Toniolo and Linder, Equation (10-)

Average Error: 47.3 → 0.4

Time: 33.4s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -8.4 \cdot 10^{-44} \lor \neg \left(k \leq 1.3 \cdot 10^{-66}\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \cos k\right) \cdot \frac{\frac{{\sin k}^{-2}}{\frac{k}{\ell \cdot 2}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

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 (* k (/ k l))))
   (if (or (<= k -8.4e-44) (not (<= k 1.3e-66)))
     (* (* (/ l k) (cos k)) (/ (/ (pow (sin k) -2.0) (/ k (* l 2.0))) t))
     (/ 2.0 (* t_1 (* t 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 = k * (k / l);
	double tmp;
	if ((k <= -8.4e-44) || !(k <= 1.3e-66)) {
		tmp = ((l / k) * cos(k)) * ((pow(sin(k), -2.0) / (k / (l * 2.0))) / t);
	} else {
		tmp = 2.0 / (t_1 * (t * 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 = k * (k / l)
    if ((k <= (-8.4d-44)) .or. (.not. (k <= 1.3d-66))) then
        tmp = ((l / k) * cos(k)) * (((sin(k) ** (-2.0d0)) / (k / (l * 2.0d0))) / t)
    else
        tmp = 2.0d0 / (t_1 * (t * 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 = k * (k / l);
	double tmp;
	if ((k <= -8.4e-44) || !(k <= 1.3e-66)) {
		tmp = ((l / k) * Math.cos(k)) * ((Math.pow(Math.sin(k), -2.0) / (k / (l * 2.0))) / t);
	} else {
		tmp = 2.0 / (t_1 * (t * 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 = k * (k / l)
	tmp = 0
	if (k <= -8.4e-44) or not (k <= 1.3e-66):
		tmp = ((l / k) * math.cos(k)) * ((math.pow(math.sin(k), -2.0) / (k / (l * 2.0))) / t)
	else:
		tmp = 2.0 / (t_1 * (t * 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(k * Float64(k / l))
	tmp = 0.0
	if ((k <= -8.4e-44) || !(k <= 1.3e-66))
		tmp = Float64(Float64(Float64(l / k) * cos(k)) * Float64(Float64((sin(k) ^ -2.0) / Float64(k / Float64(l * 2.0))) / t));
	else
		tmp = Float64(2.0 / Float64(t_1 * Float64(t * 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 = k * (k / l);
	tmp = 0.0;
	if ((k <= -8.4e-44) || ~((k <= 1.3e-66)))
		tmp = ((l / k) * cos(k)) * (((sin(k) ^ -2.0) / (k / (l * 2.0))) / t);
	else
		tmp = 2.0 / (t_1 * (t * 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[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, -8.4e-44], N[Not[LessEqual[k, 1.3e-66]], $MachinePrecision]], N[(N[(N[(l / k), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / N[(k / N[(l * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(t$95$1 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < -8.40000000000000005e-44 or 1.2999999999999999e-66 < k

   1. Initial program 45.1

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

   2. Simplified 37.1

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

      [Start] 45.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 45.1	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 45.1	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 45.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 37.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 37.1	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 18.9

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

   4. Simplified 14.3

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

      [Start] 18.9	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 18.9	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 18.9	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 18.9	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 18.9	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 18.9	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 14.3	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 9.2

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

   6. Applied egg-rr 4.2

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

   7. Simplified 4.1

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

      [Start] 4.2	\[ \left(\frac{2}{k \cdot t} \cdot \left(\ell \cdot {\sin k}^{-2}\right)\right) \cdot \frac{\ell}{\frac{k}{\cos k}} \]
      *-commutative [=>] 4.2	\[ \color{blue}{\frac{\ell}{\frac{k}{\cos k}} \cdot \left(\frac{2}{k \cdot t} \cdot \left(\ell \cdot {\sin k}^{-2}\right)\right)} \]
      associate-/r/ [=>] 4.1	\[ \color{blue}{\left(\frac{\ell}{k} \cdot \cos k\right)} \cdot \left(\frac{2}{k \cdot t} \cdot \left(\ell \cdot {\sin k}^{-2}\right)\right) \]
      *-commutative [=>] 4.1	\[ \left(\frac{\ell}{k} \cdot \cos k\right) \cdot \color{blue}{\left(\left(\ell \cdot {\sin k}^{-2}\right) \cdot \frac{2}{k \cdot t}\right)} \]

   8. Applied egg-rr 0.4

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

   if -8.40000000000000005e-44 < k < 1.2999999999999999e-66

   1. Initial program 63.4

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

   2. Simplified 53.6

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

      [Start] 63.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 63.4	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 63.4	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 63.7	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 62.5	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 62.5	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 62.5	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 53.6	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 53.6	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 53.6	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around 0 56.2

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

   4. Simplified 56.3

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

      [Start] 56.2	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      associate-*r/ [=>] 56.2	\[ \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]
      *-commutative [=>] 56.2	\[ \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]
      times-frac [=>] 56.3	\[ \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
      unpow2 [=>] 56.3	\[ \frac{2}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4}} \]

   5. Applied egg-rr 22.3

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

   6. Applied egg-rr 0.7

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -8.4 \cdot 10^{-44} \lor \neg \left(k \leq 1.3 \cdot 10^{-66}\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \cos k\right) \cdot \frac{\frac{{\sin k}^{-2}}{\frac{k}{\ell \cdot 2}}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	4.0
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.35 \cdot 10^{-40} \lor \neg \left(k \leq 2.6 \cdot 10^{-120}\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \cos k\right) \cdot \left(\left(\ell \cdot {\sin k}^{-2}\right) \cdot \frac{2}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	20489

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.5 \cdot 10^{-43} \lor \neg \left(k \leq 3.3 \cdot 10^{-65}\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \cos k\right) \cdot \frac{\frac{2}{k} \cdot \left(\ell \cdot {\sin k}^{-2}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	4.0
Cost	20488

\[\begin{array}{l} t_1 := \frac{2}{k \cdot t}\\ t_2 := \frac{\ell}{k} \cdot \cos k\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3.5 \cdot 10^{-40}:\\ \;\;\;\;t_2 \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot t_1\right)\\ \mathbf{elif}\;k \leq 2.85 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(t \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\left(\ell \cdot {\sin k}^{-2}\right) \cdot t_1\right)\\ \end{array} \]

Alternative 4
Error	3.9
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \cos k\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3.5 \cdot 10^{-40}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{2}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{2}{\frac{k \cdot t}{\ell \cdot {\sin k}^{-2}}}\\ \end{array} \]

Alternative 5
Error	6.9
Cost	14540

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := k \cdot \frac{k}{\ell}\\ t_3 := 0.5 - \frac{\cos \left(k + k\right)}{2}\\ t_4 := \frac{2}{\frac{k \cdot t}{\frac{\ell}{t_3} \cdot \left(\ell \cdot t_1\right)}}\\ \mathbf{if}\;k \leq -0.000215:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{+127}:\\ \;\;\;\;\frac{2}{\frac{k}{t_1} \cdot \left(\frac{t}{\ell} \cdot \frac{t_3}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

Alternative 6
Error	6.9
Cost	14540

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := k \cdot \frac{k}{\ell}\\ t_3 := 0.5 - \frac{\cos \left(k + k\right)}{2}\\ t_4 := \frac{\ell}{t_3}\\ \mathbf{if}\;k \leq -0.00155:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{t_4 \cdot \frac{\ell \cdot \cos k}{k}}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+131}:\\ \;\;\;\;\frac{2}{\frac{k}{t_1} \cdot \left(\frac{t}{\ell} \cdot \frac{t_3}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{t_4 \cdot \left(\ell \cdot t_1\right)}}\\ \end{array} \]

Alternative 7
Error	3.7
Cost	14540

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := k \cdot \frac{k}{\ell}\\ t_3 := \frac{\ell}{0.5 - \frac{\cos \left(k + k\right)}{2}}\\ t_4 := \frac{2}{\frac{k \cdot \frac{t}{\ell}}{t_3 \cdot t_1}}\\ \mathbf{if}\;k \leq -0.000102:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;k \leq 5.4 \cdot 10^{+239}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{t_3 \cdot \left(\ell \cdot t_1\right)}}\\ \end{array} \]

Alternative 8
Error	12.6
Cost	14408

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -4 \cdot 10^{-18}:\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\sin k} \cdot \frac{\ell}{t}}}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\ell}\right)}\\ \end{array} \]

Alternative 9
Error	12.9
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -7 \cdot 10^{-22} \lor \neg \left(k \leq 1.16 \cdot 10^{-63}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\sin k} \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 10
Error	20.4
Cost	8009

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.05 \cdot 10^{-42} \lor \neg \left(k \leq 1.85 \cdot 10^{-120}\right):\\ \;\;\;\;\left(\frac{\ell}{k} \cdot \cos k\right) \cdot \left(\frac{2}{k \cdot t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 11
Error	25.8
Cost	960

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

Alternative 12
Error	22.4
Cost	960

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

Reproduce

herbie shell --seed 2022356 
(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))))