Toniolo and Linder, Equation (10+)

Average Error: 33.1 → 17.3

Time: 32.0s

Precision: binary64

Cost: 27080

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -1.86 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3} \cdot \frac{\tan k}{\ell}}}{\frac{\sin k \cdot t_1}{\ell}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \left(t \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \frac{\frac{\ell}{\sin k}}{t_1}\\ \end{array} \]

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 (+ 2.0 (pow (/ k t) 2.0))))
   (if (<= t -1.86e-38)
     (/ (/ 2.0 (* (pow t 3.0) (/ (tan k) l))) (/ (* (sin k) t_1) l))
     (if (<= t 3.8e-95)
       (/ (* l (/ 2.0 (sin k))) (* (tan k) (* t (/ (pow k 2.0) l))))
       (* (/ (/ l (/ (tan k) 2.0)) (pow t 3.0)) (/ (/ l (sin 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 = 2.0 + pow((k / t), 2.0);
	double tmp;
	if (t <= -1.86e-38) {
		tmp = (2.0 / (pow(t, 3.0) * (tan(k) / l))) / ((sin(k) * t_1) / l);
	} else if (t <= 3.8e-95) {
		tmp = (l * (2.0 / sin(k))) / (tan(k) * (t * (pow(k, 2.0) / l)));
	} else {
		tmp = ((l / (tan(k) / 2.0)) / pow(t, 3.0)) * ((l / sin(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) :: tmp
    t_1 = 2.0d0 + ((k / t) ** 2.0d0)
    if (t <= (-1.86d-38)) then
        tmp = (2.0d0 / ((t ** 3.0d0) * (tan(k) / l))) / ((sin(k) * t_1) / l)
    else if (t <= 3.8d-95) then
        tmp = (l * (2.0d0 / sin(k))) / (tan(k) * (t * ((k ** 2.0d0) / l)))
    else
        tmp = ((l / (tan(k) / 2.0d0)) / (t ** 3.0d0)) * ((l / sin(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 = 2.0 + Math.pow((k / t), 2.0);
	double tmp;
	if (t <= -1.86e-38) {
		tmp = (2.0 / (Math.pow(t, 3.0) * (Math.tan(k) / l))) / ((Math.sin(k) * t_1) / l);
	} else if (t <= 3.8e-95) {
		tmp = (l * (2.0 / Math.sin(k))) / (Math.tan(k) * (t * (Math.pow(k, 2.0) / l)));
	} else {
		tmp = ((l / (Math.tan(k) / 2.0)) / Math.pow(t, 3.0)) * ((l / Math.sin(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 = 2.0 + math.pow((k / t), 2.0)
	tmp = 0
	if t <= -1.86e-38:
		tmp = (2.0 / (math.pow(t, 3.0) * (math.tan(k) / l))) / ((math.sin(k) * t_1) / l)
	elif t <= 3.8e-95:
		tmp = (l * (2.0 / math.sin(k))) / (math.tan(k) * (t * (math.pow(k, 2.0) / l)))
	else:
		tmp = ((l / (math.tan(k) / 2.0)) / math.pow(t, 3.0)) * ((l / math.sin(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 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	tmp = 0.0
	if (t <= -1.86e-38)
		tmp = Float64(Float64(2.0 / Float64((t ^ 3.0) * Float64(tan(k) / l))) / Float64(Float64(sin(k) * t_1) / l));
	elseif (t <= 3.8e-95)
		tmp = Float64(Float64(l * Float64(2.0 / sin(k))) / Float64(tan(k) * Float64(t * Float64((k ^ 2.0) / l))));
	else
		tmp = Float64(Float64(Float64(l / Float64(tan(k) / 2.0)) / (t ^ 3.0)) * Float64(Float64(l / sin(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 = 2.0 + ((k / t) ^ 2.0);
	tmp = 0.0;
	if (t <= -1.86e-38)
		tmp = (2.0 / ((t ^ 3.0) * (tan(k) / l))) / ((sin(k) * t_1) / l);
	elseif (t <= 3.8e-95)
		tmp = (l * (2.0 / sin(k))) / (tan(k) * (t * ((k ^ 2.0) / l)));
	else
		tmp = ((l / (tan(k) / 2.0)) / (t ^ 3.0)) * ((l / sin(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[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.86e-38], N[(N[(2.0 / N[(N[Power[t, 3.0], $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * t$95$1), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 3.8e-95], N[(N[(l * N[(2.0 / N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Tan[k], $MachinePrecision] * N[(t * N[(N[Power[k, 2.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(N[Tan[k], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] / N[Power[t, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.85999999999999995e-38

   1. Initial program 22.4

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

   2. Simplified 18.2

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

      [Start] 22.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-46 [=>] 22.4	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-46 [=>] 22.4	\[ \frac{\color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-44 [=>] 22.4	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 22.4	\[ \frac{\color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-61 [=>] 21.7	\[ \frac{\frac{\color{blue}{\frac{\ell \cdot \ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-49 [=>] 18.2	\[ \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-1 [=>] 18.2	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-1 [=>] 18.2	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]
      rational.json-simplify-41 [=>] 18.2	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]
      metadata-eval [=>] 18.2	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \]
      rational.json-simplify-1 [=>] 18.2	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 23.7

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

   4. Simplified 16.3

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

      [Start] 23.7	\[ \ell \cdot \frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\tan k \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 23.7	\[ \color{blue}{\ell \cdot \frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\tan k \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-44 [=>] 23.7	\[ \ell \cdot \color{blue}{\frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k}} \]
      rational.json-simplify-7 [<=] 23.7	\[ \ell \cdot \frac{\frac{\color{blue}{\frac{\frac{\ell + \ell}{{t}^{3}}}{1}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      metadata-eval [<=] 23.7	\[ \ell \cdot \frac{\frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\color{blue}{\frac{2}{2}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-61 [<=] 23.7	\[ \ell \cdot \frac{\frac{\color{blue}{\frac{2}{\frac{2}{\frac{\ell + \ell}{{t}^{3}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-61 [=>] 23.7	\[ \ell \cdot \frac{\frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell + \ell}{2}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      metadata-eval [<=] 23.7	\[ \ell \cdot \frac{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell + \ell}{\color{blue}{1 + 1}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-35 [<=] 23.7	\[ \ell \cdot \frac{\frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\frac{\ell}{1}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-7 [=>] 23.7	\[ \ell \cdot \frac{\frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-44 [<=] 23.7	\[ \ell \cdot \color{blue}{\frac{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\tan k \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-47 [<=] 18.2	\[ \ell \cdot \frac{\color{blue}{\frac{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\tan k}}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-44 [<=] 18.3	\[ \ell \cdot \frac{\frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}}}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

   5. Applied egg-rr 16.3

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

   if -1.85999999999999995e-38 < t < 3.7999999999999997e-95

   1. Initial program 57.7

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

   2. Simplified 56.9

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

      [Start] 57.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-46 [=>] 57.8	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-2 [=>] 57.8	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 57.8	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-2 [=>] 57.8	\[ \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 57.8	\[ \frac{\color{blue}{\frac{\frac{\frac{2}{\tan k}}{\sin k}}{\frac{{t}^{3}}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 56.9	\[ \frac{\frac{\frac{\frac{2}{\tan k}}{\sin k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-61 [=>] 56.9	\[ \frac{\color{blue}{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\tan k}}{\sin k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-44 [=>] 56.9	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\color{blue}{\frac{\frac{2}{\sin k}}{\tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-1 [=>] 56.9	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-1 [=>] 56.9	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]
      rational.json-simplify-41 [=>] 56.9	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]
      metadata-eval [=>] 56.9	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \]
      rational.json-simplify-1 [=>] 56.9	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 56.8

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

   4. Applied egg-rr 56.0

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

   5. Simplified 55.5

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

      [Start] 56.0	\[ \frac{\frac{\frac{\ell \cdot \frac{2}{\sin k}}{\tan k}}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 56.0	\[ \color{blue}{\frac{\frac{\frac{\ell \cdot \frac{2}{\sin k}}{\tan k}}{\frac{{t}^{3}}{\ell}}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-47 [=>] 55.6	\[ \color{blue}{\frac{\frac{\ell \cdot \frac{2}{\sin k}}{\tan k}}{\frac{{t}^{3}}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-47 [=>] 55.5	\[ \color{blue}{\frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \left(\frac{{t}^{3}}{\ell} \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   6. Taylor expanded in t around 0 21.6

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

   7. Simplified 19.9

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

      [Start] 21.6	\[ \frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \frac{{k}^{2} \cdot t}{\ell}} \]
      rational.json-simplify-49 [=>] 19.9	\[ \frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \color{blue}{\left(t \cdot \frac{{k}^{2}}{\ell}\right)}} \]

   if 3.7999999999999997e-95 < t

   1. Initial program 23.9

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

   2. Simplified 19.3

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

      [Start] 23.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-46 [=>] 23.9	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-46 [=>] 23.9	\[ \frac{\color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-44 [=>] 23.9	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 23.9	\[ \frac{\color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-61 [=>] 23.4	\[ \frac{\frac{\color{blue}{\frac{\ell \cdot \ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-49 [=>] 19.3	\[ \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-1 [=>] 19.3	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-1 [=>] 19.3	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]
      rational.json-simplify-41 [=>] 19.3	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]
      metadata-eval [=>] 19.3	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \]
      rational.json-simplify-1 [=>] 19.3	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 23.2

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

   4. Simplified 16.3

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

      [Start] 23.2	\[ \ell \cdot \frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\tan k \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} + 0 \]
      rational.json-simplify-4 [=>] 23.2	\[ \color{blue}{\ell \cdot \frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\tan k \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-44 [=>] 23.2	\[ \ell \cdot \color{blue}{\frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k}} \]
      rational.json-simplify-7 [<=] 23.2	\[ \ell \cdot \frac{\frac{\color{blue}{\frac{\frac{\ell + \ell}{{t}^{3}}}{1}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      metadata-eval [<=] 23.2	\[ \ell \cdot \frac{\frac{\frac{\frac{\ell + \ell}{{t}^{3}}}{\color{blue}{\frac{2}{2}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-61 [<=] 23.3	\[ \ell \cdot \frac{\frac{\color{blue}{\frac{2}{\frac{2}{\frac{\ell + \ell}{{t}^{3}}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-61 [=>] 23.3	\[ \ell \cdot \frac{\frac{\frac{2}{\color{blue}{\frac{{t}^{3}}{\frac{\ell + \ell}{2}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      metadata-eval [<=] 23.3	\[ \ell \cdot \frac{\frac{\frac{2}{\frac{{t}^{3}}{\frac{\ell + \ell}{\color{blue}{1 + 1}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-35 [<=] 23.3	\[ \ell \cdot \frac{\frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\frac{\ell}{1}}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-7 [=>] 23.3	\[ \ell \cdot \frac{\frac{\frac{2}{\frac{{t}^{3}}{\color{blue}{\ell}}}}{2 + {\left(\frac{k}{t}\right)}^{2}}}{\tan k \cdot \sin k} \]
      rational.json-simplify-44 [<=] 23.3	\[ \ell \cdot \color{blue}{\frac{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\tan k \cdot \sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]
      rational.json-simplify-47 [<=] 18.3	\[ \ell \cdot \frac{\color{blue}{\frac{\frac{\frac{2}{\frac{{t}^{3}}{\ell}}}{\tan k}}{\sin k}}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]
      rational.json-simplify-44 [<=] 18.3	\[ \ell \cdot \frac{\frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell}}}}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 17.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.86 \cdot 10^{-38}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3} \cdot \frac{\tan k}{\ell}}}{\frac{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}{\ell}}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \left(t \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \frac{\frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Alternatives

Alternative 1
Error	18.3
Cost	27080

\[\begin{array}{l} t_1 := \ell \cdot \left(2 \cdot \frac{\frac{\frac{\frac{\ell}{\sin k}}{{t}^{3}}}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\\ \mathbf{if}\;t \leq -51:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-91}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \left(t \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	17.3
Cost	27080

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \frac{\frac{\ell}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 7.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \left(t \cdot \frac{{k}^{2}}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	22.1
Cost	20752

\[\begin{array}{l} t_1 := \left(2 \cdot \left(\frac{\cos k \cdot \ell}{{\left(k \cdot \sin k\right)}^{2}} \cdot \frac{1}{t}\right)\right) \cdot \ell\\ t_2 := \frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}}\\ \mathbf{if}\;k \leq -2.85 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-66}:\\ \;\;\;\;t_2 \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-25}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \left({k}^{2} \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 4 \cdot 10^{+39}:\\ \;\;\;\;t_2 \cdot \left(0.5 \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.6
Cost	20624

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ t_2 := {\left(k \cdot \sin k\right)}^{2}\\ t_3 := \cos k \cdot \frac{\frac{\ell}{t_2}}{\frac{t}{\ell + \ell}}\\ \mathbf{if}\;k \leq -3 \cdot 10^{-34}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{-25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 5.3 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \frac{\frac{\cos k}{t}}{t_2}\right)\right) \cdot \ell\\ \end{array} \]

Alternative 5
Error	21.3
Cost	20624

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ t_2 := {\left(k \cdot \sin k\right)}^{2}\\ \mathbf{if}\;k \leq -3 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \frac{\cos k \cdot \frac{\ell}{t}}{t_2}\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{-25}:\\ \;\;\;\;\cos k \cdot \frac{\frac{\ell}{t_2}}{\frac{t}{\ell + \ell}}\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \frac{\frac{\cos k}{t}}{t_2}\right)\right) \cdot \ell\\ \end{array} \]

Alternative 6
Error	21.3
Cost	20624

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}}\\ t_2 := {\left(k \cdot \sin k\right)}^{2}\\ \mathbf{if}\;k \leq -3 \cdot 10^{-34}:\\ \;\;\;\;\left(\ell + \ell\right) \cdot \frac{\cos k \cdot \frac{\ell}{t}}{t_2}\\ \mathbf{elif}\;k \leq 2.3 \cdot 10^{-66}:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 6.6 \cdot 10^{-25}:\\ \;\;\;\;\cos k \cdot \frac{\frac{\ell}{t_2}}{\frac{t}{\ell + \ell}}\\ \mathbf{elif}\;k \leq 0.0105:\\ \;\;\;\;t_1 \cdot \left(0.5 \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\ell \cdot \frac{\frac{\cos k}{t}}{t_2}\right)\right) \cdot \ell\\ \end{array} \]

Alternative 7
Error	22.1
Cost	20624

\[\begin{array}{l} t_1 := \frac{\ell \cdot \frac{2}{\sin k}}{\tan k \cdot \left(t \cdot \frac{{k}^{2}}{\ell}\right)}\\ t_2 := \frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}}\\ \mathbf{if}\;k \leq -2.85 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;t_2 \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 3.5 \cdot 10^{-25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{+41}:\\ \;\;\;\;t_2 \cdot \left(0.5 \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	22.0
Cost	20624

\[\begin{array}{l} t_1 := \ell \cdot \frac{2}{\sin k}\\ t_2 := \frac{t_1}{\tan k \cdot \left(t \cdot \frac{{k}^{2}}{\ell}\right)}\\ t_3 := \frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}}\\ \mathbf{if}\;k \leq -9.5 \cdot 10^{-36}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;t_3 \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{t_1}{\tan k \cdot \left({k}^{2} \cdot \frac{t}{\ell}\right)}\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{+39}:\\ \;\;\;\;t_3 \cdot \left(0.5 \cdot \frac{\ell}{\sin k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 9
Error	22.2
Cost	20360

\[\begin{array}{l} t_1 := \cos k \cdot \frac{\frac{\ell}{{\left(k \cdot \sin k\right)}^{2}}}{\frac{t}{\ell + \ell}}\\ \mathbf{if}\;k \leq -1.9 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	26.9
Cost	20296

\[\begin{array}{l} \mathbf{if}\;k \leq -2.85 \cdot 10^{-34}:\\ \;\;\;\;\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \ell\\ \mathbf{elif}\;k \leq 5.6 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \frac{\frac{\ell}{{k}^{2}}}{{\sin k}^{2} \cdot t}\right) \cdot \ell\\ \end{array} \]

Alternative 11
Error	26.7
Cost	14280

\[\begin{array}{l} t_1 := \frac{\ell}{{k}^{4} \cdot t}\\ \mathbf{if}\;k \leq -3 \cdot 10^{-34}:\\ \;\;\;\;\left(2 \cdot t_1\right) \cdot \ell\\ \mathbf{elif}\;k \leq 0.00033:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(t_1 + \frac{\ell}{{k}^{2} \cdot t} \cdot -0.16666666666666666\right)\right) \cdot \ell\\ \end{array} \]

Alternative 12
Error	26.7
Cost	13960

\[\begin{array}{l} t_1 := \left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \ell\\ \mathbf{if}\;k \leq -2.9 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6800:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	26.5
Cost	13960

\[\begin{array}{l} \mathbf{if}\;t \leq -3.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{\tan k}{2}}}{{t}^{3}} \cdot \left(0.5 \cdot \frac{\ell}{k}\right)\\ \mathbf{elif}\;t \leq 380000000:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\sin k}}{{t}^{3} \cdot \left(2 \cdot \frac{k}{\ell}\right)}\\ \end{array} \]

Alternative 14
Error	26.5
Cost	13960

\[\begin{array}{l} t_1 := 2 \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-43}:\\ \;\;\;\;\frac{\frac{2}{{t}^{3} \cdot \frac{\tan k}{\ell}}}{t_1}\\ \mathbf{elif}\;t \leq 380000000:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{\sin k}}{{t}^{3} \cdot t_1}\\ \end{array} \]

Alternative 15
Error	30.4
Cost	13896

\[\begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{\ell}{{t}^{3} \cdot {k}^{2}} \cdot \ell\\ \mathbf{elif}\;t \leq 6.7 \cdot 10^{-75}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{2 \cdot \left({t}^{3} \cdot \frac{{k}^{2}}{\ell}\right)}\\ \end{array} \]

Alternative 16
Error	30.4
Cost	13896

\[\begin{array}{l} \mathbf{if}\;t \leq -2.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{\ell}{{t}^{3} \cdot {k}^{2}} \cdot \ell\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell + \ell}{{k}^{2} \cdot \frac{{t}^{3}}{\ell \cdot 0.5}}\\ \end{array} \]

Alternative 17
Error	30.5
Cost	13640

\[\begin{array}{l} t_1 := \frac{\ell}{{t}^{3} \cdot {k}^{2}} \cdot \ell\\ \mathbf{if}\;t \leq -2.05 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-73}:\\ \;\;\;\;\frac{\ell + \ell}{t \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 18
Error	37.5
Cost	7040

\[\left(2 \cdot \frac{\ell}{{k}^{4} \cdot t}\right) \cdot \ell \]

Alternative 19
Error	36.9
Cost	7040

\[\frac{\ell + \ell}{t \cdot \frac{{k}^{4}}{\ell}} \]

Alternative 20
Error	36.8
Cost	7040

\[\frac{\ell + \ell}{{k}^{4} \cdot \frac{t}{\ell}} \]

Reproduce

herbie shell --seed 2023075 
(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))))