Toniolo and Linder, Equation (10+)

Percentage Accurate: 54.4% → 78.9%

Time: 17.4s

Precision: binary64

Cost: 53828

• 28.0% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{\sin k}^{2}}\right)\\ \end{array} \]

FPCore

(FPCore (t l k)
 :precision binary64
 (/
  2.0
  (*
   (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
   (+ (+ 1.0 (pow (/ k t) 2.0)) 1.0))))

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (/ k t) 2.0)))
   (if (<=
        (/
         2.0
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ 1.0 (+ 1.0 t_1))))
        5e+130)
     (* (* l l) (/ 2.0 (* (tan k) (* (+ 2.0 t_1) (* (pow t 3.0) (sin k))))))
     (* 2.0 (* (/ (cos k) t) (/ (pow (/ l k) 2.0) (pow (sin k) 2.0)))))))

C

double code(double t, double l, double k) {
	return 2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + pow((k / t), 2.0)) + 1.0));
}

↓

double code(double t, double l, double k) {
	double t_1 = pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 5e+130) {
		tmp = (l * l) * (2.0 / (tan(k) * ((2.0 + t_1) * (pow(t, 3.0) * sin(k)))));
	} else {
		tmp = 2.0 * ((cos(k) / t) * (pow((l / k), 2.0) / pow(sin(k), 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    code = 2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * ((1.0d0 + ((k / t) ** 2.0d0)) + 1.0d0))
end function

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (k / t) ** 2.0d0
    if ((2.0d0 / (((((t ** 3.0d0) / (l * l)) * sin(k)) * tan(k)) * (1.0d0 + (1.0d0 + t_1)))) <= 5d+130) then
        tmp = (l * l) * (2.0d0 / (tan(k) * ((2.0d0 + t_1) * ((t ** 3.0d0) * sin(k)))))
    else
        tmp = 2.0d0 * ((cos(k) / t) * (((l / k) ** 2.0d0) / (sin(k) ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double t, double l, double k) {
	return 2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + Math.pow((k / t), 2.0)) + 1.0));
}

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.pow((k / t), 2.0);
	double tmp;
	if ((2.0 / ((((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 5e+130) {
		tmp = (l * l) * (2.0 / (Math.tan(k) * ((2.0 + t_1) * (Math.pow(t, 3.0) * Math.sin(k)))));
	} else {
		tmp = 2.0 * ((Math.cos(k) / t) * (Math.pow((l / k), 2.0) / Math.pow(Math.sin(k), 2.0)));
	}
	return tmp;
}

Python

def code(t, l, k):
	return 2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + math.pow((k / t), 2.0)) + 1.0))

↓

def code(t, l, k):
	t_1 = math.pow((k / t), 2.0)
	tmp = 0
	if (2.0 / ((((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * (1.0 + (1.0 + t_1)))) <= 5e+130:
		tmp = (l * l) * (2.0 / (math.tan(k) * ((2.0 + t_1) * (math.pow(t, 3.0) * math.sin(k)))))
	else:
		tmp = 2.0 * ((math.cos(k) / t) * (math.pow((l / k), 2.0) / math.pow(math.sin(k), 2.0)))
	return tmp

Julia

function code(t, l, k)
	return Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + (Float64(k / t) ^ 2.0)) + 1.0)))
end

↓

function code(t, l, k)
	t_1 = Float64(k / t) ^ 2.0
	tmp = 0.0
	if (Float64(2.0 / Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(1.0 + Float64(1.0 + t_1)))) <= 5e+130)
		tmp = Float64(Float64(l * l) * Float64(2.0 / Float64(tan(k) * Float64(Float64(2.0 + t_1) * Float64((t ^ 3.0) * sin(k))))));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / t) * Float64((Float64(l / k) ^ 2.0) / (sin(k) ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp = code(t, l, k)
	tmp = 2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + ((k / t) ^ 2.0)) + 1.0));
end

↓

function tmp_2 = code(t, l, k)
	t_1 = (k / t) ^ 2.0;
	tmp = 0.0;
	if ((2.0 / (((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * (1.0 + (1.0 + t_1)))) <= 5e+130)
		tmp = (l * l) * (2.0 / (tan(k) * ((2.0 + t_1) * ((t ^ 3.0) * sin(k)))));
	else
		tmp = 2.0 * ((cos(k) / t) * (((l / k) ^ 2.0) / (sin(k) ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[t_, l_, k_] := N[(2.0 / N[(N[(N[(N[(N[Power[t, 3.0], $MachinePrecision] / N[(l * l), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[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[(1.0 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+130], N[(N[(l * l), $MachinePrecision] * N[(2.0 / N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + t$95$1), $MachinePrecision] * N[(N[Power[t, 3.0], $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / t), $MachinePrecision] * N[(N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))) < 4.9999999999999996e130

   1. Initial program 85.2%

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

   2. Simplified 86.4%

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

      [Start] 85.2%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 85.1%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 86.4%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 86.4%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 86.4%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 86.4%	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 86.4%	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 86.4%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 86.4%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 86.4%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 86.4%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   if 4.9999999999999996e130 < (/.f64 2 (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))

   1. Initial program 21.2%

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

   2. Simplified 21.3%

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

      [Start] 21.2%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l/ [<=] 21.2%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}} \]
      associate-*l/ [=>] 21.2%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}} \cdot \tan k} \]
      associate-*l/ [=>] 21.2%	\[ \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\color{blue}{\frac{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}{\ell \cdot \ell}}} \]
      associate-/r/ [=>] 21.3%	\[ \color{blue}{\frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k} \cdot \left(\ell \cdot \ell\right)} \]
      *-commutative [=>] 21.3%	\[ \color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\frac{2}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}}{\left({t}^{3} \cdot \sin k\right) \cdot \tan k}} \]
      associate-/l/ [=>] 21.3%	\[ \left(\ell \cdot \ell\right) \cdot \color{blue}{\frac{2}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}} \]
      associate-*r* [<=] 21.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left({t}^{3} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      *-commutative [=>] 21.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\left({t}^{3} \cdot \sin k\right) \cdot \color{blue}{\left(\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \tan k\right)}} \]
      associate-*r* [=>] 21.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right) \cdot \tan k}} \]
      *-commutative [=>] 21.3%	\[ \left(\ell \cdot \ell\right) \cdot \frac{2}{\color{blue}{\tan k \cdot \left(\left({t}^{3} \cdot \sin k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]

   3. Taylor expanded in k around inf 59.7%

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

   4. Simplified 80.0%

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

      [Start] 59.7%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 59.7%	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 60.1%	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 60.1%	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 60.1%	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 80.0%	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 80.0%	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Taylor expanded in l around 0 59.7%

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

   6. Simplified 81.6%

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

      [Start] 59.7%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 59.7%	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [<=] 59.7%	\[ 2 \cdot \frac{{\ell}^{2} \cdot \cos k}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]
      times-frac [=>] 60.1%	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)} \]
      unpow2 [=>] 60.1%	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]
      unpow2 [=>] 60.1%	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]
      times-frac [=>] 80.0%	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]
      unpow2 [<=] 80.0%	\[ 2 \cdot \left(\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]
      associate-*r/ [=>] 80.0%	\[ 2 \cdot \color{blue}{\frac{{\left(\frac{\ell}{k}\right)}^{2} \cdot \cos k}{t \cdot {\sin k}^{2}}} \]
      times-frac [=>] 81.6%	\[ 2 \cdot \color{blue}{\left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)} \]

   7. Applied egg-rr 81.6%

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

      [Start] 81.6%	\[ 2 \cdot \left(\frac{{\left(\frac{\ell}{k}\right)}^{2}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]
      unpow2 [=>] 81.6%	\[ 2 \cdot \left(\frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right) \]

   8. Taylor expanded in l around 0 59.7%

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

   9. Simplified 81.7%

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

      [Start] 59.7%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/r* [=>] 60.2%	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2}}}{{\sin k}^{2} \cdot t}} \]
      associate-*r/ [<=] 60.2%	\[ 2 \cdot \frac{\color{blue}{\cos k \cdot \frac{{\ell}^{2}}{{k}^{2}}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 60.2%	\[ 2 \cdot \frac{\cos k \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}{{\sin k}^{2} \cdot t} \]
      unpow2 [=>] 60.2%	\[ 2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}{{\sin k}^{2} \cdot t} \]
      times-frac [=>] 80.0%	\[ 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}}{{\sin k}^{2} \cdot t} \]
      unpow2 [<=] 80.0%	\[ 2 \cdot \frac{\cos k \cdot \color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}{{\sin k}^{2} \cdot t} \]
      *-commutative [<=] 80.0%	\[ 2 \cdot \frac{\cos k \cdot {\left(\frac{\ell}{k}\right)}^{2}}{\color{blue}{t \cdot {\sin k}^{2}}} \]
      times-frac [=>] 81.7%	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{t} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{\sin k}^{2}}\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	78.9%
Cost	53828

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + \left(1 + t_1\right)\right)} \leq 5 \cdot 10^{+130}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(\left(2 + t_1\right) \cdot \left({t}^{3} \cdot \sin k\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{t} \cdot \frac{{\left(\frac{\ell}{k}\right)}^{2}}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 2
Accuracy	76.7%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;t \leq -1.65 \cdot 10^{+166} \lor \neg \left(t \leq 4.2 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \end{array} \]

Alternative 3
Accuracy	77.1%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;t \leq -9.6 \cdot 10^{+168} \lor \neg \left(t \leq 4 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{\cos k}{{\sin k}^{2}}\right)\\ \end{array} \]

Alternative 4
Accuracy	79.4%
Cost	14540

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ \mathbf{if}\;k \leq -190000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{+127}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	79.4%
Cost	14540

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\frac{\cos k}{0.5 + \cos \left(k \cdot -2\right) \cdot -0.5}}{t}\right)\\ \mathbf{if}\;k \leq -190000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{elif}\;k \leq 3 \cdot 10^{+125}:\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	71.9%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -190000000000 \lor \neg \left(k \leq 2.9 \cdot 10^{-21}\right):\\ \;\;\;\;\left(\ell \cdot \ell\right) \cdot \frac{2}{\tan k \cdot \left(t \cdot \left(\sin k \cdot \left(k \cdot k\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \end{array} \]

Alternative 7
Accuracy	66.0%
Cost	7305

\[\begin{array}{l} \mathbf{if}\;t \leq -3.55 \cdot 10^{+165} \lor \neg \left(t \leq 1.85 \cdot 10^{-28}\right):\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{{t}^{3} \cdot k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	63.2%
Cost	1481

\[\begin{array}{l} \mathbf{if}\;t \leq -3.3 \cdot 10^{+165} \lor \neg \left(t \leq 2.05 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \left(\frac{1}{k \cdot k} + -0.16666666666666666\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	60.5%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{+192} \lor \neg \left(t \leq 4 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{k \cdot \left(t \cdot k\right)}\right)\\ \end{array} \]

Alternative 10
Accuracy	60.5%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+189} \lor \neg \left(t \leq 4 \cdot 10^{+38}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{1}{t \cdot \left(k \cdot k\right)}\right)\\ \end{array} \]

Alternative 11
Accuracy	61.9%
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+194} \lor \neg \left(t \leq 5.8 \cdot 10^{+40}\right):\\ \;\;\;\;\frac{\ell \cdot \ell}{k \cdot \left(k \cdot \left(t \cdot \left(t \cdot t\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t} \cdot \frac{1}{k \cdot k}\right)\\ \end{array} \]

Alternative 12
Accuracy	54.8%
Cost	832

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

Reproduce

herbie shell --seed 2023165 
(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))))