Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 0.8

Time: 31.6s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -1.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{t_1}{\ell}}{\cos k} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t_1}{\ell} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \end{array} \]

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 (sin k) 2.0)))
   (if (<= k -1.25e-141)
     (/ 2.0 (* (/ (* k (/ t_1 l)) (cos k)) (/ t (/ l k))))
     (if (<= k 7.5e-24)
       (* 2.0 (/ (/ (/ l k) k) (* t (* k (/ k l)))))
       (/ 2.0 (* (/ (* k t_1) l) (/ (- t) (* l (- (/ (cos k) 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(sin(k), 2.0);
	double tmp;
	if (k <= -1.25e-141) {
		tmp = 2.0 / (((k * (t_1 / l)) / cos(k)) * (t / (l / k)));
	} else if (k <= 7.5e-24) {
		tmp = 2.0 * (((l / k) / k) / (t * (k * (k / l))));
	} else {
		tmp = 2.0 / (((k * t_1) / l) * (-t / (l * -(cos(k) / k))));
	}
	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 = sin(k) ** 2.0d0
    if (k <= (-1.25d-141)) then
        tmp = 2.0d0 / (((k * (t_1 / l)) / cos(k)) * (t / (l / k)))
    else if (k <= 7.5d-24) then
        tmp = 2.0d0 * (((l / k) / k) / (t * (k * (k / l))))
    else
        tmp = 2.0d0 / (((k * t_1) / l) * (-t / (l * -(cos(k) / k))))
    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 = Math.pow(Math.sin(k), 2.0);
	double tmp;
	if (k <= -1.25e-141) {
		tmp = 2.0 / (((k * (t_1 / l)) / Math.cos(k)) * (t / (l / k)));
	} else if (k <= 7.5e-24) {
		tmp = 2.0 * (((l / k) / k) / (t * (k * (k / l))));
	} else {
		tmp = 2.0 / (((k * t_1) / l) * (-t / (l * -(Math.cos(k) / k))));
	}
	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 = math.pow(math.sin(k), 2.0)
	tmp = 0
	if k <= -1.25e-141:
		tmp = 2.0 / (((k * (t_1 / l)) / math.cos(k)) * (t / (l / k)))
	elif k <= 7.5e-24:
		tmp = 2.0 * (((l / k) / k) / (t * (k * (k / l))))
	else:
		tmp = 2.0 / (((k * t_1) / l) * (-t / (l * -(math.cos(k) / 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 = sin(k) ^ 2.0
	tmp = 0.0
	if (k <= -1.25e-141)
		tmp = Float64(2.0 / Float64(Float64(Float64(k * Float64(t_1 / l)) / cos(k)) * Float64(t / Float64(l / k))));
	elseif (k <= 7.5e-24)
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / k) / Float64(t * Float64(k * Float64(k / l)))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(k * t_1) / l) * Float64(Float64(-t) / Float64(l * Float64(-Float64(cos(k) / k))))));
	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 = sin(k) ^ 2.0;
	tmp = 0.0;
	if (k <= -1.25e-141)
		tmp = 2.0 / (((k * (t_1 / l)) / cos(k)) * (t / (l / k)));
	elseif (k <= 7.5e-24)
		tmp = 2.0 * (((l / k) / k) / (t * (k * (k / l))));
	else
		tmp = 2.0 / (((k * t_1) / l) * (-t / (l * -(cos(k) / k))));
	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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, -1.25e-141], N[(2.0 / N[(N[(N[(k * N[(t$95$1 / l), $MachinePrecision]), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 7.5e-24], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] / N[(t * N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(k * t$95$1), $MachinePrecision] / l), $MachinePrecision] * N[((-t) / N[(l * (-N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if k < -1.25e-141

   1. Initial program 46.7

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

   2. Taylor expanded in t around 0 21.1

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

   3. Simplified 15.3

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

      [Start] 21.1	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 20.8	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 20.8	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 20.8	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 20.8	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 15.3	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   4. Applied egg-rr 9.3

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

   5. Simplified 1.1

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

      [Start] 9.3	\[ \frac{2}{\frac{k \cdot \left(-t\right)}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}} \]
      times-frac [=>] 1.1	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}}} \]

   6. Applied egg-rr 1.1

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

   7. Applied egg-rr 1.0

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

   if -1.25e-141 < k < 7.50000000000000007e-24

   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 53.5

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

      [Start] 62.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 62.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-*l* [=>] 62.3	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]
      associate-/r* [=>] 62.3	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 61.5	\[ \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-/r/ [=>] 61.5	\[ \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-*r* [=>] 61.5	\[ \frac{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      times-frac [=>] 62.2	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      associate-/r* [<=] 62.2	\[ \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      *-commutative [=>] 62.2	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. Taylor expanded in k around 0 48.1

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

   4. Simplified 44.8

      \[\leadsto \color{blue}{2 \cdot \frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]
      Proof

      [Start] 48.1	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 48.1	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      associate-/l* [=>] 44.8	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

   5. Applied egg-rr 22.3

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

   6. Applied egg-rr 0.5

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

   if 7.50000000000000007e-24 < k

   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. Taylor expanded in t around 0 19.6

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

   3. Simplified 15.3

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

      [Start] 19.6	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 19.5	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 19.5	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 19.5	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 19.5	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 15.3	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   4. Applied egg-rr 9.6

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

   5. Simplified 0.8

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

      [Start] 9.6	\[ \frac{2}{\frac{k \cdot \left(-t\right)}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}} \]
      times-frac [=>] 0.8	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\ell}{{\sin k}^{2}}} \cdot \frac{-t}{\frac{\cos k}{k} \cdot \left(-\ell\right)}}} \]

   6. Taylor expanded in k around inf 0.7

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.25 \cdot 10^{-141}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{{\sin k}^{2}}{\ell}}{\cos k} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-24}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot {\sin k}^{2}}{\ell} \cdot \frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.8
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -1.25 \cdot 10^{-141} \lor \neg \left(k \leq 10^{-27}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{{\sin k}^{2}}{\ell}}{\cos k} \cdot \frac{t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 2
Error	1.0
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00013 \lor \neg \left(k \leq 9.5 \cdot 10^{-7}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(k \cdot \frac{0.5 + \cos \left(k + k\right) \cdot -0.5}{\ell}\right)}{\frac{\cos k}{\frac{k}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 3
Error	11.1
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.15 \cdot 10^{-7} \lor \neg \left(k \leq 1.55 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\frac{k \cdot \left(k \cdot \sin k\right)}{\ell} \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 4
Error	20.2
Cost	8137

\[\begin{array}{l} \mathbf{if}\;k \leq -1.25 \cdot 10^{-141} \lor \neg \left(k \leq 4 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{2}{\frac{-t}{\ell \cdot \left(-\frac{\cos k}{k}\right)} \cdot \frac{k}{\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t \cdot \left(k \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 5
Error	24.2
Cost	1092

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;t \leq -0.04:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\frac{\ell}{k}}{k}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right)\\ \end{array} \]

Alternative 6
Error	26.0
Cost	960

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

Alternative 7
Error	24.6
Cost	960

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

Alternative 8
Error	23.4
Cost	960

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

Alternative 9
Error	22.9
Cost	960

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

Reproduce

herbie shell --seed 2023066 
(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))))