Toniolo and Linder, Equation (10-)

Average Error: 47.2 → 0.6

Time: 30.7s

Precision: binary64

Cost: 14025

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -5.6 \cdot 10^{-142} \lor \neg \left(k \leq 7.6 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell \cdot \frac{\frac{2}{k}}{\tan k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)}\\ \end{array} \]

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
 (if (or (<= k -5.6e-142) (not (<= k 7.6e-150)))
   (* (/ l (* k (sin k))) (/ (* l (/ (/ 2.0 k) (tan k))) t))
   (/ (* 2.0 (/ l k)) (* (* k (/ k l)) (* k t)))))

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 tmp;
	if ((k <= -5.6e-142) || !(k <= 7.6e-150)) {
		tmp = (l / (k * sin(k))) * ((l * ((2.0 / k) / tan(k))) / t);
	} else {
		tmp = (2.0 * (l / k)) / ((k * (k / l)) * (k * t));
	}
	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) :: tmp
    if ((k <= (-5.6d-142)) .or. (.not. (k <= 7.6d-150))) then
        tmp = (l / (k * sin(k))) * ((l * ((2.0d0 / k) / tan(k))) / t)
    else
        tmp = (2.0d0 * (l / k)) / ((k * (k / l)) * (k * t))
    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 tmp;
	if ((k <= -5.6e-142) || !(k <= 7.6e-150)) {
		tmp = (l / (k * Math.sin(k))) * ((l * ((2.0 / k) / Math.tan(k))) / t);
	} else {
		tmp = (2.0 * (l / k)) / ((k * (k / l)) * (k * t));
	}
	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):
	tmp = 0
	if (k <= -5.6e-142) or not (k <= 7.6e-150):
		tmp = (l / (k * math.sin(k))) * ((l * ((2.0 / k) / math.tan(k))) / t)
	else:
		tmp = (2.0 * (l / k)) / ((k * (k / l)) * (k * t))
	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)
	tmp = 0.0
	if ((k <= -5.6e-142) || !(k <= 7.6e-150))
		tmp = Float64(Float64(l / Float64(k * sin(k))) * Float64(Float64(l * Float64(Float64(2.0 / k) / tan(k))) / t));
	else
		tmp = Float64(Float64(2.0 * Float64(l / k)) / Float64(Float64(k * Float64(k / l)) * Float64(k * t)));
	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)
	tmp = 0.0;
	if ((k <= -5.6e-142) || ~((k <= 7.6e-150)))
		tmp = (l / (k * sin(k))) * ((l * ((2.0 / k) / tan(k))) / t);
	else
		tmp = (2.0 * (l / k)) / ((k * (k / l)) * (k * t));
	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_] := If[Or[LessEqual[k, -5.6e-142], N[Not[LessEqual[k, 7.6e-150]], $MachinePrecision]], N[(N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[(N[(2.0 / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[(N[(k * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -5.60000000000000009e-142 or 7.5999999999999997e-150 < k

   1. Initial program 46.5

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

   2. Simplified 38.3

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

      [Start] 46.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 46.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 46.5	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 46.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 38.3	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 38.3	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in t around 0 21.1

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

   4. Simplified 14.5

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

      [Start] 21.1	\[ \frac{2}{\tan k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{{\ell}^{2}}} \]
      associate-*r* [=>] 21.6	\[ \frac{2}{\tan k \cdot \frac{\color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}}{{\ell}^{2}}} \]
      unpow2 [=>] 21.6	\[ \frac{2}{\tan k \cdot \frac{\left({k}^{2} \cdot \sin k\right) \cdot t}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 14.5	\[ \frac{2}{\tan k \cdot \color{blue}{\left(\frac{{k}^{2} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)}} \]
      unpow2 [=>] 14.5	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{\left(k \cdot k\right)} \cdot \sin k}{\ell} \cdot \frac{t}{\ell}\right)} \]
      associate-*l* [=>] 14.5	\[ \frac{2}{\tan k \cdot \left(\frac{\color{blue}{k \cdot \left(k \cdot \sin k\right)}}{\ell} \cdot \frac{t}{\ell}\right)} \]

   5. Applied egg-rr 5.0

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

   6. Simplified 5.0

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

      [Start] 5.0	\[ \frac{\frac{2}{\tan k}}{k \cdot \frac{t}{\ell}} \cdot \frac{\frac{\ell}{k}}{\sin k} \]
      *-commutative [=>] 5.0	\[ \color{blue}{\frac{\frac{\ell}{k}}{\sin k} \cdot \frac{\frac{2}{\tan k}}{k \cdot \frac{t}{\ell}}} \]
      associate-/r* [=>] 5.0	\[ \frac{\frac{\ell}{k}}{\sin k} \cdot \color{blue}{\frac{\frac{\frac{2}{\tan k}}{k}}{\frac{t}{\ell}}} \]
      associate-/l/ [=>] 5.0	\[ \color{blue}{\frac{\ell}{\sin k \cdot k}} \cdot \frac{\frac{\frac{2}{\tan k}}{k}}{\frac{t}{\ell}} \]
      *-commutative [<=] 5.0	\[ \frac{\ell}{\color{blue}{k \cdot \sin k}} \cdot \frac{\frac{\frac{2}{\tan k}}{k}}{\frac{t}{\ell}} \]
      associate-/l/ [=>] 5.0	\[ \frac{\ell}{k \cdot \sin k} \cdot \frac{\color{blue}{\frac{2}{k \cdot \tan k}}}{\frac{t}{\ell}} \]

   7. Applied egg-rr 0.4

      \[\leadsto \frac{\ell}{k \cdot \sin k} \cdot \color{blue}{\left(\frac{2}{t} \cdot \frac{\frac{1}{k \cdot \tan k}}{\frac{1}{\ell}}\right)} \]

   8. Applied egg-rr 25.3

      \[\leadsto \frac{\ell}{k \cdot \sin k} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{2}{\frac{t}{\ell} \cdot \left(k \cdot \tan k\right)}\right)} - 1\right)} \]

   9. Simplified 0.3

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

      [Start] 25.3	\[ \frac{\ell}{k \cdot \sin k} \cdot \left(e^{\mathsf{log1p}\left(\frac{2}{\frac{t}{\ell} \cdot \left(k \cdot \tan k\right)}\right)} - 1\right) \]
      expm1-def [=>] 11.8	\[ \frac{\ell}{k \cdot \sin k} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{t}{\ell} \cdot \left(k \cdot \tan k\right)}\right)\right)} \]
      expm1-log1p [=>] 5.0	\[ \frac{\ell}{k \cdot \sin k} \cdot \color{blue}{\frac{2}{\frac{t}{\ell} \cdot \left(k \cdot \tan k\right)}} \]
      associate-*l/ [=>] 5.4	\[ \frac{\ell}{k \cdot \sin k} \cdot \frac{2}{\color{blue}{\frac{t \cdot \left(k \cdot \tan k\right)}{\ell}}} \]
      associate-*r/ [<=] 0.3	\[ \frac{\ell}{k \cdot \sin k} \cdot \frac{2}{\color{blue}{t \cdot \frac{k \cdot \tan k}{\ell}}} \]
      associate-/l/ [<=] 0.3	\[ \frac{\ell}{k \cdot \sin k} \cdot \color{blue}{\frac{\frac{2}{\frac{k \cdot \tan k}{\ell}}}{t}} \]
      associate-/r/ [=>] 0.3	\[ \frac{\ell}{k \cdot \sin k} \cdot \frac{\color{blue}{\frac{2}{k \cdot \tan k} \cdot \ell}}{t} \]
      associate-/r* [=>] 0.3	\[ \frac{\ell}{k \cdot \sin k} \cdot \frac{\color{blue}{\frac{\frac{2}{k}}{\tan k}} \cdot \ell}{t} \]

   if -5.60000000000000009e-142 < k < 7.5999999999999997e-150

   1. Initial program 64.0

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

   2. Simplified 57.3

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

      [Start] 64.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 64.0	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 64.0	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 64.0	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 64.0	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 64.0	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 64.0	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 57.3	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 57.3	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 57.3	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around 0 64.0

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

   4. Simplified 64.0

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

      [Start] 64.0	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      associate-*r/ [=>] 64.0	\[ \color{blue}{\frac{2 \cdot {\ell}^{2}}{{k}^{4} \cdot t}} \]
      *-commutative [=>] 64.0	\[ \frac{2 \cdot {\ell}^{2}}{\color{blue}{t \cdot {k}^{4}}} \]
      times-frac [=>] 64.0	\[ \color{blue}{\frac{2}{t} \cdot \frac{{\ell}^{2}}{{k}^{4}}} \]
      unpow2 [=>] 64.0	\[ \frac{2}{t} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4}} \]

   5. Applied egg-rr 55.5

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

   6. Applied egg-rr 6.4

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5.6 \cdot 10^{-142} \lor \neg \left(k \leq 7.6 \cdot 10^{-150}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell \cdot \frac{\frac{2}{k}}{\tan k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	6.6
Cost	14156

\[\begin{array}{l} t_1 := \frac{\frac{2}{\tan k}}{k} \cdot \left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell}{t}\right)\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \leq -6.4 \cdot 10^{-201}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3.5 \cdot 10^{-235}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;\ell \leq 4.6 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{2}{t \cdot \left(\tan k \cdot \left(\sin k \cdot \frac{k}{\frac{\ell}{k}}\right)\right)}\\ \end{array} \]

Alternative 2
Error	5.2
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -5 \cdot 10^{-26} \lor \neg \left(k \leq 1.95 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k}}{k} \cdot \left(\frac{\ell}{k \cdot \sin k} \cdot \frac{\ell}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Error	3.8
Cost	14024

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -4.6 \cdot 10^{-28}:\\ \;\;\;\;\frac{\ell}{k \cdot \sin k} \cdot \left(\ell \cdot \frac{\frac{2}{k}}{\tan k \cdot t}\right)\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{\frac{2}{\tan k}}{k} \cdot \frac{\frac{\ell}{t}}{\sin k}\right)\\ \end{array} \]

Alternative 4
Error	3.7
Cost	14024

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot \sin k}\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -4.7 \cdot 10^{-30}:\\ \;\;\;\;t_1 \cdot \left(\ell \cdot \frac{\frac{2}{k}}{\tan k \cdot t}\right)\\ \mathbf{elif}\;k \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;\frac{2}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{\frac{2}{k}}{\tan k} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 5
Error	21.8
Cost	8068

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;\ell \cdot \ell \leq 0.004:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{\frac{t \cdot {k}^{4}}{\ell}} + \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.16666666666666666\right)\\ \end{array} \]

Alternative 6
Error	26.3
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 7
Error	25.6
Cost	960

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

Alternative 8
Error	22.7
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 2023047 
(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))))