Toniolo and Linder, Equation (10-)

Average Accuracy: 25.6% → 98.7%

Time: 33.9s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1.65 \cdot 10^{-96} \lor \neg \left(k \leq 2.9 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

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 (* k (/ k l))))
   (if (or (<= k -1.65e-96) (not (<= k 2.9e-108)))
     (/ 2.0 (/ (* (/ k l) t) (* (/ l (pow (sin k) 2.0)) (/ (cos k) k))))
     (* 2.0 (/ 1.0 (* t_1 (* t 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 = k * (k / l);
	double tmp;
	if ((k <= -1.65e-96) || !(k <= 2.9e-108)) {
		tmp = 2.0 / (((k / l) * t) / ((l / pow(sin(k), 2.0)) * (cos(k) / k)));
	} else {
		tmp = 2.0 * (1.0 / (t_1 * (t * 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 * (k / l)
    if ((k <= (-1.65d-96)) .or. (.not. (k <= 2.9d-108))) then
        tmp = 2.0d0 / (((k / l) * t) / ((l / (sin(k) ** 2.0d0)) * (cos(k) / k)))
    else
        tmp = 2.0d0 * (1.0d0 / (t_1 * (t * 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 = k * (k / l);
	double tmp;
	if ((k <= -1.65e-96) || !(k <= 2.9e-108)) {
		tmp = 2.0 / (((k / l) * t) / ((l / Math.pow(Math.sin(k), 2.0)) * (Math.cos(k) / k)));
	} else {
		tmp = 2.0 * (1.0 / (t_1 * (t * 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 = k * (k / l)
	tmp = 0
	if (k <= -1.65e-96) or not (k <= 2.9e-108):
		tmp = 2.0 / (((k / l) * t) / ((l / math.pow(math.sin(k), 2.0)) * (math.cos(k) / k)))
	else:
		tmp = 2.0 * (1.0 / (t_1 * (t * 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(k * Float64(k / l))
	tmp = 0.0
	if ((k <= -1.65e-96) || !(k <= 2.9e-108))
		tmp = Float64(2.0 / Float64(Float64(Float64(k / l) * t) / Float64(Float64(l / (sin(k) ^ 2.0)) * Float64(cos(k) / k))));
	else
		tmp = Float64(2.0 * Float64(1.0 / Float64(t_1 * Float64(t * 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 * (k / l);
	tmp = 0.0;
	if ((k <= -1.65e-96) || ~((k <= 2.9e-108)))
		tmp = 2.0 / (((k / l) * t) / ((l / (sin(k) ^ 2.0)) * (cos(k) / k)));
	else
		tmp = 2.0 * (1.0 / (t_1 * (t * 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[(k * N[(k / l), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, -1.65e-96], N[Not[LessEqual[k, 2.9e-108]], $MachinePrecision]], N[(2.0 / N[(N[(N[(k / l), $MachinePrecision] * t), $MachinePrecision] / N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(1.0 / N[(t$95$1 * N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < -1.64999999999999995e-96 or 2.9000000000000001e-108 < k

   1. Initial program 27.5%

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

   2. Simplified 40.8%

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

      [Start] 27.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 27.5	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 27.5	\[ \frac{2}{\color{blue}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      +-commutative [=>] 27.5	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1\right)\right)} \]
      associate--l+ [=>] 40.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)\right)}\right)} \]
      metadata-eval [=>] 40.8	\[ \frac{2}{\tan k \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]

   3. Taylor expanded in k around inf 67.9%

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

   4. Simplified 77.2%

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

      [Start] 67.9	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 68.5	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 68.5	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      associate-/l* [=>] 68.5	\[ \frac{2}{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 68.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 68.5	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 77.2	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 93.3%

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

      [Start] 77.2	\[ \frac{2}{\frac{k}{\frac{\cos k}{k}} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)} \]
      associate-*r* [=>] 80.9	\[ \frac{2}{\color{blue}{\left(\frac{k}{\frac{\cos k}{k}} \cdot \frac{t}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\ell}}} \]
      clear-num [=>] 80.9	\[ \frac{2}{\left(\frac{k}{\frac{\cos k}{k}} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{{\sin k}^{2}}}}} \]
      un-div-inv [=>] 80.9	\[ \frac{2}{\color{blue}{\frac{\frac{k}{\frac{\cos k}{k}} \cdot \frac{t}{\ell}}{\frac{\ell}{{\sin k}^{2}}}}} \]
      associate-*l/ [=>] 87.7	\[ \frac{2}{\frac{\color{blue}{\frac{k \cdot \frac{t}{\ell}}{\frac{\cos k}{k}}}}{\frac{\ell}{{\sin k}^{2}}}} \]
      associate-/l/ [=>] 93.3	\[ \frac{2}{\color{blue}{\frac{k \cdot \frac{t}{\ell}}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}}} \]

   6. Taylor expanded in k around 0 92.0%

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

   7. Simplified 98.6%

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

      [Start] 92.0	\[ \frac{2}{\frac{\frac{k \cdot t}{\ell}}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}} \]
      associate-*l/ [<=] 98.6	\[ \frac{2}{\frac{\color{blue}{\frac{k}{\ell} \cdot t}}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}} \]

   if -1.64999999999999995e-96 < k < 2.9000000000000001e-108

   1. Initial program 0.0%

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

   2. Simplified 10.0%

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

      [Start] 0.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)\right)}} \]
      associate-/r* [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 0.0	\[ \frac{\frac{2}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-/r/ [=>] 0.0	\[ \frac{\color{blue}{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}}{\sin k \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} \]
      associate-*r* [=>] 0.0	\[ \frac{\frac{2}{\frac{{t}^{3}}{\ell}} \cdot \ell}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\sin k \cdot \tan k} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      associate-/r* [<=] 0.0	\[ \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell} \cdot \left(\sin k \cdot \tan k\right)}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      *-commutative [=>] 0.0	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \tan k\right) \cdot \frac{{t}^{3}}{\ell}}} \cdot \frac{\ell}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]

   3. Taylor expanded in k around 0 0.0%

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

   4. Simplified 0.0%

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

      [Start] 0.0	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 0.0	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      associate-/l* [=>] 0.0	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}}} \]

   5. Applied egg-rr 45.8%

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

      [Start] 0.0	\[ 2 \cdot \frac{\ell}{\frac{{k}^{4} \cdot t}{\ell}} \]
      associate-/l* [=>] 0.0	\[ 2 \cdot \frac{\ell}{\color{blue}{\frac{{k}^{4}}{\frac{\ell}{t}}}} \]
      sqr-pow [=>] 0.0	\[ 2 \cdot \frac{\ell}{\frac{\color{blue}{{k}^{\left(\frac{4}{2}\right)} \cdot {k}^{\left(\frac{4}{2}\right)}}}{\frac{\ell}{t}}} \]
      associate-/l* [=>] 20.2	\[ 2 \cdot \frac{\ell}{\color{blue}{\frac{{k}^{\left(\frac{4}{2}\right)}}{\frac{\frac{\ell}{t}}{{k}^{\left(\frac{4}{2}\right)}}}}} \]
      associate-/r/ [=>] 39.3	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{{k}^{\left(\frac{4}{2}\right)}} \cdot \frac{\frac{\ell}{t}}{{k}^{\left(\frac{4}{2}\right)}}\right)} \]
      metadata-eval [=>] 39.3	\[ 2 \cdot \left(\frac{\ell}{{k}^{\color{blue}{2}}} \cdot \frac{\frac{\ell}{t}}{{k}^{\left(\frac{4}{2}\right)}}\right) \]
      unpow2 [=>] 39.3	\[ 2 \cdot \left(\frac{\ell}{\color{blue}{k \cdot k}} \cdot \frac{\frac{\ell}{t}}{{k}^{\left(\frac{4}{2}\right)}}\right) \]
      associate-/l/ [=>] 45.8	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \color{blue}{\frac{\ell}{{k}^{\left(\frac{4}{2}\right)} \cdot t}}\right) \]
      *-commutative [=>] 45.8	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{\color{blue}{t \cdot {k}^{\left(\frac{4}{2}\right)}}}\right) \]
      metadata-eval [=>] 45.8	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot {k}^{\color{blue}{2}}}\right) \]
      unpow2 [=>] 45.8	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \color{blue}{\left(k \cdot k\right)}}\right) \]

   6. Applied egg-rr 98.8%

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

      [Start] 45.8	\[ 2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{\ell}{t \cdot \left(k \cdot k\right)}\right) \]
      associate-/r/ [<=] 38.1	\[ 2 \cdot \color{blue}{\frac{\ell}{\frac{k \cdot k}{\frac{\ell}{t \cdot \left(k \cdot k\right)}}}} \]
      associate-/r/ [=>] 38.0	\[ 2 \cdot \frac{\ell}{\color{blue}{\frac{k \cdot k}{\ell} \cdot \left(t \cdot \left(k \cdot k\right)\right)}} \]
      associate-/l/ [<=] 45.8	\[ 2 \cdot \color{blue}{\frac{\frac{\ell}{t \cdot \left(k \cdot k\right)}}{\frac{k \cdot k}{\ell}}} \]
      clear-num [=>] 45.8	\[ 2 \cdot \frac{\color{blue}{\frac{1}{\frac{t \cdot \left(k \cdot k\right)}{\ell}}}}{\frac{k \cdot k}{\ell}} \]
      clear-num [=>] 45.8	\[ 2 \cdot \frac{\color{blue}{\frac{1}{\frac{\frac{t \cdot \left(k \cdot k\right)}{\ell}}{1}}}}{\frac{k \cdot k}{\ell}} \]
      associate-/l/ [=>] 45.7	\[ 2 \cdot \color{blue}{\frac{1}{\frac{k \cdot k}{\ell} \cdot \frac{\frac{t \cdot \left(k \cdot k\right)}{\ell}}{1}}} \]
      associate-/l* [=>] 45.8	\[ 2 \cdot \frac{1}{\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot \frac{\frac{t \cdot \left(k \cdot k\right)}{\ell}}{1}} \]
      associate-/r/ [=>] 45.8	\[ 2 \cdot \frac{1}{\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot \frac{\frac{t \cdot \left(k \cdot k\right)}{\ell}}{1}} \]
      *-commutative [=>] 45.8	\[ 2 \cdot \frac{1}{\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{\frac{\color{blue}{\left(k \cdot k\right) \cdot t}}{\ell}}{1}} \]
      associate-/l* [=>] 39.4	\[ 2 \cdot \frac{1}{\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{\color{blue}{\frac{k \cdot k}{\frac{\ell}{t}}}}{1}} \]
      associate-/l/ [=>] 39.4	\[ 2 \cdot \frac{1}{\left(\frac{k}{\ell} \cdot k\right) \cdot \color{blue}{\frac{k \cdot k}{1 \cdot \frac{\ell}{t}}}} \]
      *-un-lft-identity [<=] 39.4	\[ 2 \cdot \frac{1}{\left(\frac{k}{\ell} \cdot k\right) \cdot \frac{k \cdot k}{\color{blue}{\frac{\ell}{t}}}} \]
      associate-/r/ [=>] 58.1	\[ 2 \cdot \frac{1}{\left(\frac{k}{\ell} \cdot k\right) \cdot \color{blue}{\left(\frac{k \cdot k}{\ell} \cdot t\right)}} \]
      associate-/l* [=>] 99.0	\[ 2 \cdot \frac{1}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\color{blue}{\frac{k}{\frac{\ell}{k}}} \cdot t\right)} \]
      associate-/r/ [=>] 98.8	\[ 2 \cdot \frac{1}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(\color{blue}{\left(\frac{k}{\ell} \cdot k\right)} \cdot t\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.65 \cdot 10^{-96} \lor \neg \left(k \leq 2.9 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{\ell}{{\sin k}^{2}} \cdot \frac{\cos k}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(t \cdot \left(k \cdot \frac{k}{\ell}\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.3%
Cost	14416

\[\begin{array}{l} t_1 := \sin k \cdot \tan k\\ t_2 := \frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)}{t_1}\\ t_3 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -8 \cdot 10^{-15}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 3.4 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{1}{t_3 \cdot \left(t \cdot t_3\right)}\\ \mathbf{elif}\;k \leq 8 \cdot 10^{+204}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+272}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(\frac{\ell}{\frac{k}{\ell}} \cdot \frac{1}{k \cdot t}\right)}{t_1}\\ \end{array} \]

Alternative 2
Accuracy	98.2%
Cost	14409

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1050 \lor \neg \left(k \leq 8.5 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{\cos k}{k} \cdot \frac{\ell}{0.5 - \frac{\cos \left(k + k\right)}{2}}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 3
Accuracy	92.2%
Cost	14289

\[\begin{array}{l} t_1 := \frac{2 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)}{\sin k \cdot \tan k}\\ t_2 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3.7 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-25}:\\ \;\;\;\;2 \cdot \frac{1}{t_2 \cdot \left(t \cdot t_2\right)}\\ \mathbf{elif}\;k \leq 7.3 \cdot 10^{+204} \lor \neg \left(k \leq 2.8 \cdot 10^{+272}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	92.4%
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -1050 \lor \neg \left(k \leq 1.2 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\frac{2}{\tan k}}{k} \cdot \frac{\frac{\ell}{\sin k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 5
Accuracy	94.0%
Cost	14025

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -6.8 \cdot 10^{-112} \lor \neg \left(k \leq 2 \cdot 10^{-30}\right):\\ \;\;\;\;\frac{2}{\tan k \cdot \left(\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 6
Accuracy	68.1%
Cost	8009

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;k \leq -3.55 \cdot 10^{-97} \lor \neg \left(k \leq 3.8 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{2}{\frac{\frac{k}{\ell} \cdot t}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 7
Accuracy	65.2%
Cost	8008

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -8.5 \cdot 10^{+80}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t_1 \cdot \left(k \cdot t\right)}\\ \mathbf{elif}\;t \leq 6.3 \cdot 10^{+150}:\\ \;\;\;\;\frac{2}{\frac{k \cdot \frac{t}{\ell}}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{k \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 8
Accuracy	63.9%
Cost	1088

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ 2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)} \end{array} \]

Alternative 9
Accuracy	59.0%
Cost	960

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

Alternative 10
Accuracy	61.1%
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ 2 \cdot \left(t_1 \cdot \frac{t_1}{t}\right) \end{array} \]

Alternative 11
Accuracy	63.1%
Cost	960

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

Alternative 12
Accuracy	62.6%
Cost	960

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

Reproduce

herbie shell --seed 2023139 
(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))))