Toniolo and Linder, Equation (10-)

Average Error: 47.6 → 13.2

Time: 37.2s

Precision: binary64

Cost: 26756

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{\cos k \cdot \ell}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{\cos k}{\sin k \cdot \frac{t}{\frac{\ell}{\sin k}}}\right) \cdot 2\\ \end{array} \]

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
 (if (<= k -8.5e+130)
   (* (/ l (* (sin k) (/ (* (pow k 2.0) (* (sin k) t)) (* (cos k) l)))) 2.0)
   (* (* (/ l (pow k 2.0)) (/ (cos k) (* (sin k) (/ t (/ l (sin k)))))) 2.0)))

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 tmp;
	if (k <= -8.5e+130) {
		tmp = (l / (sin(k) * ((pow(k, 2.0) * (sin(k) * t)) / (cos(k) * l)))) * 2.0;
	} else {
		tmp = ((l / pow(k, 2.0)) * (cos(k) / (sin(k) * (t / (l / sin(k)))))) * 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) - 1.0d0))
end function

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (k <= (-8.5d+130)) then
        tmp = (l / (sin(k) * (((k ** 2.0d0) * (sin(k) * t)) / (cos(k) * l)))) * 2.0d0
    else
        tmp = ((l / (k ** 2.0d0)) * (cos(k) / (sin(k) * (t / (l / sin(k)))))) * 2.0d0
    end if
    code = tmp
end function

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 tmp;
	if (k <= -8.5e+130) {
		tmp = (l / (Math.sin(k) * ((Math.pow(k, 2.0) * (Math.sin(k) * t)) / (Math.cos(k) * l)))) * 2.0;
	} else {
		tmp = ((l / Math.pow(k, 2.0)) * (Math.cos(k) / (Math.sin(k) * (t / (l / Math.sin(k)))))) * 2.0;
	}
	return tmp;
}

Python

def code(t, l, 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))

↓

def code(t, l, k):
	tmp = 0
	if k <= -8.5e+130:
		tmp = (l / (math.sin(k) * ((math.pow(k, 2.0) * (math.sin(k) * t)) / (math.cos(k) * l)))) * 2.0
	else:
		tmp = ((l / math.pow(k, 2.0)) * (math.cos(k) / (math.sin(k) * (t / (l / math.sin(k)))))) * 2.0
	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)
	tmp = 0.0
	if (k <= -8.5e+130)
		tmp = Float64(Float64(l / Float64(sin(k) * Float64(Float64((k ^ 2.0) * Float64(sin(k) * t)) / Float64(cos(k) * l)))) * 2.0);
	else
		tmp = Float64(Float64(Float64(l / (k ^ 2.0)) * Float64(cos(k) / Float64(sin(k) * Float64(t / Float64(l / sin(k)))))) * 2.0);
	end
	return tmp
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) - 1.0));
end

↓

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (k <= -8.5e+130)
		tmp = (l / (sin(k) * (((k ^ 2.0) * (sin(k) * t)) / (cos(k) * l)))) * 2.0;
	else
		tmp = ((l / (k ^ 2.0)) * (cos(k) / (sin(k) * (t / (l / sin(k)))))) * 2.0;
	end
	tmp_2 = 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_] := If[LessEqual[k, -8.5e+130], N[(N[(l / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision], N[(N[(N[(l / N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[(t / N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -8.49999999999999965e130

   1. Initial program 40.6

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

   2. Simplified 34.8

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

      [Start] 40.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational.json-simplify-28 [=>] 40.7	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      rational.json-simplify-5 [=>] 40.7	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]
      rational.json-simplify-3 [=>] 34.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 34.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      rational.json-simplify-8 [=>] 34.9	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-14 [=>] 34.8	\[ \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-50 [=>] 34.8	\[ \frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational.json-simplify-50 [=>] 34.8	\[ \frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]
      rational.json-simplify-24 [=>] 34.8	\[ \frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
      rational.json-simplify-24 [=>] 34.8	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
      rational.json-simplify-50 [=>] 34.8	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \color{blue}{\left(\sin k \cdot \tan k\right)}\right)} \]

   3. Applied egg-rr 32.2

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

   4. Taylor expanded in t around 0 22.3

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

   if -8.49999999999999965e130 < k

   1. Initial program 49.8

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

   2. Simplified 42.0

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

      [Start] 49.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational.json-simplify-28 [=>] 49.8	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      rational.json-simplify-5 [=>] 49.8	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{1 + \left({\left(\frac{k}{t}\right)}^{2} - 1\right)}} \]
      rational.json-simplify-3 [=>] 41.5	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 41.5	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      rational.json-simplify-8 [=>] 41.5	\[ \frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-14 [=>] 41.4	\[ \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-50 [=>] 41.4	\[ \frac{2}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational.json-simplify-50 [=>] 41.4	\[ \frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]
      rational.json-simplify-24 [=>] 41.4	\[ \frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
      rational.json-simplify-24 [=>] 42.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
      rational.json-simplify-50 [=>] 42.0	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} \cdot \color{blue}{\left(\sin k \cdot \tan k\right)}\right)} \]

   3. Applied egg-rr 38.5

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

   4. Taylor expanded in t around 0 15.0

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

   5. Simplified 11.6

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

      [Start] 15.0	\[ \frac{\ell}{\sin k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{\cos k \cdot \ell}} \cdot 2 \]
      rational.json-simplify-43 [=>] 11.6	\[ \frac{\ell}{\sin k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{{k}^{2}}{\cos k \cdot \ell}\right)}} \cdot 2 \]

   6. Applied egg-rr 10.3

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

4. Final simplification 13.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\ell}{\sin k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{\cos k \cdot \ell}} \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \frac{\cos k}{\sin k \cdot \frac{t}{\frac{\ell}{\sin k}}}\right) \cdot 2\\ \end{array} \]

Alternatives

Alternative 1
Error	12.9
Cost	26824

\[\begin{array}{l} t_1 := \left(\frac{\ell}{{k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\right) \cdot 2\\ \mathbf{if}\;k \leq -4 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{-15}:\\ \;\;\;\;\left(\frac{0.5}{\frac{\frac{{k}^{2}}{\ell}}{\frac{\frac{\frac{\ell}{\sin k}}{\sin k}}{t}}} \cdot 2\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	12.8
Cost	26824

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ \mathbf{if}\;k \leq -4 \cdot 10^{-64}:\\ \;\;\;\;\left(\ell \cdot \frac{\cos k}{\left(\sin k \cdot t\right) \cdot \frac{{k}^{2}}{t_1}}\right) \cdot 2\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{-12}:\\ \;\;\;\;\left(\frac{0.5}{\frac{\frac{{k}^{2}}{\ell}}{\frac{\frac{t_1}{\sin k}}{t}}} \cdot 2\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\right) \cdot 2\\ \end{array} \]

Alternative 3
Error	12.9
Cost	26824

\[\begin{array}{l} t_1 := \frac{{k}^{2}}{\ell}\\ \mathbf{if}\;k \leq -9.5 \cdot 10^{-65}:\\ \;\;\;\;\left(\frac{\frac{\cos k}{t_1}}{\sin k \cdot \left(\sin k \cdot t\right)} \cdot \ell\right) \cdot 2\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\left(\frac{0.5}{\frac{t_1}{\frac{\frac{\frac{\ell}{\sin k}}{\sin k}}{t}}} \cdot 2\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\ell}{{k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)\right) \cdot 2\\ \end{array} \]

Alternative 4
Error	17.4
Cost	26696

\[\begin{array}{l} t_1 := \frac{-0.16666666666666666}{\frac{t}{\ell}}\\ t_2 := 2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\sin k \cdot \tan k}\\ t_3 := \frac{\ell}{{k}^{2}}\\ \mathbf{if}\;k \leq -0.022:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 0.0295:\\ \;\;\;\;\left(t_3 \cdot \left(\frac{t_3}{t} + \left({k}^{2} \cdot \left(\frac{\ell \cdot 0.041666666666666664}{t} - \left(-0.3333333333333333 \cdot t_1 + 0.044444444444444446 \cdot \frac{\ell}{t}\right)\right) + t_1\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	17.4
Cost	26696

\[\begin{array}{l} t_1 := \frac{-0.16666666666666666}{\frac{t}{\ell}}\\ t_2 := \frac{\ell}{{k}^{2}}\\ \mathbf{if}\;k \leq -0.0115:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\sin k \cdot \tan k}\\ \mathbf{elif}\;k \leq 0.0265:\\ \;\;\;\;\left(t_2 \cdot \left(\frac{t_2}{t} + \left({k}^{2} \cdot \left(\frac{\ell \cdot 0.041666666666666664}{t} - \left(-0.3333333333333333 \cdot t_1 + 0.044444444444444446 \cdot \frac{\ell}{t}\right)\right) + t_1\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k} \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left(\tan k \cdot t\right)}\\ \end{array} \]

Alternative 6
Error	17.3
Cost	26696

\[\begin{array}{l} t_1 := \frac{-0.16666666666666666}{\frac{t}{\ell}}\\ t_2 := \frac{\ell}{{k}^{2}}\\ \mathbf{if}\;k \leq -0.04:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2} \cdot t}}{\sin k \cdot \tan k}\\ \mathbf{elif}\;k \leq 0.031:\\ \;\;\;\;\left(t_2 \cdot \left(\frac{t_2}{t} + \left({k}^{2} \cdot \left(\frac{\ell \cdot 0.041666666666666664}{t} - \left(-0.3333333333333333 \cdot t_1 + 0.044444444444444446 \cdot \frac{\ell}{t}\right)\right) + t_1\right)\right)\right) \cdot 2\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{2} \cdot \frac{\frac{\frac{\frac{2}{t}}{{k}^{2}}}{\sin k}}{\tan k}\\ \end{array} \]

Alternative 7
Error	13.8
Cost	26560

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

Alternative 8
Error	23.3
Cost	20096

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

Alternative 9
Error	23.5
Cost	14016

\[\begin{array}{l} t_1 := \frac{\ell}{{k}^{2}}\\ \left(t_1 \cdot \left(\frac{t_1}{t} + \frac{-0.16666666666666666}{\frac{t}{\ell}}\right)\right) \cdot 2 \end{array} \]

Alternative 10
Error	25.4
Cost	13632

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

Alternative 11
Error	24.2
Cost	13632

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

Alternative 12
Error	27.0
Cost	13568

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

Alternative 13
Error	31.5
Cost	13376

\[2 \cdot \frac{\frac{{\ell}^{2}}{t}}{{k}^{4}} \]

Alternative 14
Error	31.4
Cost	13376

\[\frac{2}{t \cdot \frac{{k}^{4}}{{\ell}^{2}}} \]

Alternative 15
Error	31.4
Cost	13376

\[\frac{\frac{{\ell}^{2}}{{k}^{4} \cdot 0.5}}{t} \]

Reproduce

herbie shell --seed 2023073 
(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))))