left(t \cdot k\right) \cdot k\right) \cdot \frac{\left(t \cdot k\right) \cdot t}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left({k}^{-4} \cdot \ell\right) \cdot \left(\frac{\ell}{t} \cdot 2\right)\\ \end{array} \]

Alternative 28
Error	28.1
Cost	7304

\[\begin{array}{l} \mathbf{if}\;k \leq -8.6 \cdot 10^{-78}:\\ \;\;\;\;2 \cdot \left(\ell \cdot \left(\ell \cdot \left(-\frac{{k}^{-4}}{-t}\right)\right)\right)\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell \cdot \left(\ell \cdot t\right)}}}{\left(\left(t \cdot k\right) \cdot k\right) \cdot \frac{\left(t \cdot k\right) \cdot t}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left({k}^{-4} \cdot \ell\right) \cdot \left(\frac{\ell}{t} \cdot 2\right)\\ \end{array} \]

Alternative 29
Error	31.5
Cost	1732

\[\begin{array}{l} t_1 := \left(k \cdot k\right) \cdot t\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{2}{\frac{k}{\ell \cdot \left(\ell \cdot t\right)}}}{t_1 \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\frac{\frac{\frac{k}{\ell}}{\ell}}{t}}}{t_1 \cdot \frac{\left(t \cdot k\right) \cdot t}{t}}\\ \end{array} \]

Alternative 30
Error	33.4
Cost	1216

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

Reproduce

herbie shell --seed 2023010 
(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))))