Toniolo and Linder, Equation (10+)

Average Accuracy: 49.2% → 87.1%

Time: 56.2s

Precision: binary64

Cost: 27344

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ t_3 := \left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}\\ \mathbf{if}\;t \leq -1.85 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{2 \cdot {t}^{-3}}{\frac{\tan k}{\ell}}}{t_3}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (/ l (* t k)))
        (t_2 (* t_1 (* (/ 1.0 t) t_1)))
        (t_3 (* (+ 2.0 (pow (/ k t) 2.0)) (/ (sin k) l))))
   (if (<= t -1.85e+110)
     t_2
     (if (<= t -1.6e-82)
       (/ (/ (* 2.0 (pow t -3.0)) (/ (tan k) l)) t_3)
       (if (<= t 8.8e-47)
         (/ 2.0 (/ (* t k) (* (/ l (pow (sin k) 2.0)) (* l (/ (cos k) k)))))
         (if (<= t 2.3e+98)
           (/ (* l (* 2.0 (/ (pow t -3.0) (tan k)))) t_3)
           t_2))))))

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 = l / (t * k);
	double t_2 = t_1 * ((1.0 / t) * t_1);
	double t_3 = (2.0 + pow((k / t), 2.0)) * (sin(k) / l);
	double tmp;
	if (t <= -1.85e+110) {
		tmp = t_2;
	} else if (t <= -1.6e-82) {
		tmp = ((2.0 * pow(t, -3.0)) / (tan(k) / l)) / t_3;
	} else if (t <= 8.8e-47) {
		tmp = 2.0 / ((t * k) / ((l / pow(sin(k), 2.0)) * (l * (cos(k) / k))));
	} else if (t <= 2.3e+98) {
		tmp = (l * (2.0 * (pow(t, -3.0) / tan(k)))) / t_3;
	} else {
		tmp = t_2;
	}
	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) :: t_3
    real(8) :: tmp
    t_1 = l / (t * k)
    t_2 = t_1 * ((1.0d0 / t) * t_1)
    t_3 = (2.0d0 + ((k / t) ** 2.0d0)) * (sin(k) / l)
    if (t <= (-1.85d+110)) then
        tmp = t_2
    else if (t <= (-1.6d-82)) then
        tmp = ((2.0d0 * (t ** (-3.0d0))) / (tan(k) / l)) / t_3
    else if (t <= 8.8d-47) then
        tmp = 2.0d0 / ((t * k) / ((l / (sin(k) ** 2.0d0)) * (l * (cos(k) / k))))
    else if (t <= 2.3d+98) then
        tmp = (l * (2.0d0 * ((t ** (-3.0d0)) / tan(k)))) / t_3
    else
        tmp = t_2
    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 = l / (t * k);
	double t_2 = t_1 * ((1.0 / t) * t_1);
	double t_3 = (2.0 + Math.pow((k / t), 2.0)) * (Math.sin(k) / l);
	double tmp;
	if (t <= -1.85e+110) {
		tmp = t_2;
	} else if (t <= -1.6e-82) {
		tmp = ((2.0 * Math.pow(t, -3.0)) / (Math.tan(k) / l)) / t_3;
	} else if (t <= 8.8e-47) {
		tmp = 2.0 / ((t * k) / ((l / Math.pow(Math.sin(k), 2.0)) * (l * (Math.cos(k) / k))));
	} else if (t <= 2.3e+98) {
		tmp = (l * (2.0 * (Math.pow(t, -3.0) / Math.tan(k)))) / t_3;
	} else {
		tmp = t_2;
	}
	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 = l / (t * k)
	t_2 = t_1 * ((1.0 / t) * t_1)
	t_3 = (2.0 + math.pow((k / t), 2.0)) * (math.sin(k) / l)
	tmp = 0
	if t <= -1.85e+110:
		tmp = t_2
	elif t <= -1.6e-82:
		tmp = ((2.0 * math.pow(t, -3.0)) / (math.tan(k) / l)) / t_3
	elif t <= 8.8e-47:
		tmp = 2.0 / ((t * k) / ((l / math.pow(math.sin(k), 2.0)) * (l * (math.cos(k) / k))))
	elif t <= 2.3e+98:
		tmp = (l * (2.0 * (math.pow(t, -3.0) / math.tan(k)))) / t_3
	else:
		tmp = t_2
	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(l / Float64(t * k))
	t_2 = Float64(t_1 * Float64(Float64(1.0 / t) * t_1))
	t_3 = Float64(Float64(2.0 + (Float64(k / t) ^ 2.0)) * Float64(sin(k) / l))
	tmp = 0.0
	if (t <= -1.85e+110)
		tmp = t_2;
	elseif (t <= -1.6e-82)
		tmp = Float64(Float64(Float64(2.0 * (t ^ -3.0)) / Float64(tan(k) / l)) / t_3);
	elseif (t <= 8.8e-47)
		tmp = Float64(2.0 / Float64(Float64(t * k) / Float64(Float64(l / (sin(k) ^ 2.0)) * Float64(l * Float64(cos(k) / k)))));
	elseif (t <= 2.3e+98)
		tmp = Float64(Float64(l * Float64(2.0 * Float64((t ^ -3.0) / tan(k)))) / t_3);
	else
		tmp = t_2;
	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 = l / (t * k);
	t_2 = t_1 * ((1.0 / t) * t_1);
	t_3 = (2.0 + ((k / t) ^ 2.0)) * (sin(k) / l);
	tmp = 0.0;
	if (t <= -1.85e+110)
		tmp = t_2;
	elseif (t <= -1.6e-82)
		tmp = ((2.0 * (t ^ -3.0)) / (tan(k) / l)) / t_3;
	elseif (t <= 8.8e-47)
		tmp = 2.0 / ((t * k) / ((l / (sin(k) ^ 2.0)) * (l * (cos(k) / k))));
	elseif (t <= 2.3e+98)
		tmp = (l * (2.0 * ((t ^ -3.0) / tan(k)))) / t_3;
	else
		tmp = t_2;
	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[(l / N[(t * k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[(1.0 / t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.85e+110], t$95$2, If[LessEqual[t, -1.6e-82], N[(N[(N[(2.0 * N[Power[t, -3.0], $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t, 8.8e-47], N[(2.0 / N[(N[(t * k), $MachinePrecision] / N[(N[(l / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(l * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.3e+98], N[(N[(l * N[(2.0 * N[(N[Power[t, -3.0], $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.85000000000000006e110 or 2.30000000000000013e98 < t

   1. Initial program 63.3%

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

   2. Simplified 54.5%

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

      [Start] 63.3	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 63.3	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 63.3	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 63.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 63.3	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 54.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Taylor expanded in k around 0 54.5%

      \[\leadsto \color{blue}{\frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}}} \]

   4. Simplified 68.3%

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

      [Start] 54.5	\[ \frac{{\ell}^{2}}{{k}^{2} \cdot {t}^{3}} \]
      unpow2 [=>] 54.5	\[ \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2} \cdot {t}^{3}} \]
      associate-/l* [=>] 59.3	\[ \color{blue}{\frac{\ell}{\frac{{k}^{2} \cdot {t}^{3}}{\ell}}} \]
      unpow2 [=>] 59.3	\[ \frac{\ell}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{3}}{\ell}} \]
      associate-*l* [=>] 68.3	\[ \frac{\ell}{\frac{\color{blue}{k \cdot \left(k \cdot {t}^{3}\right)}}{\ell}} \]

   5. Applied egg-rr 67.8%

      \[\leadsto \color{blue}{{\left({\left(\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}\right)}^{3}\right)}^{0.3333333333333333}} \]
      Proof

      [Start] 68.3	\[ \frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}} \]
      add-cbrt-cube [=>] 68.1	\[ \color{blue}{\sqrt[3]{\left(\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}} \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\right) \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}}} \]
      pow1/3 [=>] 68.1	\[ \color{blue}{{\left(\left(\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}} \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\right) \cdot \frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\right)}^{0.3333333333333333}} \]
      pow3 [=>] 68.1	\[ {\color{blue}{\left({\left(\frac{\ell}{\frac{k \cdot \left(k \cdot {t}^{3}\right)}{\ell}}\right)}^{3}\right)}}^{0.3333333333333333} \]
      *-commutative [=>] 68.1	\[ {\left({\left(\frac{\ell}{\frac{\color{blue}{\left(k \cdot {t}^{3}\right) \cdot k}}{\ell}}\right)}^{3}\right)}^{0.3333333333333333} \]
      associate-/l* [=>] 67.8	\[ {\left({\left(\frac{\ell}{\color{blue}{\frac{k \cdot {t}^{3}}{\frac{\ell}{k}}}}\right)}^{3}\right)}^{0.3333333333333333} \]
      associate-/r/ [=>] 67.8	\[ {\left({\color{blue}{\left(\frac{\ell}{k \cdot {t}^{3}} \cdot \frac{\ell}{k}\right)}}^{3}\right)}^{0.3333333333333333} \]
      associate-/r* [=>] 67.8	\[ {\left({\left(\color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}}} \cdot \frac{\ell}{k}\right)}^{3}\right)}^{0.3333333333333333} \]

   6. Applied egg-rr 74.4%

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

      [Start] 67.8	\[ {\left({\left(\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}\right)}^{3}\right)}^{0.3333333333333333} \]
      pow-pow [=>] 68.0	\[ \color{blue}{{\left(\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}\right)}^{\left(3 \cdot 0.3333333333333333\right)}} \]
      metadata-eval [=>] 68.0	\[ {\left(\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}\right)}^{\color{blue}{1}} \]
      pow1 [<=] 68.0	\[ \color{blue}{\frac{\frac{\ell}{k}}{{t}^{3}} \cdot \frac{\ell}{k}} \]
      associate-*l/ [=>] 63.8	\[ \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}} \]
      unpow3 [=>] 63.8	\[ \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{\color{blue}{\left(t \cdot t\right) \cdot t}} \]
      times-frac [=>] 73.2	\[ \color{blue}{\frac{\frac{\ell}{k}}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{t}} \]
      associate-/l/ [=>] 73.2	\[ \color{blue}{\frac{\ell}{\left(t \cdot t\right) \cdot k}} \cdot \frac{\frac{\ell}{k}}{t} \]
      associate-/l/ [=>] 74.4	\[ \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \color{blue}{\frac{\ell}{t \cdot k}} \]

   7. Applied egg-rr 89.1%

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

      [Start] 74.4	\[ \frac{\ell}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{t \cdot k} \]
      *-un-lft-identity [=>] 74.4	\[ \frac{\color{blue}{1 \cdot \ell}}{\left(t \cdot t\right) \cdot k} \cdot \frac{\ell}{t \cdot k} \]
      associate-*l* [=>] 83.7	\[ \frac{1 \cdot \ell}{\color{blue}{t \cdot \left(t \cdot k\right)}} \cdot \frac{\ell}{t \cdot k} \]
      times-frac [=>] 89.1	\[ \color{blue}{\left(\frac{1}{t} \cdot \frac{\ell}{t \cdot k}\right)} \cdot \frac{\ell}{t \cdot k} \]

   if -1.85000000000000006e110 < t < -1.6000000000000001e-82

   1. Initial program 65.6%

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

   2. Simplified 73.1%

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

      [Start] 65.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 65.8	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 65.6	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 68.0	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 68.1	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 68.7	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 73.1	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 73.1	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 73.1	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 73.1	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Applied egg-rr 76.6%

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

      [Start] 73.1	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
      associate-*l/ [=>] 73.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k}} \]
      associate-/l* [=>] 72.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\frac{\tan k}{\frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}}} \]
      div-inv [=>] 72.0	\[ \frac{\color{blue}{2 \cdot \frac{1}{{t}^{3}}}}{\frac{\tan k}{\frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]
      pow-flip [=>] 72.9	\[ \frac{2 \cdot \color{blue}{{t}^{\left(-3\right)}}}{\frac{\tan k}{\frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]
      metadata-eval [=>] 72.9	\[ \frac{2 \cdot {t}^{\color{blue}{-3}}}{\frac{\tan k}{\frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}} \]
      associate-/l/ [=>] 76.6	\[ \frac{2 \cdot {t}^{-3}}{\frac{\tan k}{\color{blue}{\frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}}}} \]
      *-commutative [=>] 76.6	\[ \frac{2 \cdot {t}^{-3}}{\frac{\tan k}{\frac{\ell}{\color{blue}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}}} \]

   4. Simplified 82.5%

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

      [Start] 76.6	\[ \frac{2 \cdot {t}^{-3}}{\frac{\tan k}{\frac{\ell}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}} \]
      associate-/r/ [=>] 77.6	\[ \color{blue}{\frac{2 \cdot {t}^{-3}}{\tan k} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      times-frac [<=] 78.6	\[ \color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      *-commutative [<=] 78.6	\[ \frac{\color{blue}{\ell \cdot \left(2 \cdot {t}^{-3}\right)}}{\tan k \cdot \left(\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      associate-/r* [=>] 82.5	\[ \color{blue}{\frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      *-commutative [=>] 82.5	\[ \frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{\color{blue}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}} \]

   5. Applied egg-rr 53.0%

      \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell}{\tan k} \cdot \left(2 \cdot {t}^{-3}\right)\right)} - 1}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      Proof

      [Start] 82.5	\[ \frac{\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      expm1-log1p-u [=>] 70.0	\[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}\right)\right)}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      expm1-udef [=>] 53.0	\[ \frac{\color{blue}{e^{\mathsf{log1p}\left(\frac{\ell \cdot \left(2 \cdot {t}^{-3}\right)}{\tan k}\right)} - 1}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      associate-/l* [=>] 51.7	\[ \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell}{\frac{\tan k}{2 \cdot {t}^{-3}}}}\right)} - 1}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      associate-/r/ [=>] 53.0	\[ \frac{e^{\mathsf{log1p}\left(\color{blue}{\frac{\ell}{\tan k} \cdot \left(2 \cdot {t}^{-3}\right)}\right)} - 1}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]

   6. Simplified 86.3%

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

      [Start] 53.0	\[ \frac{e^{\mathsf{log1p}\left(\frac{\ell}{\tan k} \cdot \left(2 \cdot {t}^{-3}\right)\right)} - 1}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      expm1-def [=>] 73.8	\[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\ell}{\tan k} \cdot \left(2 \cdot {t}^{-3}\right)\right)\right)}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      expm1-log1p [=>] 86.4	\[ \frac{\color{blue}{\frac{\ell}{\tan k} \cdot \left(2 \cdot {t}^{-3}\right)}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      *-commutative [=>] 86.4	\[ \frac{\color{blue}{\left(2 \cdot {t}^{-3}\right) \cdot \frac{\ell}{\tan k}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      associate-*r/ [=>] 82.5	\[ \frac{\color{blue}{\frac{\left(2 \cdot {t}^{-3}\right) \cdot \ell}{\tan k}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      associate-/l* [=>] 86.3	\[ \frac{\color{blue}{\frac{2 \cdot {t}^{-3}}{\frac{\tan k}{\ell}}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]

   if -1.6000000000000001e-82 < t < 8.80000000000000075e-47

   1. Initial program 10.9%

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

   2. Simplified 9.9%

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

      [Start] 10.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 10.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 10.9	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 9.9	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 9.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 9.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 9.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 9.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 58.2%

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

   4. Simplified 66.3%

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

      [Start] 58.2	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 55.5	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 55.5	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 55.5	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 55.5	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 66.3	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Applied egg-rr 84.4%

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

      [Start] 66.3	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)} \]
      associate-*r* [=>] 71.1	\[ \frac{2}{\color{blue}{\left(\frac{k \cdot k}{\cos k} \cdot \frac{t}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\ell}}} \]
      clear-num [=>] 71.1	\[ \frac{2}{\left(\frac{k \cdot k}{\cos k} \cdot \frac{t}{\ell}\right) \cdot \color{blue}{\frac{1}{\frac{\ell}{{\sin k}^{2}}}}} \]
      un-div-inv [=>] 71.1	\[ \frac{2}{\color{blue}{\frac{\frac{k \cdot k}{\cos k} \cdot \frac{t}{\ell}}{\frac{\ell}{{\sin k}^{2}}}}} \]
      associate-/l* [=>] 71.1	\[ \frac{2}{\frac{\color{blue}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{t}{\ell}}{\frac{\ell}{{\sin k}^{2}}}} \]
      frac-times [=>] 90.7	\[ \frac{2}{\frac{\color{blue}{\frac{k \cdot t}{\frac{\cos k}{k} \cdot \ell}}}{\frac{\ell}{{\sin k}^{2}}}} \]
      associate-/l/ [=>] 84.4	\[ \frac{2}{\color{blue}{\frac{k \cdot t}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\frac{\cos k}{k} \cdot \ell\right)}}} \]

   if 8.80000000000000075e-47 < t < 2.30000000000000013e98

   1. Initial program 66.1%

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

   2. Simplified 77.4%

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

      [Start] 66.1	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 66.1	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      associate-/r* [=>] 66.1	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-/r/ [<=] 69.9	\[ \frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [=>] 69.9	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \frac{\ell \cdot \ell}{\sin k}}}{\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      times-frac [=>] 71.1	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      associate-/l* [=>] 77.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      +-commutative [=>] 77.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-+r+ [=>] 77.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}}} \]
      metadata-eval [=>] 77.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}} \]

   3. Applied egg-rr 87.7%

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

      [Start] 77.4	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
      associate-/l/ [=>] 81.7	\[ \frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \color{blue}{\frac{\ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}} \]
      associate-*r/ [=>] 87.7	\[ \color{blue}{\frac{\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}} \]
      div-inv [=>] 87.7	\[ \frac{\frac{\color{blue}{2 \cdot \frac{1}{{t}^{3}}}}{\tan k} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      *-un-lft-identity [=>] 87.7	\[ \frac{\frac{2 \cdot \frac{1}{{t}^{3}}}{\color{blue}{1 \cdot \tan k}} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      times-frac [=>] 87.7	\[ \frac{\color{blue}{\left(\frac{2}{1} \cdot \frac{\frac{1}{{t}^{3}}}{\tan k}\right)} \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      metadata-eval [=>] 87.7	\[ \frac{\left(\color{blue}{2} \cdot \frac{\frac{1}{{t}^{3}}}{\tan k}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      pow-flip [=>] 87.7	\[ \frac{\left(2 \cdot \frac{\color{blue}{{t}^{\left(-3\right)}}}{\tan k}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      metadata-eval [=>] 87.7	\[ \frac{\left(2 \cdot \frac{{t}^{\color{blue}{-3}}}{\tan k}\right) \cdot \ell}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}} \]
      *-commutative [=>] 87.7	\[ \frac{\left(2 \cdot \frac{{t}^{-3}}{\tan k}\right) \cdot \ell}{\color{blue}{\frac{\sin k}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+110}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \left(\frac{1}{t} \cdot \frac{\ell}{t \cdot k}\right)\\ \mathbf{elif}\;t \leq -1.6 \cdot 10^{-82}:\\ \;\;\;\;\frac{\frac{2 \cdot {t}^{-3}}{\frac{\tan k}{\ell}}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+98}:\\ \;\;\;\;\frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot k} \cdot \left(\frac{1}{t} \cdot \frac{\ell}{t \cdot k}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.1%
Cost	27344

\[\begin{array}{l} t_1 := \frac{\ell \cdot \left(2 \cdot \frac{{t}^{-3}}{\tan k}\right)}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\sin k}{\ell}}\\ t_2 := \frac{\ell}{t \cdot k}\\ t_3 := t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+110}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{-46}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Accuracy	83.2%
Cost	27080

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3}}}{\tan k} \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	83.2%
Cost	27080

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ t_2 := t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -5.9 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}}}\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-33}:\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Accuracy	84.0%
Cost	20620

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ t_3 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -3 \cdot 10^{+157}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -2.35 \cdot 10^{+30}:\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\cos k}{k}}{k \cdot t_1} \cdot \left(\ell \cdot \frac{2}{t}\right)\right)\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-27}:\\ \;\;\;\;t_3 \cdot \left(\frac{1}{t} \cdot t_3\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	83.0%
Cost	20489

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -7.5 \cdot 10^{+31} \lor \neg \left(k \leq 5.2 \cdot 10^{-27}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \end{array} \]

Alternative 6
Accuracy	83.7%
Cost	20489

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -2.02 \cdot 10^{+30} \lor \neg \left(k \leq 5.2 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot k}{\frac{\ell}{{\sin k}^{2}} \cdot \left(\ell \cdot \frac{\cos k}{k}\right)}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \end{array} \]

Alternative 7
Accuracy	83.0%
Cost	20488

\[\begin{array}{l} t_1 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_2 := \frac{\ell}{t \cdot k}\\ t_3 := t \cdot {\sin k}^{2}\\ \mathbf{if}\;k \leq -3.7 \cdot 10^{+30}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_1}}\\ \mathbf{elif}\;k \leq 1.52 \cdot 10^{-27}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_1 \cdot \frac{\cos k}{t_3}\right)\\ \end{array} \]

Alternative 8
Accuracy	72.9%
Cost	14540

\[\begin{array}{l} t_1 := \frac{2}{\frac{t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}{\frac{\frac{\ell}{k}}{t}}}\\ t_2 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -5.2 \cdot 10^{-148}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.1 \cdot 10^{-290}:\\ \;\;\;\;t_2 \cdot \left(\frac{1}{t} \cdot t_2\right)\\ \mathbf{elif}\;k \leq 0.021:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{1 - \cos \left(k + k\right)}{\ell \cdot \frac{\ell \cdot 2}{t}}}\\ \end{array} \]

Alternative 9
Accuracy	69.7%
Cost	8328

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -5 \cdot 10^{-148}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-27}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t \cdot \left(\frac{\frac{\ell}{t}}{k} \cdot \left(\ell \cdot -0.16666666666666666\right)\right) + k \cdot \left(\ell \cdot \frac{\ell}{{k}^{4}}\right)}{t \cdot k}\\ \end{array} \]

Alternative 10
Accuracy	70.5%
Cost	8073

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -5 \cdot 10^{-148} \lor \neg \left(k \leq 3.1 \cdot 10^{-290}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \end{array} \]

Alternative 11
Accuracy	71.2%
Cost	7433

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-67} \lor \neg \left(t \leq 4.6 \cdot 10^{-131}\right):\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\frac{{k}^{3}}{\ell}}{\frac{\frac{\ell}{k}}{t}}}\\ \end{array} \]

Alternative 12
Accuracy	68.9%
Cost	7305

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -1.22 \cdot 10^{+32} \lor \neg \left(k \leq 4.5 \cdot 10^{-27}\right):\\ \;\;\;\;2 \cdot \left(\ell \cdot \frac{\ell}{t \cdot {k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \end{array} \]

Alternative 13
Accuracy	70.6%
Cost	7305

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-67} \lor \neg \left(t \leq 4.6 \cdot 10^{-131}\right):\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 14
Accuracy	70.6%
Cost	7305

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{-73} \lor \neg \left(t \leq 4.6 \cdot 10^{-131}\right):\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \end{array} \]

Alternative 15
Accuracy	65.1%
Cost	1352

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -5.8 \cdot 10^{-133}:\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 6.2 \cdot 10^{-292}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(t \cdot \frac{k}{\frac{\ell}{2}}\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \end{array} \]

Alternative 16
Accuracy	66.6%
Cost	1225

\[\begin{array}{l} t_1 := \frac{\ell}{t \cdot k}\\ \mathbf{if}\;k \leq -9.6 \cdot 10^{-131} \lor \neg \left(k \leq 4 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot \left(k \cdot \frac{t}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{1}{t} \cdot t_1\right)\\ \end{array} \]

Alternative 17
Accuracy	63.1%
Cost	1097

\[\begin{array}{l} t_1 := k \cdot \frac{t}{\ell}\\ \mathbf{if}\;k \leq -1.15 \cdot 10^{-164} \lor \neg \left(k \leq 1.65 \cdot 10^{-286}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t_1 \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]

Alternative 18
Accuracy	64.4%
Cost	1097

\[\begin{array}{l} t_1 := k \cdot \frac{t}{\ell}\\ \mathbf{if}\;k \leq -1.16 \cdot 10^{-119} \lor \neg \left(k \leq 4.2 \cdot 10^{-156}\right):\\ \;\;\;\;\frac{\frac{\ell}{t}}{t \cdot \left(k \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{t \cdot \left(t \cdot k\right)}}{t_1}\\ \end{array} \]

Alternative 19
Accuracy	59.5%
Cost	964

\[\begin{array}{l} \mathbf{if}\;k \leq -3.8 \cdot 10^{-61}:\\ \;\;\;\;\frac{\frac{\ell}{t} \cdot \frac{\ell}{t}}{k \cdot \left(t \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\left(k \cdot \frac{t}{\ell}\right) \cdot \left(t \cdot \left(t \cdot k\right)\right)}\\ \end{array} \]

Alternative 20
Accuracy	55.1%
Cost	832

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

Alternative 21
Accuracy	58.8%
Cost	832

\[\frac{\ell}{t \cdot k} \cdot \frac{\frac{\frac{\ell}{t}}{t}}{k} \]

Alternative 22
Accuracy	58.8%
Cost	832

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

Reproduce

herbie shell --seed 2023133 
(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))))