Toniolo and Linder, Equation (10-)

Average Error: 47.1 → 22.1

Time: 40.3s

Precision: binary64

Cost: 33092

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{{k}^{2}}{\cos k \cdot {\ell}^{2}}\right)}\\ \end{array} \]

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 (<= l -1e-134)
   (* 2.0 (* (cos k) (/ (pow l 2.0) (* (pow (sin k) 2.0) (* t (pow k 2.0))))))
   (/
    2.0
    (* (sin k) (* (* (sin k) t) (/ (pow k 2.0) (* (cos k) (pow l 2.0))))))))

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 tmp;
	if (l <= -1e-134) {
		tmp = 2.0 * (cos(k) * (pow(l, 2.0) / (pow(sin(k), 2.0) * (t * pow(k, 2.0)))));
	} else {
		tmp = 2.0 / (sin(k) * ((sin(k) * t) * (pow(k, 2.0) / (cos(k) * pow(l, 2.0)))));
	}
	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) :: tmp
    if (l <= (-1d-134)) then
        tmp = 2.0d0 * (cos(k) * ((l ** 2.0d0) / ((sin(k) ** 2.0d0) * (t * (k ** 2.0d0)))))
    else
        tmp = 2.0d0 / (sin(k) * ((sin(k) * t) * ((k ** 2.0d0) / (cos(k) * (l ** 2.0d0)))))
    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 tmp;
	if (l <= -1e-134) {
		tmp = 2.0 * (Math.cos(k) * (Math.pow(l, 2.0) / (Math.pow(Math.sin(k), 2.0) * (t * Math.pow(k, 2.0)))));
	} else {
		tmp = 2.0 / (Math.sin(k) * ((Math.sin(k) * t) * (Math.pow(k, 2.0) / (Math.cos(k) * Math.pow(l, 2.0)))));
	}
	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):
	tmp = 0
	if l <= -1e-134:
		tmp = 2.0 * (math.cos(k) * (math.pow(l, 2.0) / (math.pow(math.sin(k), 2.0) * (t * math.pow(k, 2.0)))))
	else:
		tmp = 2.0 / (math.sin(k) * ((math.sin(k) * t) * (math.pow(k, 2.0) / (math.cos(k) * math.pow(l, 2.0)))))
	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)
	tmp = 0.0
	if (l <= -1e-134)
		tmp = Float64(2.0 * Float64(cos(k) * Float64((l ^ 2.0) / Float64((sin(k) ^ 2.0) * Float64(t * (k ^ 2.0))))));
	else
		tmp = Float64(2.0 / Float64(sin(k) * Float64(Float64(sin(k) * t) * Float64((k ^ 2.0) / Float64(cos(k) * (l ^ 2.0))))));
	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)
	tmp = 0.0;
	if (l <= -1e-134)
		tmp = 2.0 * (cos(k) * ((l ^ 2.0) / ((sin(k) ^ 2.0) * (t * (k ^ 2.0)))));
	else
		tmp = 2.0 / (sin(k) * ((sin(k) * t) * ((k ^ 2.0) / (cos(k) * (l ^ 2.0)))));
	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_] := If[LessEqual[l, -1e-134], N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(N[Power[l, 2.0], $MachinePrecision] / N[(N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision] * N[(t * N[Power[k, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[Sin[k], $MachinePrecision] * N[(N[(N[Sin[k], $MachinePrecision] * t), $MachinePrecision] * N[(N[Power[k, 2.0], $MachinePrecision] / N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < -1.00000000000000004e-134

   1. Initial program 49.0

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

   2. Simplified 42.2

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

      [Start] 49.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational_best-simplify-2 [=>] 49.0	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best-simplify-44 [=>] 49.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \tan k\right)}} \]
      rational_best-simplify-2 [=>] 49.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      rational_best-simplify-19 [=>] 49.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)}\right)} \]
      rational_best-simplify-1 [=>] 49.0	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(-1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]
      rational_best-simplify-43 [=>] 42.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + -1\right)\right)}\right)} \]
      metadata-eval [=>] 42.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]
      rational_best-simplify-3 [=>] 42.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   3. Taylor expanded in t around 0 26.0

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

   4. Simplified 27.2

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

      [Start] 26.0	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      rational_best-simplify-2 [=>] 26.0	\[ \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}{\cos k \cdot {\ell}^{2}}} \]
      rational_best-simplify-44 [=>] 27.2	\[ \frac{2}{\frac{\color{blue}{t \cdot \left({k}^{2} \cdot {\sin k}^{2}\right)}}{\cos k \cdot {\ell}^{2}}} \]
      rational_best-simplify-2 [=>] 27.2	\[ \frac{2}{\frac{\color{blue}{\left({k}^{2} \cdot {\sin k}^{2}\right) \cdot t}}{\cos k \cdot {\ell}^{2}}} \]
      rational_best-simplify-47 [=>] 27.2	\[ \frac{2}{\color{blue}{t \cdot \frac{{k}^{2} \cdot {\sin k}^{2}}{\cos k \cdot {\ell}^{2}}}} \]
      exponential-simplify-27 [=>] 27.2	\[ \frac{2}{t \cdot \frac{\color{blue}{{\left(\sin k \cdot k\right)}^{2}}}{\cos k \cdot {\ell}^{2}}} \]
      rational_best-simplify-2 [=>] 27.2	\[ \frac{2}{t \cdot \frac{{\color{blue}{\left(k \cdot \sin k\right)}}^{2}}{\cos k \cdot {\ell}^{2}}} \]

   5. Taylor expanded in t around 0 25.8

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

   6. Simplified 27.1

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

      [Start] 25.8	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      rational_best-simplify-47 [=>] 25.9	\[ 2 \cdot \color{blue}{\left({\ell}^{2} \cdot \frac{\cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right)} \]
      rational_best-simplify-2 [=>] 25.9	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot {\ell}^{2}\right)} \]
      rational_best-simplify-44 [=>] 26.0	\[ \color{blue}{\frac{\cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot \left(2 \cdot {\ell}^{2}\right)} \]
      rational_best-simplify-2 [=>] 26.0	\[ \frac{\cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \cdot \left(2 \cdot {\ell}^{2}\right) \]
      rational_best-simplify-44 [=>] 27.1	\[ \frac{\cos k}{\color{blue}{t \cdot \left({k}^{2} \cdot {\sin k}^{2}\right)}} \cdot \left(2 \cdot {\ell}^{2}\right) \]
      exponential-simplify-27 [=>] 27.1	\[ \frac{\cos k}{t \cdot \color{blue}{{\left(\sin k \cdot k\right)}^{2}}} \cdot \left(2 \cdot {\ell}^{2}\right) \]
      rational_best-simplify-2 [<=] 27.1	\[ \frac{\cos k}{t \cdot {\color{blue}{\left(k \cdot \sin k\right)}}^{2}} \cdot \left(2 \cdot {\ell}^{2}\right) \]

   7. Taylor expanded in k around inf 25.8

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

   8. Simplified 25.8

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

      [Start] 25.8	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      rational_best-simplify-2 [=>] 25.8	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      rational_best-simplify-47 [=>] 25.8	\[ 2 \cdot \color{blue}{\left(\cos k \cdot \frac{{\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}\right)} \]
      rational_best-simplify-44 [=>] 25.8	\[ 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}}\right) \]
      rational_best-simplify-2 [=>] 25.8	\[ 2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \color{blue}{\left(t \cdot {k}^{2}\right)}}\right) \]

   if -1.00000000000000004e-134 < l

   1. Initial program 46.3

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

   2. Simplified 38.5

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

      [Start] 46.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      rational_best-simplify-2 [=>] 46.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      rational_best-simplify-44 [=>] 46.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \tan k\right)}} \]
      rational_best-simplify-2 [=>] 46.3	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \tan k\right) \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}} \]
      rational_best-simplify-44 [=>] 46.3	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \tan k\right) \cdot \sin k\right)}} \]
      rational_best-simplify-2 [<=] 46.3	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \tan k\right)\right)}} \]
      rational_best-simplify-2 [=>] 46.3	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}\right)} \]
      rational_best-simplify-19 [=>] 46.3	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + -1\right)}\right)\right)} \]
      rational_best-simplify-1 [=>] 46.3	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left(-1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)\right)} \]
      rational_best-simplify-43 [=>] 38.5	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + -1\right)\right)}\right)\right)} \]
      metadata-eval [=>] 38.5	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)\right)} \]
      rational_best-simplify-3 [=>] 38.5	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)\right)} \]

   3. Applied egg-rr 38.1

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

   4. Simplified 38.1

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

      [Start] 38.1	\[ \frac{2}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{{\left(\frac{k}{t}\right)}^{2}}{\ell \cdot \ell}\right)\right) + 0} \]
      rational_best-simplify-3 [=>] 38.1	\[ \frac{2}{\color{blue}{{t}^{3} \cdot \left(\sin k \cdot \left(\tan k \cdot \frac{{\left(\frac{k}{t}\right)}^{2}}{\ell \cdot \ell}\right)\right)}} \]
      rational_best-simplify-44 [=>] 38.1	\[ \frac{2}{\color{blue}{\sin k \cdot \left({t}^{3} \cdot \left(\tan k \cdot \frac{{\left(\frac{k}{t}\right)}^{2}}{\ell \cdot \ell}\right)\right)}} \]

   5. Taylor expanded in t around 0 20.6

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

   6. Simplified 20.4

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

      [Start] 20.6	\[ \frac{2}{\sin k \cdot \frac{{k}^{2} \cdot \left(\sin k \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      rational_best-simplify-47 [=>] 20.4	\[ \frac{2}{\sin k \cdot \color{blue}{\left(\left(\sin k \cdot t\right) \cdot \frac{{k}^{2}}{\cos k \cdot {\ell}^{2}}\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 22.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq -1 \cdot 10^{-134}:\\ \;\;\;\;2 \cdot \left(\cos k \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot \left(t \cdot {k}^{2}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{{k}^{2}}{\cos k \cdot {\ell}^{2}}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	21.9
Cost	32960

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

Alternative 2
Error	22.5
Cost	32896

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

Alternative 3
Error	22.0
Cost	26696

\[\begin{array}{l} t_1 := 2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {\left(\sin k \cdot k\right)}^{2}}\right)\\ \mathbf{if}\;k \leq -700:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.9
Cost	26696

\[\begin{array}{l} \mathbf{if}\;k \leq -700:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {\left(\sin k \cdot k\right)}^{2}}\right)\\ \mathbf{elif}\;k \leq 10^{-73}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{2}}{t \cdot {\left(k \cdot \sin k\right)}^{2}} \cdot \left(2 \cdot \cos k\right)\\ \end{array} \]

Alternative 5
Error	22.0
Cost	26696

\[\begin{array}{l} \mathbf{if}\;k \leq -700:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot {\left(\sin k \cdot k\right)}^{2}}\right)\\ \mathbf{elif}\;k \leq 4 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{{k}^{2} \cdot \left(t \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]

Alternative 6
Error	21.8
Cost	26696

\[\begin{array}{l} t_1 := {\left(\sin k \cdot k\right)}^{2}\\ \mathbf{if}\;k \leq -700:\\ \;\;\;\;2 \cdot \left({\ell}^{2} \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{\sin k \cdot \left(\left(\sin k \cdot t\right) \cdot \frac{{k}^{2}}{{\ell}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \frac{t}{\cos k \cdot {\ell}^{2}}}\\ \end{array} \]

Alternative 7
Error	29.3
Cost	19968

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

Alternative 8
Error	30.4
Cost	19904

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

Alternative 9
Error	30.3
Cost	19904

\[\frac{2}{\sin k \cdot \frac{{k}^{3} \cdot t}{{\ell}^{2}}} \]

Alternative 10
Error	31.1
Cost	13376

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

Alternative 11
Error	31.1
Cost	13376

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

Reproduce

herbie shell --seed 2023096 
(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))))