Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 23.9

Time: 28.8s

Precision: binary64

Cost: 33416

Math

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

↓

\[\begin{array}{l} t_1 := \left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\\ \mathbf{if}\;t \leq -24000:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot t_1}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \sin k}{{\ell}^{2}} \cdot t_1}\\ \end{array} \]

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) 2.0) (tan k))))
   (if (<= t -24000.0)
     (/ 2.0 (* (* (/ (pow t 3.0) (* l l)) (sin k)) t_1))
     (if (<= t 9e-82)
       (*
        2.0
        (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* (pow (sin k) 2.0) t))))
       (/ 2.0 (* (/ (* (pow t 3.0) (sin k)) (pow l 2.0)) 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 = (pow((k / t), 2.0) + 2.0) * tan(k);
	double tmp;
	if (t <= -24000.0) {
		tmp = 2.0 / (((pow(t, 3.0) / (l * l)) * sin(k)) * t_1);
	} else if (t <= 9e-82) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (pow(sin(k), 2.0) * t)));
	} else {
		tmp = 2.0 / (((pow(t, 3.0) * sin(k)) / pow(l, 2.0)) * 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 / t) ** 2.0d0) + 2.0d0) * tan(k)
    if (t <= (-24000.0d0)) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / (l * l)) * sin(k)) * t_1)
    else if (t <= 9d-82) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / ((k ** 2.0d0) * ((sin(k) ** 2.0d0) * t)))
    else
        tmp = 2.0d0 / ((((t ** 3.0d0) * sin(k)) / (l ** 2.0d0)) * 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 = (Math.pow((k / t), 2.0) + 2.0) * Math.tan(k);
	double tmp;
	if (t <= -24000.0) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * t_1);
	} else if (t <= 9e-82) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * (Math.pow(Math.sin(k), 2.0) * t)));
	} else {
		tmp = 2.0 / (((Math.pow(t, 3.0) * Math.sin(k)) / Math.pow(l, 2.0)) * 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 = (math.pow((k / t), 2.0) + 2.0) * math.tan(k)
	tmp = 0
	if t <= -24000.0:
		tmp = 2.0 / (((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * t_1)
	elif t <= 9e-82:
		tmp = 2.0 * ((math.cos(k) * math.pow(l, 2.0)) / (math.pow(k, 2.0) * (math.pow(math.sin(k), 2.0) * t)))
	else:
		tmp = 2.0 / (((math.pow(t, 3.0) * math.sin(k)) / math.pow(l, 2.0)) * 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(Float64((Float64(k / t) ^ 2.0) + 2.0) * tan(k))
	tmp = 0.0
	if (t <= -24000.0)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * t_1));
	elseif (t <= 9e-82)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * Float64((sin(k) ^ 2.0) * t))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) * sin(k)) / (l ^ 2.0)) * 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 / t) ^ 2.0) + 2.0) * tan(k);
	tmp = 0.0;
	if (t <= -24000.0)
		tmp = 2.0 / ((((t ^ 3.0) / (l * l)) * sin(k)) * t_1);
	elseif (t <= 9e-82)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / ((k ^ 2.0) * ((sin(k) ^ 2.0) * t)));
	else
		tmp = 2.0 / ((((t ^ 3.0) * sin(k)) / (l ^ 2.0)) * 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[(N[(N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision] + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -24000.0], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9e-82], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -24000

   1. Initial program 23.0

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

   2. Simplified 22.9

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

      [Start] 23.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational_best_45_simplify-91 [=>] 23.0	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best_45_simplify-25 [=>] 22.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      rational_best_45_simplify-19 [=>] 22.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - -1\right)} \cdot \tan k\right)} \]
      rational_best_45_simplify-109 [=>] 22.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - -1\right)\right)} \cdot \tan k\right)} \]
      metadata-eval [=>] 22.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right) \cdot \tan k\right)} \]

   if -24000 < t < 8.9999999999999997e-82

   1. Initial program 53.1

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

   2. Simplified 53.1

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

      [Start] 53.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational_best_45_simplify-91 [=>] 53.1	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best_45_simplify-25 [=>] 53.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      rational_best_45_simplify-19 [=>] 53.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - -1\right)} \cdot \tan k\right)} \]
      rational_best_45_simplify-109 [=>] 53.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - -1\right)\right)} \cdot \tan k\right)} \]
      metadata-eval [=>] 53.1	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right) \cdot \tan k\right)} \]

   3. Taylor expanded in t around 0 27.4

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

   if 8.9999999999999997e-82 < t

   1. Initial program 22.4

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

   2. Simplified 22.4

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

      [Start] 22.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational_best_45_simplify-91 [=>] 22.4	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best_45_simplify-25 [=>] 22.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      rational_best_45_simplify-19 [=>] 22.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - -1\right)} \cdot \tan k\right)} \]
      rational_best_45_simplify-109 [=>] 22.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - -1\right)\right)} \cdot \tan k\right)} \]
      metadata-eval [=>] 22.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right) \cdot \tan k\right)} \]

   3. Taylor expanded in t around 0 21.8

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

   4. Simplified 21.8

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

      [Start] 21.8	\[ \frac{2}{\frac{\sin k \cdot {t}^{3}}{{\ell}^{2}} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)} \]
      rational_best_45_simplify-91 [=>] 21.8	\[ \frac{2}{\frac{\color{blue}{{t}^{3} \cdot \sin k}}{{\ell}^{2}} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 23.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -24000:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)}\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot \sin k}{{\ell}^{2}} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	24.2
Cost	33160

\[\begin{array}{l} t_1 := \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)}\\ \mathbf{if}\;t \leq -24000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-81}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	33.2
Cost	27412

\[\begin{array}{l} t_1 := 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + \left(-\frac{{\ell}^{2}}{t \cdot {k}^{2}}\right)\\ t_2 := \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{if}\;k \leq -2.3 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1080:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1.3 \cdot 10^{-33}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left({k}^{4} + -0.3333333333333333 \cdot {k}^{6}\right)}\\ \mathbf{elif}\;k \leq -1 \cdot 10^{-111}:\\ \;\;\;\;\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}\\ \mathbf{elif}\;k \leq 3.7 \cdot 10^{+65}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	29.4
Cost	27344

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{t \cdot {k}^{2}}\\ t_2 := \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ t_3 := 2 \cdot \left(t_2 + t_1 \cdot -0.16666666666666666\right)\\ \mathbf{if}\;k \leq -7.5 \cdot 10^{+34}:\\ \;\;\;\;2 \cdot t_2 + \left(-t_1\right)\\ \mathbf{elif}\;k \leq -1080:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -2.7 \cdot 10^{-32}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot k}{{\ell}^{2}} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 4
Error	29.4
Cost	27280

\[\begin{array}{l} t_1 := 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + \left(-\frac{{\ell}^{2}}{t \cdot {k}^{2}}\right)\\ \mathbf{if}\;k \leq -2.5 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -1080:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ \mathbf{elif}\;k \leq -9.2 \cdot 10^{-36}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left({k}^{4} + -0.3333333333333333 \cdot {k}^{6}\right)}\\ \mathbf{elif}\;k \leq 5.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3} \cdot k}{{\ell}^{2}} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	33.4
Cost	27220

\[\begin{array}{l} t_1 := \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ t_2 := \frac{2}{\frac{{k}^{2} \cdot \left({k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{if}\;k \leq -1.36 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1080:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.2 \cdot 10^{-32}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-123}:\\ \;\;\;\;\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}\\ \mathbf{elif}\;k \leq 8.5 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	33.3
Cost	27220

\[\begin{array}{l} t_1 := \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot \frac{{\sin k}^{2}}{\cos k}\right)}\\ t_2 := \frac{2}{\frac{{k}^{2} \cdot \left({k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{if}\;k \leq -1.28 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -1080:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -2.8 \cdot 10^{-32}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \cos k}{t \cdot \left({k}^{4} + -0.3333333333333333 \cdot {k}^{6}\right)}\\ \mathbf{elif}\;k \leq -2 \cdot 10^{-121}:\\ \;\;\;\;\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}\\ \mathbf{elif}\;k \leq 9 \cdot 10^{+66}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	27.1
Cost	27080

\[\begin{array}{l} t_1 := \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)}\\ \mathbf{if}\;t \leq -9.4 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + \left(-\frac{{\ell}^{2}}{t \cdot {k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	32.3
Cost	26760

\[\begin{array}{l} t_1 := \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-61}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left({k}^{2} \cdot {k}^{2}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	32.3
Cost	26760

\[\begin{array}{l} t_1 := {\ell}^{2} \cdot \cos k\\ \mathbf{if}\;t \leq -5.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{3}\right)}{t_1}}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left({k}^{2} \cdot {k}^{2}\right)}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \end{array} \]

Alternative 10
Error	32.3
Cost	26760

\[\begin{array}{l} t_1 := \cos k \cdot {\ell}^{2}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-39}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left(2 \cdot {t}^{3}\right)}{{\ell}^{2} \cdot \cos k}}\\ \mathbf{elif}\;t \leq 4.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{{k}^{2} \cdot \left({k}^{2} \cdot t\right)}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{{k}^{2} \cdot {t}^{3}}\\ \end{array} \]

Alternative 11
Error	32.3
Cost	26504

\[\begin{array}{l} t_1 := \cos k \cdot {\ell}^{2}\\ t_2 := \frac{t_1}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{if}\;t \leq -8.2 \cdot 10^{-63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4} \cdot t}{t_1}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 12
Error	32.7
Cost	20424

\[\begin{array}{l} \mathbf{if}\;t \leq -24000:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 1.18 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4} \cdot t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot \frac{{k}^{2}}{\cos k}\right)}\\ \end{array} \]

Alternative 13
Error	32.7
Cost	20168

\[\begin{array}{l} \mathbf{if}\;t \leq -165000:\\ \;\;\;\;\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 1.08 \cdot 10^{-45}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4} \cdot t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]

Alternative 14
Error	32.7
Cost	20168

\[\begin{array}{l} \mathbf{if}\;t \leq -30000:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3} \cdot {k}^{2}}{{\ell}^{2}}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4} \cdot t}{\cos k \cdot {\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]

Alternative 15
Error	32.7
Cost	19844

\[\begin{array}{l} \mathbf{if}\;t \leq -24000:\\ \;\;\;\;\frac{{\ell}^{2}}{{t}^{3} \cdot {k}^{2}}\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-45}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]

Alternative 16
Error	33.0
Cost	13896

\[\begin{array}{l} t_1 := \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{if}\;t \leq -24000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{-46}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Error	38.7
Cost	13376

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

Reproduce

herbie shell --seed 2023098 
(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))))