Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 0.6

Time: 35.5s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -1.38 \cdot 10^{-126} \lor \neg \left(k \leq 2 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot \frac{{\sin k}^{2}}{\ell}\right)} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

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 (/ l k))))
   (if (or (<= k -1.38e-126) (not (<= k 2e-13)))
     (* (/ 2.0 (* t (* k (/ (pow (sin k) 2.0) l)))) (* l (/ (cos k) 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 / (l / k);
	double tmp;
	if ((k <= -1.38e-126) || !(k <= 2e-13)) {
		tmp = (2.0 / (t * (k * (pow(sin(k), 2.0) / l)))) * (l * (cos(k) / 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 / (l / k)
    if ((k <= (-1.38d-126)) .or. (.not. (k <= 2d-13))) then
        tmp = (2.0d0 / (t * (k * ((sin(k) ** 2.0d0) / l)))) * (l * (cos(k) / 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 / (l / k);
	double tmp;
	if ((k <= -1.38e-126) || !(k <= 2e-13)) {
		tmp = (2.0 / (t * (k * (Math.pow(Math.sin(k), 2.0) / l)))) * (l * (Math.cos(k) / 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 / (l / k)
	tmp = 0
	if (k <= -1.38e-126) or not (k <= 2e-13):
		tmp = (2.0 / (t * (k * (math.pow(math.sin(k), 2.0) / l)))) * (l * (math.cos(k) / 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(l / k))
	tmp = 0.0
	if ((k <= -1.38e-126) || !(k <= 2e-13))
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k * Float64((sin(k) ^ 2.0) / l)))) * Float64(l * Float64(cos(k) / 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 / (l / k);
	tmp = 0.0;
	if ((k <= -1.38e-126) || ~((k <= 2e-13)))
		tmp = (2.0 / (t * (k * ((sin(k) ^ 2.0) / l)))) * (l * (cos(k) / 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[(l / k), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, -1.38e-126], N[Not[LessEqual[k, 2e-13]], $MachinePrecision]], N[(N[(2.0 / N[(t * N[(k * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / k), $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 < -1.38000000000000003e-126 or 2.0000000000000001e-13 < k

   1. Initial program 45.2

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

   2. Simplified 36.8

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

      [Start] 45.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 45.2	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 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+ [=>] 36.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 36.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 19.6

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

   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] 19.6	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 19.6	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 19.6	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 19.6	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 19.6	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 19.6	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      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 4.6

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

   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 -1.38000000000000003e-126 < k < 2.0000000000000001e-13

   1. Initial program 61.9

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

   2. Simplified 55.9

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

      [Start] 61.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 61.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 61.8	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 61.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 55.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 55.9	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around 0 44.8

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

   4. Simplified 37.3

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

      [Start] 44.8	\[ \frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 44.8	\[ \frac{2}{\frac{\color{blue}{t \cdot {k}^{4}}}{{\ell}^{2}}} \]
      unpow2 [=>] 44.8	\[ \frac{2}{\frac{t \cdot {k}^{4}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 37.3	\[ \frac{2}{\color{blue}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}} \]

   5. Applied egg-rr 20.0

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

   6. Applied egg-rr 0.8

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.38 \cdot 10^{-126} \lor \neg \left(k \leq 2 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot \frac{{\sin k}^{2}}{\ell}\right)} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\ell}{k}} \cdot \left(t \cdot \frac{k}{\frac{\ell}{k}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.8
Cost	20489

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -4.6 \cdot 10^{-11} \lor \neg \left(k \leq 3.5 \cdot 10^{-11}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\ell}{t} \cdot \frac{\cos k}{k \cdot k}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 2
Error	6.8
Cost	20489

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -2.7 \cdot 10^{-17} \lor \neg \left(k \leq 2.05 \cdot 10^{-13}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{{\sin k}^{2}} \cdot \frac{\frac{\cos k \cdot \frac{\ell}{t}}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	4.1
Cost	20489

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -1.95 \cdot 10^{-13} \lor \neg \left(k \leq 1.8 \cdot 10^{-13}\right):\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{t}}{\frac{k}{\cos k}} \cdot \frac{\ell \cdot {\sin k}^{-2}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 4
Error	3.8
Cost	20489

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -5.5 \cdot 10^{-152} \lor \neg \left(k \leq 1.1 \cdot 10^{-92}\right):\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(t \cdot \frac{{\sin k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 5
Error	3.9
Cost	20488

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ t_2 := \frac{\cos k}{k}\\ t_3 := t \cdot \frac{{\sin k}^{2}}{\ell}\\ \mathbf{if}\;k \leq -6.6 \cdot 10^{-152}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot t_2}{k \cdot t_3}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{t_2}{t_3}\right)\\ \end{array} \]

Alternative 6
Error	11.0
Cost	14409

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -5.5 \cdot 10^{-5} \lor \neg \left(k \leq 9.5 \cdot 10^{-5}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{k \cdot k}\right) \cdot \frac{2 \cdot \ell}{1 - \cos \left(k + k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 7
Error	12.7
Cost	14025

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{-9} \lor \neg \left(k \leq 1.35 \cdot 10^{-10}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \frac{k \cdot k}{\frac{\ell}{t} \cdot \frac{\ell}{\sin k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 8
Error	20.1
Cost	8009

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -4.6 \cdot 10^{-11} \lor \neg \left(k \leq 6.5 \cdot 10^{-69}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right) \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 9
Error	20.3
Cost	8008

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -1.55:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \frac{\cos k}{\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right) \cdot \frac{\cos k}{k \cdot \left(k \cdot \frac{t}{\ell}\right)}\right)\\ \end{array} \]

Alternative 10
Error	21.5
Cost	8004

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{-276}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{t} \cdot \frac{\cos k}{k \cdot k}\right) \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 11
Error	20.8
Cost	8004

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-163}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right) \cdot \left(\cos k \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\right)\\ \end{array} \]

Alternative 12
Error	21.2
Cost	7492

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;k \leq -1.55:\\ \;\;\;\;2 \cdot \left(0.3333333333333333 \cdot \frac{\cos k}{\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 13
Error	24.1
Cost	1224

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ t_2 := \frac{\ell}{k \cdot k}\\ \mathbf{if}\;k \leq -2.6 \cdot 10^{-152}:\\ \;\;\;\;t_2 \cdot \frac{\frac{\ell}{t} \cdot \frac{2}{k}}{k}\\ \mathbf{elif}\;k \leq 1.1 \cdot 10^{-107}:\\ \;\;\;\;\frac{2}{t \cdot \left(t_1 \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \left(\frac{\ell}{k} \cdot \frac{2}{k \cdot t}\right)\\ \end{array} \]

Alternative 14
Error	25.4
Cost	960

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

Alternative 15
Error	24.5
Cost	960

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

Alternative 16
Error	24.7
Cost	960

\[\frac{\ell}{k \cdot k} \cdot \frac{\frac{\frac{\ell}{0.5}}{k \cdot t}}{k} \]

Alternative 17
Error	22.3
Cost	960

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

Reproduce

herbie shell --seed 2023045 
(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))))