Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.7% → 78.1%

Time: 26.2s

Precision: binary64

Cost: 26828

• 26.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{if}\;k \leq -3.4 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \frac{t_2 \cdot \cos k}{t \cdot t_1}\\ \mathbf{elif}\;k \leq -1.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot 2 + \frac{k \cdot k}{\frac{\ell \cdot \ell}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{t} \cdot \frac{\cos k}{t_1}\right)\\ \end{array} \]

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 (pow (sin k) 2.0)) (t_2 (pow (/ l k) 2.0)))
   (if (<= k -3.4e+94)
     (* 2.0 (/ (* t_2 (cos k)) (* t t_1)))
     (if (<= k -1.9e-109)
       (/
        2.0
        (*
         (+ (* (/ (pow t 3.0) (* l l)) 2.0) (/ (* k k) (/ (* l l) t)))
         (* (sin k) (tan k))))
       (if (<= k 4.1e-5)
         (/
          2.0
          (*
           (* (/ (pow t 3.0) l) (/ k l))
           (* (tan k) (+ 1.0 (+ 1.0 (pow (/ k t) 2.0))))))
         (* 2.0 (* (/ t_2 t) (/ (cos k) 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 = pow(sin(k), 2.0);
	double t_2 = pow((l / k), 2.0);
	double tmp;
	if (k <= -3.4e+94) {
		tmp = 2.0 * ((t_2 * cos(k)) / (t * t_1));
	} else if (k <= -1.9e-109) {
		tmp = 2.0 / ((((pow(t, 3.0) / (l * l)) * 2.0) + ((k * k) / ((l * l) / t))) * (sin(k) * tan(k)));
	} else if (k <= 4.1e-5) {
		tmp = 2.0 / (((pow(t, 3.0) / l) * (k / l)) * (tan(k) * (1.0 + (1.0 + pow((k / t), 2.0)))));
	} else {
		tmp = 2.0 * ((t_2 / t) * (cos(k) / 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) :: t_2
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = (l / k) ** 2.0d0
    if (k <= (-3.4d+94)) then
        tmp = 2.0d0 * ((t_2 * cos(k)) / (t * t_1))
    else if (k <= (-1.9d-109)) then
        tmp = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * 2.0d0) + ((k * k) / ((l * l) / t))) * (sin(k) * tan(k)))
    else if (k <= 4.1d-5) then
        tmp = 2.0d0 / ((((t ** 3.0d0) / l) * (k / l)) * (tan(k) * (1.0d0 + (1.0d0 + ((k / t) ** 2.0d0)))))
    else
        tmp = 2.0d0 * ((t_2 / t) * (cos(k) / 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 = Math.pow(Math.sin(k), 2.0);
	double t_2 = Math.pow((l / k), 2.0);
	double tmp;
	if (k <= -3.4e+94) {
		tmp = 2.0 * ((t_2 * Math.cos(k)) / (t * t_1));
	} else if (k <= -1.9e-109) {
		tmp = 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * 2.0) + ((k * k) / ((l * l) / t))) * (Math.sin(k) * Math.tan(k)));
	} else if (k <= 4.1e-5) {
		tmp = 2.0 / (((Math.pow(t, 3.0) / l) * (k / l)) * (Math.tan(k) * (1.0 + (1.0 + Math.pow((k / t), 2.0)))));
	} else {
		tmp = 2.0 * ((t_2 / t) * (Math.cos(k) / 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 = math.pow(math.sin(k), 2.0)
	t_2 = math.pow((l / k), 2.0)
	tmp = 0
	if k <= -3.4e+94:
		tmp = 2.0 * ((t_2 * math.cos(k)) / (t * t_1))
	elif k <= -1.9e-109:
		tmp = 2.0 / ((((math.pow(t, 3.0) / (l * l)) * 2.0) + ((k * k) / ((l * l) / t))) * (math.sin(k) * math.tan(k)))
	elif k <= 4.1e-5:
		tmp = 2.0 / (((math.pow(t, 3.0) / l) * (k / l)) * (math.tan(k) * (1.0 + (1.0 + math.pow((k / t), 2.0)))))
	else:
		tmp = 2.0 * ((t_2 / t) * (math.cos(k) / 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 = sin(k) ^ 2.0
	t_2 = Float64(l / k) ^ 2.0
	tmp = 0.0
	if (k <= -3.4e+94)
		tmp = Float64(2.0 * Float64(Float64(t_2 * cos(k)) / Float64(t * t_1)));
	elseif (k <= -1.9e-109)
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * 2.0) + Float64(Float64(k * k) / Float64(Float64(l * l) / t))) * Float64(sin(k) * tan(k))));
	elseif (k <= 4.1e-5)
		tmp = Float64(2.0 / Float64(Float64(Float64((t ^ 3.0) / l) * Float64(k / l)) * Float64(tan(k) * Float64(1.0 + Float64(1.0 + (Float64(k / t) ^ 2.0))))));
	else
		tmp = Float64(2.0 * Float64(Float64(t_2 / t) * Float64(cos(k) / 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 = sin(k) ^ 2.0;
	t_2 = (l / k) ^ 2.0;
	tmp = 0.0;
	if (k <= -3.4e+94)
		tmp = 2.0 * ((t_2 * cos(k)) / (t * t_1));
	elseif (k <= -1.9e-109)
		tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * 2.0) + ((k * k) / ((l * l) / t))) * (sin(k) * tan(k)));
	elseif (k <= 4.1e-5)
		tmp = 2.0 / ((((t ^ 3.0) / l) * (k / l)) * (tan(k) * (1.0 + (1.0 + ((k / t) ^ 2.0)))));
	else
		tmp = 2.0 * ((t_2 / t) * (cos(k) / 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[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[k, -3.4e+94], N[(2.0 * N[(N[(t$95$2 * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[(t * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1.9e-109], N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision] + N[(N[(k * k), $MachinePrecision] / N[(N[(l * l), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 4.1e-5], N[(2.0 / N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(1.0 + N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(t$95$2 / t), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -3.4000000000000002e94

   1. Initial program 42.6%

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

   2. Simplified 40.3%

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

      [Start] 42.6%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 42.6%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 42.6%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 42.6%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 40.3%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 40.3%	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 40.3%	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 40.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 40.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 40.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 40.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Taylor expanded in k around inf 68.0%

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

   4. Simplified 68.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \left(\left(k \cdot k\right) \cdot \sin k\right) \cdot t\right)}} \]
      Step-by-step derivation

      [Start] 68.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left(\sin k \cdot {t}^{3}\right) + {k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
      fma-def [=>] 68.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\mathsf{fma}\left(2, \sin k \cdot {t}^{3}, {k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
      *-commutative [=>] 68.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \color{blue}{{t}^{3} \cdot \sin k}, {k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
      associate-*r* [=>] 68.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}\right)} \]
      unpow2 [=>] 68.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \left(\color{blue}{\left(k \cdot k\right)} \cdot \sin k\right) \cdot t\right)} \]

   5. Taylor expanded in k around inf 68.0%

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

   6. Simplified 95.0%

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

      [Start] 68.0%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 68.0%	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 70.3%	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 70.3%	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 70.3%	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 95.0%	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [<=] 95.0%	\[ 2 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 95.0%	\[ 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   7. Applied egg-rr 95.0%

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

      [Start] 95.0%	\[ 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]
      associate-*r/ [=>] 95.0%	\[ 2 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]

   if -3.4000000000000002e94 < k < -1.90000000000000001e-109

   1. Initial program 69.2%

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

   2. Simplified 69.2%

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

      [Start] 69.2%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 69.2%	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      associate-*l* [=>] 69.2%	\[ \frac{2}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      associate-*r* [=>] 69.2%	\[ \frac{2}{\color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      +-commutative [=>] 69.2%	\[ \frac{2}{\left(\color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-+r+ [=>] 69.2%	\[ \frac{2}{\left(\color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      metadata-eval [=>] 69.2%	\[ \frac{2}{\left(\left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   3. Taylor expanded in k around 0 85.7%

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

   4. Simplified 85.7%

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

      [Start] 85.7%	\[ \frac{2}{\left(\frac{{k}^{2} \cdot t}{{\ell}^{2}} + 2 \cdot \frac{{t}^{3}}{{\ell}^{2}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      +-commutative [=>] 85.7%	\[ \frac{2}{\color{blue}{\left(2 \cdot \frac{{t}^{3}}{{\ell}^{2}} + \frac{{k}^{2} \cdot t}{{\ell}^{2}}\right)} \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 85.7%	\[ \frac{2}{\left(2 \cdot \frac{{t}^{3}}{\color{blue}{\ell \cdot \ell}} + \frac{{k}^{2} \cdot t}{{\ell}^{2}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      associate-/l* [=>] 85.7%	\[ \frac{2}{\left(2 \cdot \frac{{t}^{3}}{\ell \cdot \ell} + \color{blue}{\frac{{k}^{2}}{\frac{{\ell}^{2}}{t}}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 85.7%	\[ \frac{2}{\left(2 \cdot \frac{{t}^{3}}{\ell \cdot \ell} + \frac{\color{blue}{k \cdot k}}{\frac{{\ell}^{2}}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]
      unpow2 [=>] 85.7%	\[ \frac{2}{\left(2 \cdot \frac{{t}^{3}}{\ell \cdot \ell} + \frac{k \cdot k}{\frac{\color{blue}{\ell \cdot \ell}}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)} \]

   if -1.90000000000000001e-109 < k < 4.10000000000000005e-5

   1. Initial program 64.9%

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

   2. Simplified 64.9%

      \[\leadsto \color{blue}{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      Step-by-step derivation

      [Start] 64.9%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 64.9%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 64.9%	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in k around 0 66.4%

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

   4. Simplified 76.5%

      \[\leadsto \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      Step-by-step derivation

      [Start] 66.4%	\[ \frac{2}{\frac{k \cdot {t}^{3}}{{\ell}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      *-commutative [=>] 66.4%	\[ \frac{2}{\frac{\color{blue}{{t}^{3} \cdot k}}{{\ell}^{2}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow2 [=>] 66.4%	\[ \frac{2}{\frac{{t}^{3} \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 76.5%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   if 4.10000000000000005e-5 < k

   1. Initial program 51.3%

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

   2. Simplified 52.5%

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

      [Start] 51.3%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 52.5%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 52.5%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 52.5%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 52.5%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 52.5%	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 52.6%	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 52.6%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 52.6%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 52.5%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 52.5%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Taylor expanded in k around inf 73.0%

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

   4. Simplified 73.0%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \left(\left(k \cdot k\right) \cdot \sin k\right) \cdot t\right)}} \]
      Step-by-step derivation

      [Start] 73.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(2 \cdot \left(\sin k \cdot {t}^{3}\right) + {k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
      fma-def [=>] 73.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \color{blue}{\mathsf{fma}\left(2, \sin k \cdot {t}^{3}, {k}^{2} \cdot \left(\sin k \cdot t\right)\right)}} \]
      *-commutative [=>] 73.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \color{blue}{{t}^{3} \cdot \sin k}, {k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
      associate-*r* [=>] 73.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \color{blue}{\left({k}^{2} \cdot \sin k\right) \cdot t}\right)} \]
      unpow2 [=>] 73.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \left(\color{blue}{\left(k \cdot k\right)} \cdot \sin k\right) \cdot t\right)} \]

   5. Taylor expanded in k around inf 73.0%

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

   6. Simplified 76.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \color{blue}{\sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)}\right)} \]
      Step-by-step derivation

      [Start] 73.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, {k}^{2} \cdot \left(\sin k \cdot t\right)\right)} \]
      *-commutative [=>] 73.0%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \color{blue}{\left(\sin k \cdot t\right) \cdot {k}^{2}}\right)} \]
      associate-*l* [=>] 72.9%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \color{blue}{\sin k \cdot \left(t \cdot {k}^{2}\right)}\right)} \]
      *-commutative [<=] 72.9%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \sin k \cdot \color{blue}{\left({k}^{2} \cdot t\right)}\right)} \]
      unpow2 [=>] 72.9%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \sin k \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)\right)} \]
      associate-*l* [=>] 76.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \sin k \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}\right)} \]

   7. Applied egg-rr 76.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)\right)}^{1}}} \]
      Step-by-step derivation

      [Start] 76.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)} \]
      pow1 [=>] 76.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{{\left(\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)\right)}^{1}}} \]

   8. Simplified 76.3%

      \[\leadsto \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \mathsf{fma}\left(2, \sin k \cdot {t}^{3}, k \cdot \left(\left(k \cdot t\right) \cdot \sin k\right)\right)}} \]
      Step-by-step derivation

      [Start] 76.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{{\left(\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)\right)}^{1}} \]
      unpow1 [=>] 76.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \mathsf{fma}\left(2, {t}^{3} \cdot \sin k, \sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}} \]
      *-commutative [=>] 76.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \color{blue}{\sin k \cdot {t}^{3}}, \sin k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)} \]
      associate-*r* [=>] 72.9%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \sin k \cdot {t}^{3}, \sin k \cdot \color{blue}{\left(\left(k \cdot k\right) \cdot t\right)}\right)} \]
      unpow2 [<=] 72.9%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \sin k \cdot {t}^{3}, \sin k \cdot \left(\color{blue}{{k}^{2}} \cdot t\right)\right)} \]
      *-commutative [=>] 72.9%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \sin k \cdot {t}^{3}, \color{blue}{\left({k}^{2} \cdot t\right) \cdot \sin k}\right)} \]
      unpow2 [=>] 72.9%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \sin k \cdot {t}^{3}, \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right) \cdot \sin k\right)} \]
      associate-*r* [<=] 76.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \sin k \cdot {t}^{3}, \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)} \cdot \sin k\right)} \]
      associate-*l* [=>] 76.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \mathsf{fma}\left(2, \sin k \cdot {t}^{3}, \color{blue}{k \cdot \left(\left(k \cdot t\right) \cdot \sin k\right)}\right)} \]

   9. Taylor expanded in k around inf 66.6%

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

   10. Simplified 84.6%

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

       [Start] 66.6%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
       *-commutative [=>] 66.6%	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
       times-frac [=>] 65.8%	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
       unpow2 [=>] 65.8%	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
       unpow2 [=>] 65.8%	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
       times-frac [=>] 84.6%	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
       unpow2 [<=] 84.6%	\[ 2 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
       *-commutative [<=] 84.6%	\[ 2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]
       associate-*r/ [=>] 84.6%	\[ 2 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]
       times-frac [=>] 84.6%	\[ 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -3.4 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}\\ \mathbf{elif}\;k \leq -1.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot 2 + \frac{k \cdot k}{\frac{\ell \cdot \ell}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	78.1%
Cost	26828

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{if}\;k \leq -3.4 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \frac{t_2 \cdot \cos k}{t \cdot t_1}\\ \mathbf{elif}\;k \leq -1.9 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot 2 + \frac{k \cdot k}{\frac{\ell \cdot \ell}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 4.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{t_2}{t} \cdot \frac{\cos k}{t_1}\right)\\ \end{array} \]

Alternative 2
Accuracy	80.0%
Cost	80712

\[\begin{array}{l} t_1 := 1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\\ t_2 := \frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\\ t_3 := \left(t_2 \cdot \tan k\right) \cdot t_1\\ t_4 := \tan k \cdot t_1\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot t_4}\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;\frac{2}{t_2 \cdot t_4}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3
Accuracy	80.0%
Cost	80712

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

Alternative 4
Accuracy	78.1%
Cost	26828

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ \mathbf{if}\;k \leq -2.4 \cdot 10^{+94}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ \mathbf{elif}\;k \leq -7 \cdot 10^{-110}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot 2 + \frac{k \cdot k}{\frac{\ell \cdot \ell}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{t_1}\right)\\ \end{array} \]

Alternative 5
Accuracy	78.1%
Cost	21000

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{if}\;k \leq -3 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -3.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot 2 + \frac{k \cdot k}{\frac{\ell \cdot \ell}{t}}\right) \cdot \left(\sin k \cdot \tan k\right)}\\ \mathbf{elif}\;k \leq 3.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	76.5%
Cost	20808

\[\begin{array}{l} \mathbf{if}\;k \leq -7.6 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{\sin k \cdot \tan k}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{2}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 7
Accuracy	77.0%
Cost	20488

\[\begin{array}{l} \mathbf{if}\;k \leq -2.1 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{2}{t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}}{\sin k \cdot \tan k}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 8
Accuracy	66.9%
Cost	14092

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-42}:\\ \;\;\;\;2 \cdot \left({\left(\frac{\ell}{k}\right)}^{2} \cdot \frac{\cos k}{t \cdot \left(k \cdot k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	72.6%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{+20} \lor \neg \left(t \leq 10^{-41}\right):\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot 2}{\left(\sin k \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \tan k\right)}\\ \end{array} \]

Alternative 10
Accuracy	77.0%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -8.5 \cdot 10^{-40} \lor \neg \left(k \leq 0.00092\right):\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \end{array} \]

Alternative 11
Accuracy	72.4%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;k \leq -1.06 \cdot 10^{-39}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\sin k \cdot \left(k \cdot \left(t \cdot k\right)\right)\right)}\\ \mathbf{elif}\;k \leq 0.0003:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot 2}{\left(\sin k \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \tan k\right)}\\ \end{array} \]

Alternative 12
Accuracy	71.5%
Cost	14024

\[\begin{array}{l} \mathbf{if}\;k \leq -1.95 \cdot 10^{-41}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\ell \cdot 2}{\left(\sin k \cdot \left(k \cdot k\right)\right) \cdot \left(t \cdot \tan k\right)}\\ \end{array} \]

Alternative 13
Accuracy	77.0%
Cost	14024

\[\begin{array}{l} t_1 := t \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)\\ t_2 := \sin k \cdot \tan k\\ \mathbf{if}\;k \leq -4.45 \cdot 10^{-41}:\\ \;\;\;\;\frac{\frac{2}{t_1}}{t_2}\\ \mathbf{elif}\;k \leq 0.00082:\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_2 \cdot t_1}\\ \end{array} \]

Alternative 14
Accuracy	66.9%
Cost	7884

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{if}\;t \leq -7.1 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{k \cdot \left(t \cdot k\right)}{\cos k}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Accuracy	66.7%
Cost	7564

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(2 \cdot \left(\ell \cdot \frac{1}{{k}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Accuracy	66.7%
Cost	7564

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{if}\;t \leq -3.8 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-117}:\\ \;\;\;\;\frac{2}{2 \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 10^{-41}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(2 \cdot \left(\ell \cdot \frac{1}{{k}^{4}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 17
Accuracy	66.7%
Cost	7436

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{if}\;t \leq -6.6 \cdot 10^{-12}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.05 \cdot 10^{-117}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(2 \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Accuracy	66.3%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -1.38 \cdot 10^{-75} \lor \neg \left(t \leq 5.8 \cdot 10^{-42}\right):\\ \;\;\;\;\ell \cdot \frac{\ell}{k \cdot \left({t}^{3} \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \left(2 \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 19
Accuracy	55.4%
Cost	960

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

Alternative 20
Accuracy	32.3%
Cost	832

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

Reproduce

herbie shell --seed 2023271 
(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))))