Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 23.9

Time: 20.8s

Precision: binary64

Cost: 27208

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{{t}^{3}}{\ell \cdot \ell}\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t_2 \cdot \left(2 + t_1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(t_2 \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + t_1\right) + 1\right)}\\ \end{array} \]

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 (pow (/ k t) 2.0)) (t_2 (/ (pow t 3.0) (* l l))))
   (if (<= t -7.8e-47)
     (/ 2.0 (* (sin k) (* (tan k) (* t_2 (+ 2.0 t_1)))))
     (if (<= t 8.8e-114)
       (/ 2.0 (* (/ (* t (pow k 2.0)) (pow l 2.0)) (* (sin k) (tan k))))
       (/ 2.0 (* (* (* t_2 (sin k)) (tan k)) (+ (+ 1.0 t_1) 1.0)))))))

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((k / t), 2.0);
	double t_2 = pow(t, 3.0) / (l * l);
	double tmp;
	if (t <= -7.8e-47) {
		tmp = 2.0 / (sin(k) * (tan(k) * (t_2 * (2.0 + t_1))));
	} else if (t <= 8.8e-114) {
		tmp = 2.0 / (((t * pow(k, 2.0)) / pow(l, 2.0)) * (sin(k) * tan(k)));
	} else {
		tmp = 2.0 / (((t_2 * sin(k)) * tan(k)) * ((1.0 + t_1) + 1.0));
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    t_2 = (t ** 3.0d0) / (l * l)
    if (t <= (-7.8d-47)) then
        tmp = 2.0d0 / (sin(k) * (tan(k) * (t_2 * (2.0d0 + t_1))))
    else if (t <= 8.8d-114) then
        tmp = 2.0d0 / (((t * (k ** 2.0d0)) / (l ** 2.0d0)) * (sin(k) * tan(k)))
    else
        tmp = 2.0d0 / (((t_2 * sin(k)) * tan(k)) * ((1.0d0 + t_1) + 1.0d0))
    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((k / t), 2.0);
	double t_2 = Math.pow(t, 3.0) / (l * l);
	double tmp;
	if (t <= -7.8e-47) {
		tmp = 2.0 / (Math.sin(k) * (Math.tan(k) * (t_2 * (2.0 + t_1))));
	} else if (t <= 8.8e-114) {
		tmp = 2.0 / (((t * Math.pow(k, 2.0)) / Math.pow(l, 2.0)) * (Math.sin(k) * Math.tan(k)));
	} else {
		tmp = 2.0 / (((t_2 * Math.sin(k)) * Math.tan(k)) * ((1.0 + t_1) + 1.0));
	}
	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((k / t), 2.0)
	t_2 = math.pow(t, 3.0) / (l * l)
	tmp = 0
	if t <= -7.8e-47:
		tmp = 2.0 / (math.sin(k) * (math.tan(k) * (t_2 * (2.0 + t_1))))
	elif t <= 8.8e-114:
		tmp = 2.0 / (((t * math.pow(k, 2.0)) / math.pow(l, 2.0)) * (math.sin(k) * math.tan(k)))
	else:
		tmp = 2.0 / (((t_2 * math.sin(k)) * math.tan(k)) * ((1.0 + t_1) + 1.0))
	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 / t) ^ 2.0
	t_2 = Float64((t ^ 3.0) / Float64(l * l))
	tmp = 0.0
	if (t <= -7.8e-47)
		tmp = Float64(2.0 / Float64(sin(k) * Float64(tan(k) * Float64(t_2 * Float64(2.0 + t_1)))));
	elseif (t <= 8.8e-114)
		tmp = Float64(2.0 / Float64(Float64(Float64(t * (k ^ 2.0)) / (l ^ 2.0)) * Float64(sin(k) * tan(k))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t_2 * sin(k)) * tan(k)) * Float64(Float64(1.0 + t_1) + 1.0)));
	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 / t) ^ 2.0;
	t_2 = (t ^ 3.0) / (l * l);
	tmp = 0.0;
	if (t <= -7.8e-47)
		tmp = 2.0 / (sin(k) * (tan(k) * (t_2 * (2.0 + t_1))));
	elseif (t <= 8.8e-114)
		tmp = 2.0 / (((t * (k ^ 2.0)) / (l ^ 2.0)) * (sin(k) * tan(k)));
	else
		tmp = 2.0 / (((t_2 * sin(k)) * tan(k)) * ((1.0 + t_1) + 1.0));
	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[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -7.8e-47], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(t$95$2 * N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8.8e-114], N[(2.0 / N[(N[(N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(t$95$2 * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.79999999999999956e-47

   1. Initial program 22.5

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

   2. Simplified 22.5

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

      [Start] 22.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational_best.json-simplify-2 [=>] 22.5	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best.json-simplify-2 [=>] 22.5	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]
      rational_best.json-simplify-2 [=>] 22.5	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\tan k \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\right)} \]
      rational_best.json-simplify-44 [=>] 22.5	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\sin k \cdot \left(\tan k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)}} \]
      rational_best.json-simplify-44 [=>] 22.5	\[ \frac{2}{\color{blue}{\sin k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\tan k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)}} \]
      rational_best.json-simplify-44 [=>] 22.5	\[ \frac{2}{\sin k \cdot \color{blue}{\left(\tan k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)}} \]
      rational_best.json-simplify-2 [=>] 22.5	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)} \]
      rational_best.json-simplify-1 [=>] 22.5	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} \]
      rational_best.json-simplify-43 [=>] 22.5	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}\right)\right)} \]
      metadata-eval [=>] 22.5	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)\right)\right)} \]
      rational_best.json-simplify-1 [=>] 22.5	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

   if -7.79999999999999956e-47 < t < 8.80000000000000045e-114

   1. Initial program 58.3

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

   2. Simplified 58.2

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

      [Start] 58.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational_best.json-simplify-2 [=>] 58.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best.json-simplify-2 [=>] 58.3	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]
      rational_best.json-simplify-2 [=>] 58.3	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\tan k \cdot \color{blue}{\left(\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)}\right)} \]
      rational_best.json-simplify-44 [=>] 58.3	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\sin k \cdot \left(\tan k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)}} \]
      rational_best.json-simplify-44 [=>] 58.2	\[ \frac{2}{\color{blue}{\sin k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\tan k \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)}} \]
      rational_best.json-simplify-44 [=>] 58.2	\[ \frac{2}{\sin k \cdot \color{blue}{\left(\tan k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right)\right)}} \]
      rational_best.json-simplify-2 [=>] 58.2	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}\right)} \]
      rational_best.json-simplify-1 [=>] 58.2	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} \]
      rational_best.json-simplify-43 [=>] 58.2	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)\right)}\right)\right)} \]
      metadata-eval [=>] 58.2	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right)\right)\right)} \]
      rational_best.json-simplify-1 [=>] 58.2	\[ \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\right)} \]

   3. Taylor expanded in t around 0 25.6

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

   4. Applied egg-rr 25.6

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

   5. Simplified 25.6

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

      [Start] 25.6	\[ \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\tan k \cdot \sin k\right) + 0} \]
      rational_best.json-simplify-4 [=>] 25.6	\[ \frac{2}{\color{blue}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \left(\tan k \cdot \sin k\right)}} \]
      rational_best.json-simplify-2 [=>] 25.6	\[ \frac{2}{\frac{{k}^{2} \cdot t}{{\ell}^{2}} \cdot \color{blue}{\left(\sin k \cdot \tan k\right)}} \]
      rational_best.json-simplify-2 [=>] 25.6	\[ \frac{2}{\frac{\color{blue}{t \cdot {k}^{2}}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)} \]

   if 8.80000000000000045e-114 < t

   1. Initial program 24.0

      \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	23.8
Cost	27080

\[\begin{array}{l} t_1 := \frac{2}{\sin k \cdot \left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{if}\;t \leq -1.92 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	23.8
Cost	27080

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{{t}^{3}}{\ell \cdot \ell}\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \left(t_2 \cdot \left(2 + t_1\right)\right)\right)}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(t_2 \cdot \sin k\right) \cdot \left(\left(t_1 + 2\right) \cdot \tan k\right)}\\ \end{array} \]

Alternative 3
Error	25.3
Cost	26696

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-50}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{t \cdot {\left(k \cdot \sin k\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	24.9
Cost	26696

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -5 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\tan k \cdot \frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	24.9
Cost	26696

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{t \cdot {k}^{2}}{{\ell}^{2}} \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	28.8
Cost	20168

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{2}{\sin k \cdot \frac{{k}^{3} \cdot t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	29.0
Cost	13640

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4} \cdot t}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	38.5
Cost	13376

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

Alternative 9
Error	38.5
Cost	13376

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

Reproduce

herbie shell --seed 2023089 
(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))))