Toniolo and Linder, Equation (10-)

Average Accuracy: 16.5% → 63.5%

Time: 25.0s

Precision: binary64

Cost: 14152

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\sin k}{\ell}\\ t_2 := \frac{\frac{\ell}{k}}{t}\\ \mathbf{if}\;k \leq 5 \cdot 10^{-260}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \frac{1}{\left(k \cdot t_1\right) \cdot \tan k}\right)\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\ell}{t_1 \cdot \tan k}}{k \cdot t}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(\frac{\ell}{k} \cdot \frac{1}{\sin k \cdot \tan k}\right)\right)\\ \end{array} \]

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 (/ (sin k) l)) (t_2 (/ (/ l k) t)))
   (if (<= k 5e-260)
     (* 2.0 (* t_2 (/ 1.0 (* (* k t_1) (tan k)))))
     (if (<= k 1.25e-149)
       (/ (/ (* 2.0 (/ l (* t_1 (tan k)))) (* k t)) k)
       (* 2.0 (* t_2 (* (/ l k) (/ 1.0 (* (sin k) (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 = sin(k) / l;
	double t_2 = (l / k) / t;
	double tmp;
	if (k <= 5e-260) {
		tmp = 2.0 * (t_2 * (1.0 / ((k * t_1) * tan(k))));
	} else if (k <= 1.25e-149) {
		tmp = ((2.0 * (l / (t_1 * tan(k)))) / (k * t)) / k;
	} else {
		tmp = 2.0 * (t_2 * ((l / k) * (1.0 / (sin(k) * 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) :: t_2
    real(8) :: tmp
    t_1 = sin(k) / l
    t_2 = (l / k) / t
    if (k <= 5d-260) then
        tmp = 2.0d0 * (t_2 * (1.0d0 / ((k * t_1) * tan(k))))
    else if (k <= 1.25d-149) then
        tmp = ((2.0d0 * (l / (t_1 * tan(k)))) / (k * t)) / k
    else
        tmp = 2.0d0 * (t_2 * ((l / k) * (1.0d0 / (sin(k) * 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.sin(k) / l;
	double t_2 = (l / k) / t;
	double tmp;
	if (k <= 5e-260) {
		tmp = 2.0 * (t_2 * (1.0 / ((k * t_1) * Math.tan(k))));
	} else if (k <= 1.25e-149) {
		tmp = ((2.0 * (l / (t_1 * Math.tan(k)))) / (k * t)) / k;
	} else {
		tmp = 2.0 * (t_2 * ((l / k) * (1.0 / (Math.sin(k) * 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.sin(k) / l
	t_2 = (l / k) / t
	tmp = 0
	if k <= 5e-260:
		tmp = 2.0 * (t_2 * (1.0 / ((k * t_1) * math.tan(k))))
	elif k <= 1.25e-149:
		tmp = ((2.0 * (l / (t_1 * math.tan(k)))) / (k * t)) / k
	else:
		tmp = 2.0 * (t_2 * ((l / k) * (1.0 / (math.sin(k) * 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(sin(k) / l)
	t_2 = Float64(Float64(l / k) / t)
	tmp = 0.0
	if (k <= 5e-260)
		tmp = Float64(2.0 * Float64(t_2 * Float64(1.0 / Float64(Float64(k * t_1) * tan(k)))));
	elseif (k <= 1.25e-149)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / Float64(t_1 * tan(k)))) / Float64(k * t)) / k);
	else
		tmp = Float64(2.0 * Float64(t_2 * Float64(Float64(l / k) * Float64(1.0 / Float64(sin(k) * 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 = sin(k) / l;
	t_2 = (l / k) / t;
	tmp = 0.0;
	if (k <= 5e-260)
		tmp = 2.0 * (t_2 * (1.0 / ((k * t_1) * tan(k))));
	elseif (k <= 1.25e-149)
		tmp = ((2.0 * (l / (t_1 * tan(k)))) / (k * t)) / k;
	else
		tmp = 2.0 * (t_2 * ((l / k) * (1.0 / (sin(k) * 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[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]}, Block[{t$95$2 = N[(N[(l / k), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[k, 5e-260], N[(2.0 * N[(t$95$2 * N[(1.0 / N[(N[(k * t$95$1), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.25e-149], N[(N[(N[(2.0 * N[(l / N[(t$95$1 * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], N[(2.0 * N[(t$95$2 * N[(N[(l / k), $MachinePrecision] * N[(1.0 / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if k < 5.0000000000000003e-260

   1. Initial program 17.0%

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

   2. Simplified 26.2%

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

      [Start] 17.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 17.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 17.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 17.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 16.8	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 16.8	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 16.8	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 16.8	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 25.7	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 25.7	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 25.7	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 26.2	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 42.8%

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

   4. Simplified 42.0%

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

      [Start] 42.8	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*r/ [=>] 42.8	\[ \color{blue}{\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
      *-commutative [=>] 42.8	\[ \frac{2 \cdot \color{blue}{\left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*r* [=>] 42.8	\[ \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 42.8	\[ \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 42.8	\[ \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}} \]
      *-commutative [=>] 42.8	\[ \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(t \cdot {\sin k}^{2}\right)} \cdot {k}^{2}} \]
      associate-*l* [=>] 42.0	\[ \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}} \]
      unpow2 [=>] 42.0	\[ \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{t \cdot \left({\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Taylor expanded in l around 0 42.8%

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

   6. Simplified 56.4%

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

      [Start] 42.8	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/r* [=>] 43.7	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]
      *-commutative [=>] 43.7	\[ 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 43.7	\[ 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      associate-*r* [<=] 43.7	\[ 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 43.7	\[ 2 \cdot \frac{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]
      times-frac [=>] 56.4	\[ 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \cos k}{k}}}{{\sin k}^{2} \cdot t} \]
      *-commutative [=>] 56.4	\[ 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\cos k \cdot \ell}}{k}}{{\sin k}^{2} \cdot t} \]

   7. Applied egg-rr 61.8%

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

      [Start] 56.4	\[ 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k \cdot \ell}{k}}{{\sin k}^{2} \cdot t} \]
      *-commutative [=>] 56.4	\[ 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k \cdot \ell}{k}}{\color{blue}{t \cdot {\sin k}^{2}}} \]
      times-frac [=>] 61.8	\[ 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{\cos k \cdot \ell}{k}}{{\sin k}^{2}}\right)} \]
      associate-/l* [=>] 61.8	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{k}{\ell}}}}{{\sin k}^{2}}\right) \]
      div-inv [=>] 61.8	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\color{blue}{\cos k \cdot \frac{1}{\frac{k}{\ell}}}}{{\sin k}^{2}}\right) \]
      clear-num [<=] 61.8	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\cos k \cdot \color{blue}{\frac{\ell}{k}}}{{\sin k}^{2}}\right) \]

   8. Applied egg-rr 62.3%

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

      [Start] 61.8	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\cos k \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right) \]
      clear-num [=>] 61.8	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \color{blue}{\frac{1}{\frac{{\sin k}^{2}}{\cos k \cdot \frac{\ell}{k}}}}\right) \]
      inv-pow [=>] 61.8	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \color{blue}{{\left(\frac{{\sin k}^{2}}{\cos k \cdot \frac{\ell}{k}}\right)}^{-1}}\right) \]
      unpow2 [=>] 61.8	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\left(\frac{\color{blue}{\sin k \cdot \sin k}}{\cos k \cdot \frac{\ell}{k}}\right)}^{-1}\right) \]
      *-commutative [=>] 61.8	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\left(\frac{\sin k \cdot \sin k}{\color{blue}{\frac{\ell}{k} \cdot \cos k}}\right)}^{-1}\right) \]
      times-frac [=>] 62.2	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\color{blue}{\left(\frac{\sin k}{\frac{\ell}{k}} \cdot \frac{\sin k}{\cos k}\right)}}^{-1}\right) \]
      quot-tan [=>] 62.3	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\left(\frac{\sin k}{\frac{\ell}{k}} \cdot \color{blue}{\tan k}\right)}^{-1}\right) \]

   9. Simplified 62.3%

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

      [Start] 62.3	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\left(\frac{\sin k}{\frac{\ell}{k}} \cdot \tan k\right)}^{-1}\right) \]
      unpow-1 [=>] 62.3	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \color{blue}{\frac{1}{\frac{\sin k}{\frac{\ell}{k}} \cdot \tan k}}\right) \]
      associate-/r/ [=>] 62.3	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{1}{\color{blue}{\left(\frac{\sin k}{\ell} \cdot k\right)} \cdot \tan k}\right) \]

   if 5.0000000000000003e-260 < k < 1.24999999999999992e-149

   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 0.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]
      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	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 0.0	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 1.6%

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

   4. Simplified 9.0%

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

      [Start] 1.6	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 1.6	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l* [=>] 9.0	\[ \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 10.8%

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

      [Start] 9.0	\[ \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l/ [=>] 9.2	\[ \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
      *-commutative [=>] 9.2	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{\left(k \cdot t\right) \cdot k}} \]
      associate-/r* [=>] 10.8	\[ \color{blue}{\frac{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot t}}{k}} \]
      clear-num [=>] 10.8	\[ \frac{\frac{2 \cdot \left(\color{blue}{\frac{1}{\frac{\sin k}{\ell}}} \cdot \frac{\ell}{\tan k}\right)}{k \cdot t}}{k} \]
      frac-times [=>] 10.8	\[ \frac{\frac{2 \cdot \color{blue}{\frac{1 \cdot \ell}{\frac{\sin k}{\ell} \cdot \tan k}}}{k \cdot t}}{k} \]
      *-un-lft-identity [<=] 10.8	\[ \frac{\frac{2 \cdot \frac{\color{blue}{\ell}}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot t}}{k} \]

   if 1.24999999999999992e-149 < k

   1. Initial program 19.8%

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

   2. Simplified 30.1%

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

      [Start] 19.8	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 19.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 19.9	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 19.9	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 19.6	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 19.6	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 19.8	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 19.8	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 29.2	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 29.2	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 29.2	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 30.1	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 51.4%

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

   4. Simplified 49.9%

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

      [Start] 51.4	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*r/ [=>] 51.4	\[ \color{blue}{\frac{2 \cdot \left(\cos k \cdot {\ell}^{2}\right)}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
      *-commutative [=>] 51.4	\[ \frac{2 \cdot \color{blue}{\left({\ell}^{2} \cdot \cos k\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*r* [=>] 51.4	\[ \frac{\color{blue}{\left(2 \cdot {\ell}^{2}\right) \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 51.4	\[ \frac{\left(2 \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot \cos k}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 51.4	\[ \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}} \]
      *-commutative [=>] 51.4	\[ \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{\left(t \cdot {\sin k}^{2}\right)} \cdot {k}^{2}} \]
      associate-*l* [=>] 49.9	\[ \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{\color{blue}{t \cdot \left({\sin k}^{2} \cdot {k}^{2}\right)}} \]
      unpow2 [=>] 49.9	\[ \frac{\left(2 \cdot \left(\ell \cdot \ell\right)\right) \cdot \cos k}{t \cdot \left({\sin k}^{2} \cdot \color{blue}{\left(k \cdot k\right)}\right)} \]

   5. Taylor expanded in l around 0 51.4%

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

   6. Simplified 68.3%

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

      [Start] 51.4	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/r* [=>] 52.7	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]
      *-commutative [=>] 52.7	\[ 2 \cdot \frac{\frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 52.7	\[ 2 \cdot \frac{\frac{\color{blue}{\left(\ell \cdot \ell\right)} \cdot \cos k}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      associate-*r* [<=] 52.7	\[ 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \left(\ell \cdot \cos k\right)}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 52.7	\[ 2 \cdot \frac{\frac{\ell \cdot \left(\ell \cdot \cos k\right)}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]
      times-frac [=>] 68.3	\[ 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell \cdot \cos k}{k}}}{{\sin k}^{2} \cdot t} \]
      *-commutative [=>] 68.3	\[ 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\color{blue}{\cos k \cdot \ell}}{k}}{{\sin k}^{2} \cdot t} \]

   7. Applied egg-rr 77.0%

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

      [Start] 68.3	\[ 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k \cdot \ell}{k}}{{\sin k}^{2} \cdot t} \]
      *-commutative [=>] 68.3	\[ 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k \cdot \ell}{k}}{\color{blue}{t \cdot {\sin k}^{2}}} \]
      times-frac [=>] 77.1	\[ 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\frac{\cos k \cdot \ell}{k}}{{\sin k}^{2}}\right)} \]
      associate-/l* [=>] 77.1	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\color{blue}{\frac{\cos k}{\frac{k}{\ell}}}}{{\sin k}^{2}}\right) \]
      div-inv [=>] 77.1	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\color{blue}{\cos k \cdot \frac{1}{\frac{k}{\ell}}}}{{\sin k}^{2}}\right) \]
      clear-num [<=] 77.0	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\cos k \cdot \color{blue}{\frac{\ell}{k}}}{{\sin k}^{2}}\right) \]

   8. Applied egg-rr 77.1%

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

      [Start] 77.0	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{\cos k \cdot \frac{\ell}{k}}{{\sin k}^{2}}\right) \]
      clear-num [=>] 77.0	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \color{blue}{\frac{1}{\frac{{\sin k}^{2}}{\cos k \cdot \frac{\ell}{k}}}}\right) \]
      inv-pow [=>] 77.0	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \color{blue}{{\left(\frac{{\sin k}^{2}}{\cos k \cdot \frac{\ell}{k}}\right)}^{-1}}\right) \]
      unpow2 [=>] 77.0	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\left(\frac{\color{blue}{\sin k \cdot \sin k}}{\cos k \cdot \frac{\ell}{k}}\right)}^{-1}\right) \]
      *-commutative [=>] 77.0	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\left(\frac{\sin k \cdot \sin k}{\color{blue}{\frac{\ell}{k} \cdot \cos k}}\right)}^{-1}\right) \]
      times-frac [=>] 77.0	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\color{blue}{\left(\frac{\sin k}{\frac{\ell}{k}} \cdot \frac{\sin k}{\cos k}\right)}}^{-1}\right) \]
      quot-tan [=>] 77.1	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\left(\frac{\sin k}{\frac{\ell}{k}} \cdot \color{blue}{\tan k}\right)}^{-1}\right) \]

   9. Simplified 77.1%

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

      [Start] 77.1	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot {\left(\frac{\sin k}{\frac{\ell}{k}} \cdot \tan k\right)}^{-1}\right) \]
      unpow-1 [=>] 77.1	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \color{blue}{\frac{1}{\frac{\sin k}{\frac{\ell}{k}} \cdot \tan k}}\right) \]
      associate-*l/ [=>] 77.0	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{1}{\color{blue}{\frac{\sin k \cdot \tan k}{\frac{\ell}{k}}}}\right) \]
      associate-/r/ [=>] 77.1	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \color{blue}{\left(\frac{1}{\sin k \cdot \tan k} \cdot \frac{\ell}{k}\right)}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-260}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \frac{1}{\left(k \cdot \frac{\sin k}{\ell}\right) \cdot \tan k}\right)\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot t}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{\sin k \cdot \tan k}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	63.5%
Cost	14152

\[\begin{array}{l} \mathbf{if}\;k \leq 5 \cdot 10^{-260}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t \cdot \left(\tan k \cdot \frac{\sin k}{\frac{\ell}{k}}\right)}\\ \mathbf{elif}\;k \leq 1.25 \cdot 10^{-149}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot t}}{k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \left(\frac{\ell}{k} \cdot \frac{1}{\sin k \cdot \tan k}\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	60.1%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.16 \cdot 10^{-83} \lor \neg \left(k \leq 3.7 \cdot 10^{-66}\right):\\ \;\;\;\;2 \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k \cdot t}}{\sin k \cdot \tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}}{k}\\ \end{array} \]

Alternative 3
Accuracy	64.1%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -1.2 \cdot 10^{-83} \lor \neg \left(k \leq 1.85 \cdot 10^{-131}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t \cdot \left(\tan k \cdot \frac{\sin k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}}{k}\\ \end{array} \]

Alternative 4
Accuracy	64.1%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -2.45 \cdot 10^{-83} \lor \neg \left(k \leq 1.4 \cdot 10^{-129}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{t \cdot \left(\tan k \cdot \frac{\sin k}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot t}}{k}\\ \end{array} \]

Alternative 5
Accuracy	41.5%
Cost	8196

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-321}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\left(k \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{t} \cdot \left(\left(\frac{\ell}{k} \cdot -0.5 + \frac{\ell}{{k}^{3}}\right) - \frac{\ell}{k} \cdot -0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	38.9%
Cost	960

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

Alternative 7
Accuracy	39.1%
Cost	960

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

Alternative 8
Accuracy	41.5%
Cost	960

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

Alternative 9
Accuracy	32.0%
Cost	832

\[2 \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\right) \]

Alternative 10
Accuracy	32.3%
Cost	832

\[2 \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\right) \]

Alternative 11
Accuracy	32.2%
Cost	832

\[2 \cdot \left(-0.16666666666666666 \cdot \frac{\ell}{\frac{k \cdot t}{\frac{\ell}{k}}}\right) \]

Alternative 12
Accuracy	32.5%
Cost	832

\[2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{t}\right) \cdot -0.16666666666666666\right) \]

Reproduce

herbie shell --seed 2023153 
(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))))