Toniolo and Linder, Equation (10+)

Average Error: 32.6 → 10.7

Time: 39.2s

Precision: binary64

Cost: 100492

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := 1 + \left(1 + t_1\right)\\ t_3 := \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot t_2\\ t_4 := \sqrt[3]{\tan k \cdot \left(2 + t_1\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{t_4}^{2}}}{t_4}\\ \mathbf{elif}\;t_3 \leq 10^{+163}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\tan k \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot \sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0))
        (t_2 (+ 1.0 (+ 1.0 t_1)))
        (t_3 (* (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k)) t_2))
        (t_4 (* (cbrt (* (tan k) (+ 2.0 t_1))) (* t (cbrt (sin k))))))
   (if (<= t_3 (- INFINITY))
     (/ (/ (* l (* l 2.0)) (pow t_4 2.0)) t_4)
     (if (<= t_3 1e+163)
       (/ 2.0 (* t_2 (* (tan k) (/ (* (sin k) (/ (pow t 3.0) l)) l))))
       (if (<= t_3 INFINITY)
         (/ l (pow (/ (* t (cbrt k)) (cbrt (/ l k))) 3.0))
         (/ 2.0 (* (* (sin k) (tan k)) (* (/ k l) (/ t (/ l k))))))))))

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 = 1.0 + (1.0 + t_1);
	double t_3 = (((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * t_2;
	double t_4 = cbrt((tan(k) * (2.0 + t_1))) * (t * cbrt(sin(k)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = ((l * (l * 2.0)) / pow(t_4, 2.0)) / t_4;
	} else if (t_3 <= 1e+163) {
		tmp = 2.0 / (t_2 * (tan(k) * ((sin(k) * (pow(t, 3.0) / l)) / l)));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = l / pow(((t * cbrt(k)) / cbrt((l / k))), 3.0);
	} else {
		tmp = 2.0 / ((sin(k) * tan(k)) * ((k / l) * (t / (l / k))));
	}
	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 = 1.0 + (1.0 + t_1);
	double t_3 = (((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * t_2;
	double t_4 = Math.cbrt((Math.tan(k) * (2.0 + t_1))) * (t * Math.cbrt(Math.sin(k)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = ((l * (l * 2.0)) / Math.pow(t_4, 2.0)) / t_4;
	} else if (t_3 <= 1e+163) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * ((Math.sin(k) * (Math.pow(t, 3.0) / l)) / l)));
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = l / Math.pow(((t * Math.cbrt(k)) / Math.cbrt((l / k))), 3.0);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.tan(k)) * ((k / l) * (t / (l / k))));
	}
	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 = Float64(1.0 + Float64(1.0 + t_1))
	t_3 = Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * t_2)
	t_4 = Float64(cbrt(Float64(tan(k) * Float64(2.0 + t_1))) * Float64(t * cbrt(sin(k))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(l * Float64(l * 2.0)) / (t_4 ^ 2.0)) / t_4);
	elseif (t_3 <= 1e+163)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(Float64(sin(k) * Float64((t ^ 3.0) / l)) / l))));
	elseif (t_3 <= Inf)
		tmp = Float64(l / (Float64(Float64(t * cbrt(k)) / cbrt(Float64(l / k))) ^ 3.0));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * tan(k)) * Float64(Float64(k / l) * Float64(t / Float64(l / k)))));
	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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[Tan[k], $MachinePrecision] * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(l * N[(l * 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 1e+163], N[(2.0 / N[(t$95$2 * N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(l / N[Power[N[(N[(t * N[Power[k, 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(k / l), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < -inf.0

   1. Initial program 14.3

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

   2. Simplified 13.2

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

   3. Applied egg-rr 6.4

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

   if -inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < 9.9999999999999994e162

   1. Initial program 22.5

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

   2. Applied egg-rr 13.2

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

   if 9.9999999999999994e162 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 14.5

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

   2. Simplified 13.8

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

   3. Taylor expanded in k around 0 21.0

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

   4. Simplified 20.1

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

   5. Applied egg-rr 13.7

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

   6. Applied egg-rr 8.5

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 64.0

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

   2. Taylor expanded in t around 0 39.6

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

   3. Simplified 35.4

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

   4. Applied egg-rr 24.3

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

   5. Applied egg-rr 22.8

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

   6. Applied egg-rr 15.0

      \[\leadsto \frac{2}{\left(\frac{\sin k}{1} \cdot \tan k\right) \cdot \color{blue}{\left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq -\infty:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(\ell \cdot 2\right)}{{\left(\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)\right)}^{2}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(t \cdot \sqrt[3]{\sin k}\right)}\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq 10^{+163}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \frac{\sin k \cdot \frac{{t}^{3}}{\ell}}{\ell}\right)}\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \leq \infty:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot \sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{k}}}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.0
Cost	26824

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{0.75}}{\ell} \cdot \frac{{t}^{0.75}}{\frac{{2}^{-0.5}}{k}}\right)}^{2}}\\ \end{array} \]

Alternative 2
Error	11.0
Cost	20868

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{+21}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\ell} \cdot \frac{\sqrt{t}}{\frac{{2}^{-0.5}}{k}}\right)}^{2}}\\ \end{array} \]

Alternative 3
Error	10.6
Cost	20296

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{k}{\ell}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{t}{\ell} \cdot \frac{\sqrt{t}}{\frac{{2}^{-0.5}}{k}}\right)}^{2}}\\ \end{array} \]

Alternative 4
Error	11.5
Cost	20232

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{k}{\ell}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(t \cdot \left(\sqrt{t} \cdot \frac{\sqrt{2}}{\frac{\ell}{k}}\right)\right)}^{2}}\\ \end{array} \]

Alternative 5
Error	11.9
Cost	20168

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{k}{\ell}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{\left(\frac{{t}^{1.5}}{\frac{\ell}{k \cdot \sqrt{2}}}\right)}^{2}}\\ \end{array} \]

Alternative 6
Error	11.8
Cost	20104

\[\begin{array}{l} t_1 := \frac{\ell}{{\left(t \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{k}{\ell}}\right)\right)}^{3}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	11.9
Cost	20104

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell}{{\left(t \cdot \left(\sqrt[3]{k} \cdot \sqrt[3]{\frac{k}{\ell}}\right)\right)}^{3}}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{{\left(\frac{t \cdot \sqrt[3]{k}}{\sqrt[3]{\frac{\ell}{k}}}\right)}^{3}}\\ \end{array} \]

Alternative 8
Error	13.5
Cost	14408

\[\begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\ell}{\frac{1}{\frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}}}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \left(\frac{k}{\ell} \cdot \frac{{t}^{3}}{\frac{\ell}{k}}\right)}\\ \end{array} \]

Alternative 9
Error	17.5
Cost	14024

\[\begin{array}{l} t_1 := t \cdot {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{\ell}{\frac{1}{\frac{\ell}{t_1}}}\\ \mathbf{elif}\;t \leq 10^{-45}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}}{\tan k}\\ \mathbf{elif}\;t \leq 4.33158674893438 \cdot 10^{+107}:\\ \;\;\;\;{t}^{-3} \cdot {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell}}\\ \end{array} \]

Alternative 10
Error	17.5
Cost	14024

\[\begin{array}{l} t_1 := t \cdot {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{\ell}{\frac{1}{\frac{\ell}{t_1}}}\\ \mathbf{elif}\;t \leq 10^{-45}:\\ \;\;\;\;\frac{\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k}}{\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)}\\ \mathbf{elif}\;t \leq 4.33158674893438 \cdot 10^{+107}:\\ \;\;\;\;{t}^{-3} \cdot {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell}}\\ \end{array} \]

Alternative 11
Error	17.7
Cost	14024

\[\begin{array}{l} t_1 := t \cdot {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+16}:\\ \;\;\;\;\frac{\ell}{\frac{1}{\frac{\ell}{t_1}}}\\ \mathbf{elif}\;t \leq 10^{-45}:\\ \;\;\;\;\frac{\left(\ell \cdot \ell\right) \cdot 2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;t \leq 4.33158674893438 \cdot 10^{+107}:\\ \;\;\;\;{t}^{-3} \cdot {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell}}\\ \end{array} \]

Alternative 12
Error	13.3
Cost	14024

\[\begin{array}{l} t_1 := t \cdot {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\ell}{\frac{1}{\frac{\ell}{t_1}}}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\frac{\frac{t \cdot \left(\sin k \cdot \tan k\right)}{\frac{\ell}{k}}}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 4.33158674893438 \cdot 10^{+107}:\\ \;\;\;\;{t}^{-3} \cdot {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell}}\\ \end{array} \]

Alternative 13
Error	12.2
Cost	14024

\[\begin{array}{l} t_1 := t \cdot {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{+64}:\\ \;\;\;\;\frac{\ell}{\frac{1}{\frac{\ell}{t_1}}}\\ \mathbf{elif}\;t \leq 10^{-35}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \frac{t}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;t \leq 4.33158674893438 \cdot 10^{+107}:\\ \;\;\;\;{t}^{-3} \cdot {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell}}\\ \end{array} \]

Alternative 14
Error	21.5
Cost	13644

\[\begin{array}{l} t_1 := t \cdot {\left(t \cdot k\right)}^{2}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell}{\frac{1}{\frac{\ell}{t_1}}}\\ \mathbf{elif}\;t \leq 10^{-65}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \frac{1}{\ell}}{\frac{\ell}{k}}}{\frac{1}{k \cdot k} + -0.16666666666666666}}\\ \mathbf{elif}\;t \leq 4.33158674893438 \cdot 10^{+107}:\\ \;\;\;\;{t}^{-3} \cdot {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{t_1}{\ell}}\\ \end{array} \]

Alternative 15
Error	26.2
Cost	7304

\[\begin{array}{l} t_1 := \ell \cdot \frac{{t}^{-3}}{k \cdot \frac{k}{\ell}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \frac{1}{\ell}}{\frac{\ell}{k}}}{\frac{1}{k \cdot k} + -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	22.1
Cost	7304

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \frac{1}{\ell}}{\frac{\ell}{k}}}{\frac{1}{k \cdot k} + -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	23.2
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;\ell \cdot \frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}\\ \mathbf{elif}\;t \leq 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \frac{1}{\ell}}{\frac{\ell}{k}}}{\frac{1}{k \cdot k} + -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{t}^{3} \cdot k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 18
Error	23.2
Cost	7304

\[\begin{array}{l} \mathbf{if}\;t \leq -1.02 \cdot 10^{+21}:\\ \;\;\;\;\frac{\ell}{\frac{1}{\frac{\ell}{t \cdot {\left(t \cdot k\right)}^{2}}}}\\ \mathbf{elif}\;t \leq 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \frac{1}{\ell}}{\frac{\ell}{k}}}{\frac{1}{k \cdot k} + -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{{t}^{3} \cdot k}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 19
Error	33.0
Cost	1608

\[\begin{array}{l} t_1 := \frac{1}{k \cdot k} + -0.16666666666666666\\ \mathbf{if}\;\ell \leq 1.9555940263367078 \cdot 10^{-215}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\frac{\ell}{k}}}{\frac{1}{k}} \cdot \frac{\frac{t}{\ell}}{\frac{1}{k}}}\\ \mathbf{elif}\;\ell \leq 1.3277438946028484 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \frac{1}{\ell}}{\frac{\ell}{k}}}{t_1}}\\ \end{array} \]

Alternative 20
Error	33.2
Cost	1480

\[\begin{array}{l} t_1 := \frac{1}{k \cdot k}\\ \mathbf{if}\;\ell \leq 1.9555940263367078 \cdot 10^{-215}:\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\frac{\ell}{k}}}{\frac{1}{k}} \cdot \frac{\frac{t}{\ell}}{\frac{1}{k}}}\\ \mathbf{elif}\;\ell \leq 1.3277438946028484 \cdot 10^{-78}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot k}{\frac{\ell}{\frac{t}{\ell}}}}{t_1 + -0.16666666666666666}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{\left(t \cdot k\right) \cdot \frac{1}{\ell}}{\frac{\ell}{k}}}{t_1}}\\ \end{array} \]

Alternative 21
Error	33.7
Cost	1216

\[\frac{2}{\frac{\frac{k}{\frac{\ell}{k}}}{\frac{1}{k}} \cdot \frac{\frac{t}{\ell}}{\frac{1}{k}}} \]

Alternative 22
Error	35.7
Cost	960

\[\begin{array}{l} t_1 := \frac{k \cdot k}{\ell}\\ \frac{2}{t \cdot \left(t_1 \cdot t_1\right)} \end{array} \]

Alternative 23
Error	34.2
Cost	960

\[\frac{2}{\frac{k}{\frac{\ell}{k}} \cdot \frac{k \cdot k}{\frac{\ell}{t}}} \]

Alternative 24
Error	34.0
Cost	960

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

Alternative 25
Error	34.0
Cost	960

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

Reproduce

herbie shell --seed 2022217 
(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))))