Toniolo and Linder, Equation (10+)

Average Error: 33.0 → 8.1

Time: 42.1s

Precision: binary64

Cost: 20872

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_3 := t \cdot t_1\\ t_4 := 2 \cdot \left(t_2 \cdot \frac{\cos k}{t_3}\right)\\ \mathbf{if}\;k \leq -9.2 \cdot 10^{+63}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\tan k} \cdot \frac{\ell}{t}}{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 2.95 \cdot 10^{+181}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+204}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{t_1 \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_3}{t_2}}\\ \end{array} \]

FPCore

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

↓

(FPCore (t l k)
 :precision binary64
 (let* ((t_1 (pow (sin k) 2.0))
        (t_2 (* (/ l k) (/ l k)))
        (t_3 (* t t_1))
        (t_4 (* 2.0 (* t_2 (/ (cos k) t_3)))))
   (if (<= k -9.2e+63)
     t_4
     (if (<= k 2.05e+96)
       (/
        (* (/ (/ 2.0 t) (tan k)) (/ l t))
        (* (* (sin k) (+ 2.0 (pow (/ k t) 2.0))) (/ t l)))
       (if (<= k 2.95e+181)
         t_4
         (if (<= k 5.5e+204)
           (/ (* 2.0 (* (cos k) (* l l))) (* t_1 (* k (* k t))))
           (* 2.0 (/ (cos k) (/ t_3 t_2)))))))))

C

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

↓

double code(double t, double l, double k) {
	double t_1 = pow(sin(k), 2.0);
	double t_2 = (l / k) * (l / k);
	double t_3 = t * t_1;
	double t_4 = 2.0 * (t_2 * (cos(k) / t_3));
	double tmp;
	if (k <= -9.2e+63) {
		tmp = t_4;
	} else if (k <= 2.05e+96) {
		tmp = (((2.0 / t) / tan(k)) * (l / t)) / ((sin(k) * (2.0 + pow((k / t), 2.0))) * (t / l));
	} else if (k <= 2.95e+181) {
		tmp = t_4;
	} else if (k <= 5.5e+204) {
		tmp = (2.0 * (cos(k) * (l * l))) / (t_1 * (k * (k * t)));
	} else {
		tmp = 2.0 * (cos(k) / (t_3 / t_2));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = sin(k) ** 2.0d0
    t_2 = (l / k) * (l / k)
    t_3 = t * t_1
    t_4 = 2.0d0 * (t_2 * (cos(k) / t_3))
    if (k <= (-9.2d+63)) then
        tmp = t_4
    else if (k <= 2.05d+96) then
        tmp = (((2.0d0 / t) / tan(k)) * (l / t)) / ((sin(k) * (2.0d0 + ((k / t) ** 2.0d0))) * (t / l))
    else if (k <= 2.95d+181) then
        tmp = t_4
    else if (k <= 5.5d+204) then
        tmp = (2.0d0 * (cos(k) * (l * l))) / (t_1 * (k * (k * t)))
    else
        tmp = 2.0d0 * (cos(k) / (t_3 / t_2))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double t, double l, double k) {
	double t_1 = Math.pow(Math.sin(k), 2.0);
	double t_2 = (l / k) * (l / k);
	double t_3 = t * t_1;
	double t_4 = 2.0 * (t_2 * (Math.cos(k) / t_3));
	double tmp;
	if (k <= -9.2e+63) {
		tmp = t_4;
	} else if (k <= 2.05e+96) {
		tmp = (((2.0 / t) / Math.tan(k)) * (l / t)) / ((Math.sin(k) * (2.0 + Math.pow((k / t), 2.0))) * (t / l));
	} else if (k <= 2.95e+181) {
		tmp = t_4;
	} else if (k <= 5.5e+204) {
		tmp = (2.0 * (Math.cos(k) * (l * l))) / (t_1 * (k * (k * t)));
	} else {
		tmp = 2.0 * (Math.cos(k) / (t_3 / t_2));
	}
	return tmp;
}

Python

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

↓

def code(t, l, k):
	t_1 = math.pow(math.sin(k), 2.0)
	t_2 = (l / k) * (l / k)
	t_3 = t * t_1
	t_4 = 2.0 * (t_2 * (math.cos(k) / t_3))
	tmp = 0
	if k <= -9.2e+63:
		tmp = t_4
	elif k <= 2.05e+96:
		tmp = (((2.0 / t) / math.tan(k)) * (l / t)) / ((math.sin(k) * (2.0 + math.pow((k / t), 2.0))) * (t / l))
	elif k <= 2.95e+181:
		tmp = t_4
	elif k <= 5.5e+204:
		tmp = (2.0 * (math.cos(k) * (l * l))) / (t_1 * (k * (k * t)))
	else:
		tmp = 2.0 * (math.cos(k) / (t_3 / t_2))
	return tmp

Julia

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

↓

function code(t, l, k)
	t_1 = sin(k) ^ 2.0
	t_2 = Float64(Float64(l / k) * Float64(l / k))
	t_3 = Float64(t * t_1)
	t_4 = Float64(2.0 * Float64(t_2 * Float64(cos(k) / t_3)))
	tmp = 0.0
	if (k <= -9.2e+63)
		tmp = t_4;
	elseif (k <= 2.05e+96)
		tmp = Float64(Float64(Float64(Float64(2.0 / t) / tan(k)) * Float64(l / t)) / Float64(Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0))) * Float64(t / l)));
	elseif (k <= 2.95e+181)
		tmp = t_4;
	elseif (k <= 5.5e+204)
		tmp = Float64(Float64(2.0 * Float64(cos(k) * Float64(l * l))) / Float64(t_1 * Float64(k * Float64(k * t))));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(t_3 / t_2)));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(t, l, k)
	t_1 = sin(k) ^ 2.0;
	t_2 = (l / k) * (l / k);
	t_3 = t * t_1;
	t_4 = 2.0 * (t_2 * (cos(k) / t_3));
	tmp = 0.0;
	if (k <= -9.2e+63)
		tmp = t_4;
	elseif (k <= 2.05e+96)
		tmp = (((2.0 / t) / tan(k)) * (l / t)) / ((sin(k) * (2.0 + ((k / t) ^ 2.0))) * (t / l));
	elseif (k <= 2.95e+181)
		tmp = t_4;
	elseif (k <= 5.5e+204)
		tmp = (2.0 * (cos(k) * (l * l))) / (t_1 * (k * (k * t)));
	else
		tmp = 2.0 * (cos(k) / (t_3 / t_2));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[t_, l_, k_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(t$95$2 * N[(N[Cos[k], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -9.2e+63], t$95$4, If[LessEqual[k, 2.05e+96], N[(N[(N[(N[(2.0 / t), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.95e+181], t$95$4, If[LessEqual[k, 5.5e+204], N[(N[(2.0 * N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(k * N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(t$95$3 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -9.19999999999999973e63 or 2.04999999999999999e96 < k < 2.9499999999999999e181

   1. Initial program 34.2

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

   2. Simplified 34.2

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

      [Start] 34.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 34.2	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 34.2	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 34.2	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 34.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 34.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 34.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 34.2	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 20.0

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

   4. Simplified 8.7

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

      [Start] 20.0	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 20.0	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 21.7	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 21.7	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 21.7	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 8.7	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 8.7	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   if -9.19999999999999973e63 < k < 2.04999999999999999e96

   1. Initial program 31.9

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

   2. Simplified 38.4

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

      [Start] 31.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 31.9	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 31.9	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 31.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 31.8	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 38.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 38.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 38.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 38.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 35.5

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

   4. Applied egg-rr 21.2

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

   5. Applied egg-rr 15.9

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

   6. Applied egg-rr 8.0

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

   if 2.9499999999999999e181 < k < 5.4999999999999996e204

   1. Initial program 33.6

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

   2. Simplified 33.6

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

      [Start] 33.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 33.6	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 33.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 33.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 33.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 33.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 33.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 33.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 33.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 31.7

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

   4. Applied egg-rr 18.9

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

   5. Applied egg-rr 17.3

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

   6. Taylor expanded in t around 0 27.2

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

   7. Simplified 19.6

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

      [Start] 27.2	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 27.2	\[ \color{blue}{\frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \cdot 2} \]
      associate-*l/ [=>] 27.2	\[ \color{blue}{\frac{\left(\cos k \cdot {\ell}^{2}\right) \cdot 2}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}} \]
      unpow2 [=>] 27.2	\[ \frac{\left(\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}\right) \cdot 2}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-*r* [=>] 27.2	\[ \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{\color{blue}{\left({k}^{2} \cdot {\sin k}^{2}\right) \cdot t}} \]
      *-commutative [=>] 27.2	\[ \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{\color{blue}{\left({\sin k}^{2} \cdot {k}^{2}\right)} \cdot t} \]
      associate-*l* [=>] 27.2	\[ \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{\color{blue}{{\sin k}^{2} \cdot \left({k}^{2} \cdot t\right)}} \]
      unpow2 [=>] 27.2	\[ \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{{\sin k}^{2} \cdot \left(\color{blue}{\left(k \cdot k\right)} \cdot t\right)} \]
      associate-*l* [=>] 19.6	\[ \frac{\left(\cos k \cdot \left(\ell \cdot \ell\right)\right) \cdot 2}{{\sin k}^{2} \cdot \color{blue}{\left(k \cdot \left(k \cdot t\right)\right)}} \]

   if 5.4999999999999996e204 < k

   1. Initial program 34.2

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

   2. Simplified 34.2

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

      [Start] 34.2	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 34.2	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 34.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Taylor expanded in t around 0 22.8

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

   4. Simplified 4.9

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

      [Start] 22.8	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 22.8	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{{\ell}^{2}}}} \]
      *-commutative [=>] 22.8	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot {k}^{2}}}{{\ell}^{2}}} \]
      associate-/l* [=>] 22.8	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{{\ell}^{2}}{{k}^{2}}}}} \]
      unpow2 [=>] 22.8	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}}} \]
      unpow2 [=>] 22.8	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}}} \]
      times-frac [=>] 4.9	\[ 2 \cdot \frac{\cos k}{\frac{{\sin k}^{2} \cdot t}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 8.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -9.2 \cdot 10^{+63}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq 2.05 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{\frac{2}{t}}{\tan k} \cdot \frac{\ell}{t}}{\left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 2.95 \cdot 10^{+181}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{+204}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{{\sin k}^{2} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.6
Cost	21136

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\frac{\frac{2}{t \cdot \tan k}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{t} \cdot \left(\ell \cdot \frac{\ell}{t}\right)\\ t_3 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_4 := t \cdot t_1\\ t_5 := 2 \cdot \left(t_3 \cdot \frac{\cos k}{t_4}\right)\\ t_6 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -4.2 \cdot 10^{+63}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq -2 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-161}:\\ \;\;\;\;\frac{\ell}{t_6 \cdot \frac{k}{\frac{\ell}{{t_6}^{2}}}}\\ \mathbf{elif}\;k \leq 3.9 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 5.2 \cdot 10^{+182}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+206}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{t_1 \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_3}}\\ \end{array} \]

Alternative 2
Error	9.7
Cost	21136

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\ell}{t} \cdot \frac{\frac{\frac{2}{t \cdot \tan k}}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}{\frac{t}{\ell}}\\ t_3 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_4 := t \cdot t_1\\ t_5 := 2 \cdot \left(t_3 \cdot \frac{\cos k}{t_4}\right)\\ t_6 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -1.65 \cdot 10^{+82}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq -1.9 \cdot 10^{-142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-153}:\\ \;\;\;\;\frac{\ell}{t_6 \cdot \frac{k}{\frac{\ell}{{t_6}^{2}}}}\\ \mathbf{elif}\;k \leq 2.4 \cdot 10^{+96}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+182}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 4.2 \cdot 10^{+206}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{t_1 \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_3}}\\ \end{array} \]

Alternative 3
Error	13.8
Cost	20884

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := t \cdot \sqrt[3]{k}\\ t_3 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_4 := t \cdot t_1\\ t_5 := 2 \cdot \left(t_3 \cdot \frac{\cos k}{t_4}\right)\\ \mathbf{if}\;k \leq -6.7 \cdot 10^{-40}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 5.3 \cdot 10^{-156}:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{+23}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\cos k \cdot \frac{\frac{\ell}{t}}{t}}{t_1}\\ \mathbf{elif}\;k \leq 2.45 \cdot 10^{+182}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;k \leq 5 \cdot 10^{+205}:\\ \;\;\;\;\frac{2 \cdot \left(\cos k \cdot \left(\ell \cdot \ell\right)\right)}{t_1 \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_3}}\\ \end{array} \]

Alternative 4
Error	13.5
Cost	20620

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot t_1}\right)\\ t_3 := t \cdot \sqrt[3]{k}\\ \mathbf{if}\;k \leq -6.7 \cdot 10^{-40}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{-156}:\\ \;\;\;\;\frac{\ell}{t_3 \cdot \frac{k}{\frac{\ell}{{t_3}^{2}}}}\\ \mathbf{elif}\;k \leq 2.7 \cdot 10^{+23}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\cos k \cdot \frac{\frac{\ell}{t}}{t}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	13.4
Cost	20620

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := t \cdot \sqrt[3]{k}\\ t_3 := \frac{\ell}{k} \cdot \frac{\ell}{k}\\ t_4 := t \cdot t_1\\ \mathbf{if}\;k \leq -6.1 \cdot 10^{-40}:\\ \;\;\;\;2 \cdot \left(t_3 \cdot \frac{\cos k}{t_4}\right)\\ \mathbf{elif}\;k \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\frac{\ell}{t_2 \cdot \frac{k}{\frac{\ell}{{t_2}^{2}}}}\\ \mathbf{elif}\;k \leq 1.85 \cdot 10^{+25}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\cos k \cdot \frac{\frac{\ell}{t}}{t}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t_4}{t_3}}\\ \end{array} \]

Alternative 6
Error	19.3
Cost	14544

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\frac{\frac{2}{t \cdot \tan k}}{2 \cdot \sin k}}{\frac{t}{\ell}}\\ \mathbf{if}\;t \leq -1.08 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -6.7 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 8.8 \cdot 10^{-131}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{elif}\;t \leq 21500000:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{\tan k}}{\frac{k \cdot k}{\frac{t}{\sin k}}}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	19.7
Cost	14284

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\frac{\frac{2}{t \cdot \tan k}}{2 \cdot \sin k}}{\frac{t}{\ell}}\\ \mathbf{if}\;t \leq -1.5 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.7 \cdot 10^{-58}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 2250000:\\ \;\;\;\;\left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right) \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.0
Cost	14284

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\frac{\frac{2}{t \cdot \tan k}}{2 \cdot \sin k}}{\frac{t}{\ell}}\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 7300000:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	19.8
Cost	13644

\[\begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+115}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{t}{\ell}}\\ \mathbf{elif}\;t \leq -1.55 \cdot 10^{-56}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 1.3 \cdot 10^{-36}:\\ \;\;\;\;\left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right) \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\ell}{k \cdot {t}^{1.5}}\right)}^{2}\\ \end{array} \]

Alternative 10
Error	20.6
Cost	7304

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{t}{\ell}}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{+115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -7.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 3.25 \cdot 10^{-37}:\\ \;\;\;\;\left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right) \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	25.7
Cost	1740

\[\begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 6 \cdot 10^{-306}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{-221}:\\ \;\;\;\;\frac{\ell}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\\ \mathbf{elif}\;\ell \cdot \ell \leq 5 \cdot 10^{-157}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{t_1}}{\frac{t}{\ell}}\\ \end{array} \]

Alternative 12
Error	28.5
Cost	1360

\[\begin{array}{l} t_1 := \frac{\ell}{t} \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}\\ \mathbf{if}\;k \leq -1.75 \cdot 10^{-8}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\left(k \cdot t\right) \cdot \frac{k}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq -2.7 \cdot 10^{-146}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k}\\ \mathbf{elif}\;k \leq 0.9:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{\frac{k}{\frac{-0.3333333333333333}{t}}}\\ \end{array} \]

Alternative 13
Error	21.6
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -2.85 \cdot 10^{-12} \lor \neg \left(t \leq 3.9 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{k}}{t}\right) \cdot \left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right)\\ \end{array} \]

Alternative 14
Error	21.7
Cost	1353

\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-12} \lor \neg \left(t \leq 4.5 \cdot 10^{-38}\right):\\ \;\;\;\;\frac{\ell}{t} \cdot \frac{\frac{1}{k \cdot \left(k \cdot t\right)}}{\frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\left(-0.3333333333333333 + \frac{\frac{2}{k}}{k}\right) \cdot \frac{\frac{\ell}{k}}{k \cdot \frac{t}{\ell}}\\ \end{array} \]

Alternative 15
Error	25.7
Cost	1228

\[\begin{array}{l} \mathbf{if}\;k \leq -1.75 \cdot 10^{-8}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\left(k \cdot t\right) \cdot \frac{k}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-160}:\\ \;\;\;\;\frac{\ell}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\\ \mathbf{elif}\;k \leq 0.56:\\ \;\;\;\;\frac{\frac{\frac{\ell}{t}}{k \cdot k}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{\frac{k}{\frac{-0.3333333333333333}{t}}}\\ \end{array} \]

Alternative 16
Error	26.1
Cost	1096

\[\begin{array}{l} \mathbf{if}\;k \leq -1.75 \cdot 10^{-8}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\left(k \cdot t\right) \cdot \frac{k}{\ell}}{\ell}}\\ \mathbf{elif}\;k \leq 0.225:\\ \;\;\;\;\frac{\ell}{t \cdot t} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{\frac{k}{\frac{-0.3333333333333333}{t}}}\\ \end{array} \]

Alternative 17
Error	25.4
Cost	1096

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot t}\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{\ell}{\frac{t \cdot t}{t_1}}\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{-96}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\left(k \cdot t\right) \cdot \frac{k}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{t \cdot t} \cdot t_1\\ \end{array} \]

Alternative 18
Error	33.3
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+64}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k \cdot \frac{t}{\ell}}}{k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right)\\ \end{array} \]

Alternative 19
Error	33.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+64}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k \cdot \frac{t}{\ell}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k} \cdot \frac{\ell}{\frac{k}{\frac{-0.3333333333333333}{t}}}\\ \end{array} \]

Alternative 20
Error	33.4
Cost	964

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+64}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k \cdot \frac{t}{\ell}}}{k}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{\frac{\left(k \cdot t\right) \cdot \frac{k}{\ell}}{\ell}}\\ \end{array} \]

Alternative 21
Error	35.9
Cost	704

\[-0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{k \cdot \left(k \cdot t\right)}\right) \]

Alternative 22
Error	34.1
Cost	704

\[-0.3333333333333333 \cdot \frac{\ell}{k \cdot \left(k \cdot \frac{t}{\ell}\right)} \]

Alternative 23
Error	34.6
Cost	704

\[-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{t}}{k} \]

Reproduce

herbie shell --seed 2023041 
(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))))