Toniolo and Linder, Equation (10-)

Average Error: 47.5 → 1.1

Time: 30.1s

Precision: binary64

Cost: 20553

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot \frac{\cos k}{k}\\ \mathbf{if}\;k \leq -9.2 \cdot 10^{-115} \lor \neg \left(k \leq 4.6 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot \frac{{\sin k}^{2}}{\ell}\right)} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \left(\sin k \cdot \frac{\sin k}{\ell}\right)\right)}{t_1}}\\ \end{array} \]

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 (* l (/ (cos k) k))))
   (if (or (<= k -9.2e-115) (not (<= k 4.6e+117)))
     (* (/ 2.0 (* t (* k (/ (pow (sin k) 2.0) l)))) t_1)
     (/ 2.0 (/ (* k (* t (* (sin k) (/ (sin k) l)))) t_1)))))

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 = l * (cos(k) / k);
	double tmp;
	if ((k <= -9.2e-115) || !(k <= 4.6e+117)) {
		tmp = (2.0 / (t * (k * (pow(sin(k), 2.0) / l)))) * t_1;
	} else {
		tmp = 2.0 / ((k * (t * (sin(k) * (sin(k) / l)))) / t_1);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = l * (cos(k) / k)
    if ((k <= (-9.2d-115)) .or. (.not. (k <= 4.6d+117))) then
        tmp = (2.0d0 / (t * (k * ((sin(k) ** 2.0d0) / l)))) * t_1
    else
        tmp = 2.0d0 / ((k * (t * (sin(k) * (sin(k) / l)))) / t_1)
    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 t_1 = l * (Math.cos(k) / k);
	double tmp;
	if ((k <= -9.2e-115) || !(k <= 4.6e+117)) {
		tmp = (2.0 / (t * (k * (Math.pow(Math.sin(k), 2.0) / l)))) * t_1;
	} else {
		tmp = 2.0 / ((k * (t * (Math.sin(k) * (Math.sin(k) / l)))) / t_1);
	}
	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):
	t_1 = l * (math.cos(k) / k)
	tmp = 0
	if (k <= -9.2e-115) or not (k <= 4.6e+117):
		tmp = (2.0 / (t * (k * (math.pow(math.sin(k), 2.0) / l)))) * t_1
	else:
		tmp = 2.0 / ((k * (t * (math.sin(k) * (math.sin(k) / l)))) / t_1)
	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(l * Float64(cos(k) / k))
	tmp = 0.0
	if ((k <= -9.2e-115) || !(k <= 4.6e+117))
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k * Float64((sin(k) ^ 2.0) / l)))) * t_1);
	else
		tmp = Float64(2.0 / Float64(Float64(k * Float64(t * Float64(sin(k) * Float64(sin(k) / l)))) / t_1));
	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)
	t_1 = l * (cos(k) / k);
	tmp = 0.0;
	if ((k <= -9.2e-115) || ~((k <= 4.6e+117)))
		tmp = (2.0 / (t * (k * ((sin(k) ^ 2.0) / l)))) * t_1;
	else
		tmp = 2.0 / ((k * (t * (sin(k) * (sin(k) / l)))) / t_1);
	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_] := Block[{t$95$1 = N[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, -9.2e-115], N[Not[LessEqual[k, 4.6e+117]], $MachinePrecision]], N[(N[(2.0 / N[(t * N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(2.0 / N[(N[(k * N[(t * N[(N[Sin[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < -9.19999999999999938e-115 or 4.59999999999999976e117 < k

   1. Initial program 44.2

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

   2. Simplified 36.3

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 44.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 44.2	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 44.2	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 44.2	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 36.3	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 36.3	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 21.3

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

   4. Simplified 17.4

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

      [Start] 21.3	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 21.1	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 21.1	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 21.1	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 21.1	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 21.1	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 17.4	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 5.0

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

   6. Applied egg-rr 0.5

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

   if -9.19999999999999938e-115 < k < 4.59999999999999976e117

   1. Initial program 56.2

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

   2. Simplified 47.7

      \[\leadsto \color{blue}{\frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + 0\right)\right)}} \]
      Proof

      [Start] 56.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 56.2	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 56.1	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 56.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 47.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 47.7	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 26.3

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

   4. Simplified 14.5

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

      [Start] 26.3	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 24.0	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 24.0	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 24.0	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 24.0	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 24.0	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 14.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 8.2

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

   6. Applied egg-rr 2.9

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

4. Final simplification 1.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.2 \cdot 10^{-115} \lor \neg \left(k \leq 4.6 \cdot 10^{+117}\right):\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot \frac{{\sin k}^{2}}{\ell}\right)} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(t \cdot \left(\sin k \cdot \frac{\sin k}{\ell}\right)\right)}{\ell \cdot \frac{\cos k}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.1
Cost	20552

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -2 \cdot 10^{-15}:\\ \;\;\;\;\left(\left(2 \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot t_1}\\ \mathbf{elif}\;k \leq 1.95 \cdot 10^{-132}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t}}{-k} \cdot \frac{\ell}{-k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot \frac{-2}{\frac{k}{\ell}}}{\left(t_1 \cdot \frac{k}{\ell}\right) \cdot \left(-t\right)}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -2.6 \cdot 10^{-10} \lor \neg \left(k \leq 2.4 \cdot 10^{-20}\right):\\ \;\;\;\;\left(\left(2 \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot {\sin k}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t}}{-k} \cdot \frac{\ell}{-k}\right)\\ \end{array} \]

Alternative 3
Error	5.1
Cost	20488

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00052:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{k}}{t \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\ell}}\right)\\ \mathbf{elif}\;k \leq 9.5 \cdot 10^{-98}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t}}{-k} \cdot \frac{\ell}{-k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k \cdot \frac{\ell}{k}}{k \cdot t} \cdot \frac{\ell}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4
Error	1.0
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.85 \cdot 10^{-10}:\\ \;\;\;\;\left(\left(2 \cdot \cos k\right) \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\ell}{k}}{t \cdot t_1}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-47}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t}}{-k} \cdot \frac{\ell}{-k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot \frac{t_1}{\ell}\right)} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)\\ \end{array} \]

Alternative 5
Error	4.4
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00052 \lor \neg \left(k \leq 0.000102\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{k}}{t \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\ell}}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t}}{-k} \cdot \frac{\ell}{-k}\right)\\ \end{array} \]

Alternative 6
Error	13.9
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00052 \lor \neg \left(k \leq 1.06 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\sin k} \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t}}{-k} \cdot \frac{\ell}{-k}\right)\\ \end{array} \]

Alternative 7
Error	22.3
Cost	8009

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00052 \lor \neg \left(k \leq 1.12 \cdot 10^{-132}\right):\\ \;\;\;\;\frac{\cos k}{k} \cdot \left(\frac{\frac{2}{k}}{\frac{t}{\ell}} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\frac{\frac{\ell}{k}}{k}}{t}}{-k} \cdot \frac{\ell}{-k}\right)\\ \end{array} \]

Alternative 8
Error	23.9
Cost	1088

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

Alternative 9
Error	26.3
Cost	960

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

Alternative 10
Error	26.3
Cost	960

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

Alternative 11
Error	26.7
Cost	960

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

Alternative 12
Error	25.5
Cost	960

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

Alternative 13
Error	25.0
Cost	960

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

Reproduce

herbie shell --seed 2023059 
(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))))