Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 0.7

Time: 28.7s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -2 \cdot 10^{-75} \lor \neg \left(k \leq 2.35 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(t \cdot \frac{\frac{k}{\ell}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

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 -2e-75) (not (<= k 2.35e-8)))
     (/ 2.0 (* (* (/ k l) (pow (sin k) 2.0)) (* t (/ (/ k l) (cos k)))))
     (/ 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 <= -2e-75) || !(k <= 2.35e-8)) {
		tmp = 2.0 / (((k / l) * pow(sin(k), 2.0)) * (t * ((k / l) / cos(k))));
	} 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 <= (-2d-75)) .or. (.not. (k <= 2.35d-8))) then
        tmp = 2.0d0 / (((k / l) * (sin(k) ** 2.0d0)) * (t * ((k / l) / cos(k))))
    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 <= -2e-75) || !(k <= 2.35e-8)) {
		tmp = 2.0 / (((k / l) * Math.pow(Math.sin(k), 2.0)) * (t * ((k / l) / Math.cos(k))));
	} 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 <= -2e-75) or not (k <= 2.35e-8):
		tmp = 2.0 / (((k / l) * math.pow(math.sin(k), 2.0)) * (t * ((k / l) / math.cos(k))))
	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 <= -2e-75) || !(k <= 2.35e-8))
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * (sin(k) ^ 2.0)) * Float64(t * Float64(Float64(k / l) / cos(k)))));
	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 <= -2e-75) || ~((k <= 2.35e-8)))
		tmp = 2.0 / (((k / l) * (sin(k) ^ 2.0)) * (t * ((k / l) / cos(k))));
	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, -2e-75], N[Not[LessEqual[k, 2.35e-8]], $MachinePrecision]], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(t * N[(N[(k / l), $MachinePrecision] / N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < -1.9999999999999999e-75 or 2.3499999999999999e-8 < k

   1. Initial program 44.8

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

   2. Simplified 36.6

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

      [Start] 44.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 44.8	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 44.8	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 44.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 36.6	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 36.6	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 19.2

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

   4. Simplified 15.0

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

      [Start] 19.2	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 19.1	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 19.1	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 19.1	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 19.1	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 19.1	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 15.0	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 16.3

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

   6. Simplified 4.2

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

      [Start] 16.3	\[ \frac{2}{\frac{\left(\frac{t}{\ell \cdot \ell} \cdot {\sin k}^{2}\right) \cdot \left(-k\right)}{\frac{-\cos k}{k}}} \]
      associate-*r/ [<=] 19.1	\[ \frac{2}{\color{blue}{\left(\frac{t}{\ell \cdot \ell} \cdot {\sin k}^{2}\right) \cdot \frac{-k}{\frac{-\cos k}{k}}}} \]
      *-commutative [=>] 19.1	\[ \frac{2}{\color{blue}{\frac{-k}{\frac{-\cos k}{k}} \cdot \left(\frac{t}{\ell \cdot \ell} \cdot {\sin k}^{2}\right)}} \]
      associate-/r* [=>] 15.3	\[ \frac{2}{\frac{-k}{\frac{-\cos k}{k}} \cdot \left(\color{blue}{\frac{\frac{t}{\ell}}{\ell}} \cdot {\sin k}^{2}\right)} \]
      associate-*l/ [=>] 15.0	\[ \frac{2}{\frac{-k}{\frac{-\cos k}{k}} \cdot \color{blue}{\frac{\frac{t}{\ell} \cdot {\sin k}^{2}}{\ell}}} \]
      times-frac [<=] 4.3	\[ \frac{2}{\color{blue}{\frac{\left(-k\right) \cdot \left(\frac{t}{\ell} \cdot {\sin k}^{2}\right)}{\frac{-\cos k}{k} \cdot \ell}}} \]
      associate-*r* [=>] 4.2	\[ \frac{2}{\frac{\color{blue}{\left(\left(-k\right) \cdot \frac{t}{\ell}\right) \cdot {\sin k}^{2}}}{\frac{-\cos k}{k} \cdot \ell}} \]
      associate-*r/ [=>] 4.4	\[ \frac{2}{\frac{\color{blue}{\frac{\left(-k\right) \cdot t}{\ell}} \cdot {\sin k}^{2}}{\frac{-\cos k}{k} \cdot \ell}} \]
      associate-/r/ [<=] 4.4	\[ \frac{2}{\frac{\color{blue}{\frac{\left(-k\right) \cdot t}{\frac{\ell}{{\sin k}^{2}}}}}{\frac{-\cos k}{k} \cdot \ell}} \]
      distribute-frac-neg [=>] 4.4	\[ \frac{2}{\frac{\frac{\left(-k\right) \cdot t}{\frac{\ell}{{\sin k}^{2}}}}{\color{blue}{\left(-\frac{\cos k}{k}\right)} \cdot \ell}} \]
      distribute-lft-neg-in [<=] 4.4	\[ \frac{2}{\frac{\frac{\left(-k\right) \cdot t}{\frac{\ell}{{\sin k}^{2}}}}{\color{blue}{-\frac{\cos k}{k} \cdot \ell}}} \]
      distribute-rgt-neg-out [<=] 4.4	\[ \frac{2}{\frac{\frac{\left(-k\right) \cdot t}{\frac{\ell}{{\sin k}^{2}}}}{\color{blue}{\frac{\cos k}{k} \cdot \left(-\ell\right)}}} \]
      associate-/r* [<=] 8.9	\[ \frac{2}{\color{blue}{\frac{\left(-k\right) \cdot t}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}}} \]
      distribute-lft-neg-out [=>] 8.9	\[ \frac{2}{\frac{\color{blue}{-k \cdot t}}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}} \]
      distribute-rgt-neg-out [<=] 8.9	\[ \frac{2}{\frac{\color{blue}{k \cdot \left(-t\right)}}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \left(-\ell\right)\right)}} \]

   7. Taylor expanded in t around 0 4.4

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

   8. Simplified 0.7

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

      [Start] 4.4	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{k \cdot t}{\cos k \cdot \ell}} \]
      times-frac [=>] 4.3	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \color{blue}{\left(\frac{k}{\cos k} \cdot \frac{t}{\ell}\right)}} \]
      *-commutative [=>] 4.3	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{k}{\cos k}\right)}} \]
      associate-/r/ [<=] 0.7	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \color{blue}{\frac{t}{\frac{\ell}{\frac{k}{\cos k}}}}} \]
      *-rgt-identity [<=] 0.7	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \frac{\color{blue}{t \cdot 1}}{\frac{\ell}{\frac{k}{\cos k}}}} \]
      associate-*r/ [<=] 0.7	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \color{blue}{\left(t \cdot \frac{1}{\frac{\ell}{\frac{k}{\cos k}}}\right)}} \]
      associate-/r/ [=>] 0.7	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(t \cdot \color{blue}{\left(\frac{1}{\ell} \cdot \frac{k}{\cos k}\right)}\right)} \]
      associate-*r/ [=>] 0.7	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(t \cdot \color{blue}{\frac{\frac{1}{\ell} \cdot k}{\cos k}}\right)} \]
      associate-*l/ [=>] 0.7	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(t \cdot \frac{\color{blue}{\frac{1 \cdot k}{\ell}}}{\cos k}\right)} \]
      *-lft-identity [=>] 0.7	\[ \frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(t \cdot \frac{\frac{\color{blue}{k}}{\ell}}{\cos k}\right)} \]

   if -1.9999999999999999e-75 < k < 2.3499999999999999e-8

   1. Initial program 62.1

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

   2. Simplified 50.1

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

      [Start] 62.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 62.2	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 62.2	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 62.5	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 61.4	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 61.3	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 61.3	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 50.1	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 50.1	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 50.1	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around 0 45.6

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

   4. Simplified 45.2

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

      [Start] 45.6	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      associate-*r/ [=>] 45.6	\[ \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]
      *-commutative [=>] 45.6	\[ \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]
      times-frac [=>] 45.2	\[ \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
      unpow2 [=>] 45.2	\[ \frac{2}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4}} \]

   5. Applied egg-rr 18.3

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

   6. Applied egg-rr 0.9

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-75} \lor \neg \left(k \leq 2.35 \cdot 10^{-8}\right):\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(t \cdot \frac{\frac{k}{\ell}}{\cos k}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.2
Cost	14409

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3100 \lor \neg \left(k \leq 0.000105\right):\\ \;\;\;\;\frac{2}{\left(t \cdot \frac{\frac{k}{\ell}}{\cos k}\right) \cdot \left(\frac{k}{\ell} \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 2
Error	1.2
Cost	14408

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := t \cdot \frac{\frac{k}{\ell}}{\cos k}\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3100:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(k \cdot \frac{1 - t_1}{2 \cdot \ell}\right)}\\ \mathbf{elif}\;k \leq 0.000105:\\ \;\;\;\;\frac{2}{t_3 \cdot \left(t \cdot t_3\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(\frac{k}{\ell} \cdot \left(0.5 - \frac{t_1}{2}\right)\right)}\\ \end{array} \]

Alternative 3
Error	21.1
Cost	14216

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+213}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right) \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(\frac{k}{\ell} \cdot t\right)}\\ \end{array} \]

Alternative 4
Error	12.9
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -7.4 \cdot 10^{-16} \lor \neg \left(k \leq 1.05 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{\sin k} \cdot \frac{\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 5
Error	21.7
Cost	13956

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-120}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot {\sin k}^{2}\right) \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 6
Error	21.5
Cost	8004

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot t}{\left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right) \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \end{array} \]

Alternative 7
Error	26.3
Cost	960

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

Alternative 8
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 9
Error	25.1
Cost	960

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

Alternative 10
Error	22.8
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 2023017 
(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))))