Toniolo and Linder, Equation (10+)

Average Error: 32.7 → 24.5

Time: 21.1s

Precision: binary64

Cost: 33160

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.32 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(2 + t_1\right) \cdot \left(\sin k \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(t_1 + 2\right) \cdot \tan k\right)}\\ \end{array} \]

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)))
   (if (<= t -1.32e-30)
     (/ 2.0 (* (+ 2.0 t_1) (* (sin k) (/ (* k (pow t 3.0)) (pow l 2.0)))))
     (if (<= t 2.7e-38)
       (*
        2.0
        (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* (pow (sin k) 2.0) t))))
       (/
        2.0
        (* (* (/ (pow t 3.0) (* l l)) (sin k)) (* (+ t_1 2.0) (tan k))))))))

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 tmp;
	if (t <= -1.32e-30) {
		tmp = 2.0 / ((2.0 + t_1) * (sin(k) * ((k * pow(t, 3.0)) / pow(l, 2.0))));
	} else if (t <= 2.7e-38) {
		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) / (l * l)) * sin(k)) * ((t_1 + 2.0) * tan(k)));
	}
	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
    if (t <= (-1.32d-30)) then
        tmp = 2.0d0 / ((2.0d0 + t_1) * (sin(k) * ((k * (t ** 3.0d0)) / (l ** 2.0d0))))
    else if (t <= 2.7d-38) then
        tmp = 2.0d0 * ((cos(k) * (l ** 2.0d0)) / ((k ** 2.0d0) * ((sin(k) ** 2.0d0) * t)))
    else
        tmp = 2.0d0 / ((((t ** 3.0d0) / (l * l)) * sin(k)) * ((t_1 + 2.0d0) * tan(k)))
    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 tmp;
	if (t <= -1.32e-30) {
		tmp = 2.0 / ((2.0 + t_1) * (Math.sin(k) * ((k * Math.pow(t, 3.0)) / Math.pow(l, 2.0))));
	} else if (t <= 2.7e-38) {
		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) / (l * l)) * Math.sin(k)) * ((t_1 + 2.0) * Math.tan(k)));
	}
	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)
	tmp = 0
	if t <= -1.32e-30:
		tmp = 2.0 / ((2.0 + t_1) * (math.sin(k) * ((k * math.pow(t, 3.0)) / math.pow(l, 2.0))))
	elif t <= 2.7e-38:
		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) / (l * l)) * math.sin(k)) * ((t_1 + 2.0) * math.tan(k)))
	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
	tmp = 0.0
	if (t <= -1.32e-30)
		tmp = Float64(2.0 / Float64(Float64(2.0 + t_1) * Float64(sin(k) * Float64(Float64(k * (t ^ 3.0)) / (l ^ 2.0)))));
	elseif (t <= 2.7e-38)
		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) / Float64(l * l)) * sin(k)) * Float64(Float64(t_1 + 2.0) * tan(k))));
	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;
	tmp = 0.0;
	if (t <= -1.32e-30)
		tmp = 2.0 / ((2.0 + t_1) * (sin(k) * ((k * (t ^ 3.0)) / (l ^ 2.0))));
	elseif (t <= 2.7e-38)
		tmp = 2.0 * ((cos(k) * (l ^ 2.0)) / ((k ^ 2.0) * ((sin(k) ^ 2.0) * t)));
	else
		tmp = 2.0 / ((((t ^ 3.0) / (l * l)) * sin(k)) * ((t_1 + 2.0) * tan(k)));
	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]}, If[LessEqual[t, -1.32e-30], N[(2.0 / N[(N[(2.0 + t$95$1), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(k * N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e-38], 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[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 + 2.0), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.32e-30

   1. Initial program 23.1

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

   2. Simplified 27.7

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

      [Start] 23.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 23.1	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 23.1	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.7	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.7	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\tan k \cdot \sin k\right)\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-11 [=>] 27.7	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - -1\right)} \cdot \left(\tan k \cdot \sin k\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 27.7	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - -1\right)\right)} \cdot \left(\tan k \cdot \sin k\right)\right)} \]
      metadata-eval [=>] 27.7	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right) \cdot \left(\tan k \cdot \sin k\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-35 [=>] 27.7	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \left(\tan k \cdot \sin k\right)\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 27.7	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sin k \cdot \tan k\right)}\right)} \]

   3. Applied egg-rr 27.7

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

   4. Simplified 23.1

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

      [Start] 27.7	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right) + 0} \]
      rational_best_oopsla_all_46_json_45_simplify-85 [=>] 27.7	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 27.7	\[ \frac{2}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 23.1	\[ \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \color{blue}{\left(\sin k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \tan k\right)\right)}} \]

   5. Taylor expanded in k around 0 24.5

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

   if -1.32e-30 < t < 2.70000000000000005e-38

   1. Initial program 53.5

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

   2. Simplified 53.5

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

      [Start] 53.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-74 [=>] 53.5	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-7 [=>] 53.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      rational_best_oopsla_all_46_json_45_simplify-11 [=>] 53.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - -1\right)} \cdot \tan k\right)} \]
      rational_best_oopsla_all_46_json_45_simplify-107 [=>] 53.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - -1\right)\right)} \cdot \tan k\right)} \]
      metadata-eval [=>] 53.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}\right) \cdot \tan k\right)} \]

   3. Taylor expanded in t around 0 26.8

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

   if 2.70000000000000005e-38 < 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_oopsla_all_46_json_45_simplify-74 [=>] 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_oopsla_all_46_json_45_simplify-7 [=>] 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_oopsla_all_46_json_45_simplify-11 [=>] 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_oopsla_all_46_json_45_simplify-107 [=>] 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. Recombined 3 regimes into one program.

4. Final simplification 24.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.32 \cdot 10^{-30}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left({\left(\frac{k}{t}\right)}^{2} + 2\right) \cdot \tan k\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	27.4
Cost	27080

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\sin k \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}\right)}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{-64}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{4}} - 0.16666666666666666 \cdot \frac{1}{{k}^{2}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(t_1 \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \tan k\right)\right)}\\ \end{array} \]

Alternative 2
Error	28.3
Cost	27016

\[\begin{array}{l} t_1 := \frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{k \cdot {t}^{3}}{{\ell}^{2}}\right)}\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.02 \cdot 10^{-38}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{4}} - 0.16666666666666666 \cdot \frac{1}{{k}^{2}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	29.0
Cost	20616

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{\sin k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2} \cdot \left(\frac{1}{{k}^{4}} - 0.16666666666666666 \cdot \frac{1}{{k}^{2}}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	29.7
Cost	20168

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{\sin k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	29.7
Cost	20040

\[\begin{array}{l} t_1 := \frac{{\ell}^{2}}{\sin k \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-17}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	32.6
Cost	19844

\[\begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-23}:\\ \;\;\;\;\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \end{array} \]

Alternative 7
Error	32.8
Cost	13896

\[\begin{array}{l} t_1 := \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(2 \cdot {k}^{2}\right)}\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{-37}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	38.4
Cost	13376

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

Reproduce

herbie shell --seed 2023090 
(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))))