Toniolo and Linder, Equation (10+)

Average Accuracy: 49.4% → 91.4%

Time: 43.3s

Precision: binary64

Cost: 21136

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{t}\\ t_3 := \frac{\ell \cdot \frac{\frac{2}{\tan k}}{t \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot \frac{t}{\ell}}\\ t_4 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -1.85 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{t_2}{t_1}\right)\right) \cdot -2\\ \mathbf{elif}\;k \leq -5.8 \cdot 10^{-85}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-143}:\\ \;\;\;\;t_4 \cdot \frac{t_4}{t}\\ \mathbf{elif}\;k \leq 1.08 \cdot 10^{+39}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

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 (/ (cos k) t))
        (t_3
         (/
          (* l (/ (/ 2.0 (tan k)) (* t (* (sin k) (+ 2.0 (pow (/ k t) 2.0))))))
          (* t (/ t l))))
        (t_4 (/ l (* k t))))
   (if (<= k -1.85e+58)
     (* (* (/ l (- k)) (* (/ l k) (/ t_2 t_1))) -2.0)
     (if (<= k -5.8e-85)
       t_3
       (if (<= k 1.5e-143)
         (* t_4 (/ t_4 t))
         (if (<= k 1.08e+39)
           t_3
           (* 2.0 (/ (* (/ l k) t_2) (* t_1 (/ k l))))))))))

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 = cos(k) / t;
	double t_3 = (l * ((2.0 / tan(k)) / (t * (sin(k) * (2.0 + pow((k / t), 2.0)))))) / (t * (t / l));
	double t_4 = l / (k * t);
	double tmp;
	if (k <= -1.85e+58) {
		tmp = ((l / -k) * ((l / k) * (t_2 / t_1))) * -2.0;
	} else if (k <= -5.8e-85) {
		tmp = t_3;
	} else if (k <= 1.5e-143) {
		tmp = t_4 * (t_4 / t);
	} else if (k <= 1.08e+39) {
		tmp = t_3;
	} else {
		tmp = 2.0 * (((l / k) * t_2) / (t_1 * (k / l)));
	}
	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 = cos(k) / t
    t_3 = (l * ((2.0d0 / tan(k)) / (t * (sin(k) * (2.0d0 + ((k / t) ** 2.0d0)))))) / (t * (t / l))
    t_4 = l / (k * t)
    if (k <= (-1.85d+58)) then
        tmp = ((l / -k) * ((l / k) * (t_2 / t_1))) * (-2.0d0)
    else if (k <= (-5.8d-85)) then
        tmp = t_3
    else if (k <= 1.5d-143) then
        tmp = t_4 * (t_4 / t)
    else if (k <= 1.08d+39) then
        tmp = t_3
    else
        tmp = 2.0d0 * (((l / k) * t_2) / (t_1 * (k / l)))
    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.cos(k) / t;
	double t_3 = (l * ((2.0 / Math.tan(k)) / (t * (Math.sin(k) * (2.0 + Math.pow((k / t), 2.0)))))) / (t * (t / l));
	double t_4 = l / (k * t);
	double tmp;
	if (k <= -1.85e+58) {
		tmp = ((l / -k) * ((l / k) * (t_2 / t_1))) * -2.0;
	} else if (k <= -5.8e-85) {
		tmp = t_3;
	} else if (k <= 1.5e-143) {
		tmp = t_4 * (t_4 / t);
	} else if (k <= 1.08e+39) {
		tmp = t_3;
	} else {
		tmp = 2.0 * (((l / k) * t_2) / (t_1 * (k / l)));
	}
	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.cos(k) / t
	t_3 = (l * ((2.0 / math.tan(k)) / (t * (math.sin(k) * (2.0 + math.pow((k / t), 2.0)))))) / (t * (t / l))
	t_4 = l / (k * t)
	tmp = 0
	if k <= -1.85e+58:
		tmp = ((l / -k) * ((l / k) * (t_2 / t_1))) * -2.0
	elif k <= -5.8e-85:
		tmp = t_3
	elif k <= 1.5e-143:
		tmp = t_4 * (t_4 / t)
	elif k <= 1.08e+39:
		tmp = t_3
	else:
		tmp = 2.0 * (((l / k) * t_2) / (t_1 * (k / l)))
	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(cos(k) / t)
	t_3 = Float64(Float64(l * Float64(Float64(2.0 / tan(k)) / Float64(t * Float64(sin(k) * Float64(2.0 + (Float64(k / t) ^ 2.0)))))) / Float64(t * Float64(t / l)))
	t_4 = Float64(l / Float64(k * t))
	tmp = 0.0
	if (k <= -1.85e+58)
		tmp = Float64(Float64(Float64(l / Float64(-k)) * Float64(Float64(l / k) * Float64(t_2 / t_1))) * -2.0);
	elseif (k <= -5.8e-85)
		tmp = t_3;
	elseif (k <= 1.5e-143)
		tmp = Float64(t_4 * Float64(t_4 / t));
	elseif (k <= 1.08e+39)
		tmp = t_3;
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) * t_2) / Float64(t_1 * Float64(k / l))));
	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 = cos(k) / t;
	t_3 = (l * ((2.0 / tan(k)) / (t * (sin(k) * (2.0 + ((k / t) ^ 2.0)))))) / (t * (t / l));
	t_4 = l / (k * t);
	tmp = 0.0;
	if (k <= -1.85e+58)
		tmp = ((l / -k) * ((l / k) * (t_2 / t_1))) * -2.0;
	elseif (k <= -5.8e-85)
		tmp = t_3;
	elseif (k <= 1.5e-143)
		tmp = t_4 * (t_4 / t);
	elseif (k <= 1.08e+39)
		tmp = t_3;
	else
		tmp = 2.0 * (((l / k) * t_2) / (t_1 * (k / l)));
	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[Cos[k], $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(t * N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.85e+58], N[(N[(N[(l / (-k)), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] * N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision], If[LessEqual[k, -5.8e-85], t$95$3, If[LessEqual[k, 1.5e-143], N[(t$95$4 * N[(t$95$4 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 1.08e+39], t$95$3, N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] * t$95$2), $MachinePrecision] / N[(t$95$1 * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if k < -1.8500000000000001e58

   1. Initial program 48.7%

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

   2. Simplified 48.7%

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

      [Start] 48.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 48.7	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 48.7	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 48.7	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 48.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 48.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 48.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 48.7	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 68.0%

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

   4. Simplified 86.4%

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

      [Start] 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 [=>] 65.6	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 65.6	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 65.6	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 86.4	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 86.4	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 92.7%

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

      [Start] 86.4	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]
      associate-*l* [=>] 92.7	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]
      frac-2neg [=>] 92.7	\[ 2 \cdot \left(\color{blue}{\frac{-\ell}{-k}} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right) \]
      distribute-frac-neg [=>] 92.7	\[ 2 \cdot \left(\color{blue}{\left(-\frac{\ell}{-k}\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right) \]
      distribute-lft-neg-out [=>] 92.7	\[ 2 \cdot \color{blue}{\left(-\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]
      associate-/r* [=>] 92.7	\[ 2 \cdot \left(-\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}\right)\right) \]

   if -1.8500000000000001e58 < k < -5.8000000000000004e-85 or 1.49999999999999993e-143 < k < 1.07999999999999998e39

   1. Initial program 57.5%

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

   2. Simplified 57.7%

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

      [Start] 57.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 57.5	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 57.5	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 57.5	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 57.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 57.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 57.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 57.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 57.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 64.5%

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

      [Start] 57.7	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      *-commutative [=>] 57.7	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]
      associate-/r* [=>] 57.8	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{{t}^{3}}{\ell \cdot \ell}}} \]
      unpow3 [=>] 57.8	\[ \frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell}} \]
      associate-/l* [=>] 62.1	\[ \frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\color{blue}{\frac{t \cdot t}{\frac{\ell \cdot \ell}{t}}}} \]
      associate-/r/ [=>] 64.5	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t}} \]
      *-commutative [=>] 64.5	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k}}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t} \]
      associate-*l* [=>] 64.5	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t} \]

   4. Applied egg-rr 86.6%

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

      [Start] 64.5	\[ \frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t} \]
      associate-/r* [=>] 67.8	\[ \color{blue}{\frac{\frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t}}{t}} \cdot \frac{\ell \cdot \ell}{t} \]
      associate-/l* [=>] 75.1	\[ \frac{\frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t}}{t} \cdot \color{blue}{\frac{\ell}{\frac{t}{\ell}}} \]
      frac-times [=>] 86.6	\[ \color{blue}{\frac{\frac{\frac{2}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}{t} \cdot \ell}{t \cdot \frac{t}{\ell}}} \]
      associate-/r* [=>] 86.6	\[ \frac{\frac{\color{blue}{\frac{\frac{2}{\tan k}}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k}}}{t} \cdot \ell}{t \cdot \frac{t}{\ell}} \]
      associate-/l/ [=>] 86.6	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}} \cdot \ell}{t \cdot \frac{t}{\ell}} \]

   if -5.8000000000000004e-85 < k < 1.49999999999999993e-143

   1. Initial program 43.4%

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

   2. Simplified 15.1%

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

      [Start] 43.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 43.4	\[ \frac{2}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      distribute-rgt1-in [<=] 43.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right)}} \]
      *-commutative [=>] 43.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      associate-*l* [=>] 43.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-*l* [=>] 15.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}} \]
      distribute-lft-in [=>] 15.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \color{blue}{\left(\tan k \cdot 1 + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)}\right)} \]
      *-rgt-identity [=>] 15.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\color{blue}{\tan k} + \tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)} \]
      distribute-lft-in [=>] 15.9	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k + \frac{{t}^{3}}{\ell \cdot \ell} \cdot \color{blue}{\left(\sin k \cdot \tan k + \sin k \cdot \left(\tan k \cdot {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]

   3. Applied egg-rr 14.0%

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

      [Start] 15.1	\[ \frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      *-commutative [=>] 15.1	\[ \frac{2}{\color{blue}{\left(\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}}} \]
      associate-/r* [=>] 12.9	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{{t}^{3}}{\ell \cdot \ell}}} \]
      unpow3 [=>] 12.9	\[ \frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell}} \]
      associate-/l* [=>] 13.9	\[ \frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{\color{blue}{\frac{t \cdot t}{\frac{\ell \cdot \ell}{t}}}} \]
      associate-/r/ [=>] 14.0	\[ \color{blue}{\frac{\frac{2}{\sin k \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t}} \]
      *-commutative [=>] 14.0	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right) \cdot \sin k}}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t} \]
      associate-*l* [=>] 14.0	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \sin k\right)}}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t} \]

   4. Taylor expanded in k around 0 16.3%

      \[\leadsto \color{blue}{\frac{1}{{k}^{2} \cdot {t}^{2}}} \cdot \frac{\ell \cdot \ell}{t} \]

   5. Simplified 16.3%

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

      [Start] 16.3	\[ \frac{1}{{k}^{2} \cdot {t}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]
      unpow2 [=>] 16.3	\[ \frac{1}{\color{blue}{\left(k \cdot k\right)} \cdot {t}^{2}} \cdot \frac{\ell \cdot \ell}{t} \]
      unpow2 [=>] 16.3	\[ \frac{1}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot t\right)}} \cdot \frac{\ell \cdot \ell}{t} \]

   6. Applied egg-rr 62.5%

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

      [Start] 16.3	\[ \frac{1}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \cdot \frac{\ell \cdot \ell}{t} \]
      associate-*l/ [=>] 16.4	\[ \color{blue}{\frac{1 \cdot \frac{\ell \cdot \ell}{t}}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}} \]
      *-un-lft-identity [<=] 16.4	\[ \frac{\color{blue}{\frac{\ell \cdot \ell}{t}}}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)} \]
      associate-*l* [=>] 46.4	\[ \frac{\frac{\ell \cdot \ell}{t}}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot t\right)\right)}} \]
      associate-/r* [=>] 52.3	\[ \color{blue}{\frac{\frac{\frac{\ell \cdot \ell}{t}}{k}}{k \cdot \left(t \cdot t\right)}} \]
      associate-/l* [=>] 53.1	\[ \frac{\frac{\color{blue}{\frac{\ell}{\frac{t}{\ell}}}}{k}}{k \cdot \left(t \cdot t\right)} \]
      associate-/l/ [=>] 62.5	\[ \frac{\color{blue}{\frac{\ell}{k \cdot \frac{t}{\ell}}}}{k \cdot \left(t \cdot t\right)} \]

   7. Applied egg-rr 95.1%

      \[\leadsto \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{k \cdot t}}{t}} \]
      Proof

      [Start] 62.5	\[ \frac{\frac{\ell}{k \cdot \frac{t}{\ell}}}{k \cdot \left(t \cdot t\right)} \]
      div-inv [=>] 62.5	\[ \frac{\color{blue}{\ell \cdot \frac{1}{k \cdot \frac{t}{\ell}}}}{k \cdot \left(t \cdot t\right)} \]
      associate-*r* [=>] 74.9	\[ \frac{\ell \cdot \frac{1}{k \cdot \frac{t}{\ell}}}{\color{blue}{\left(k \cdot t\right) \cdot t}} \]
      times-frac [=>] 92.9	\[ \color{blue}{\frac{\ell}{k \cdot t} \cdot \frac{\frac{1}{k \cdot \frac{t}{\ell}}}{t}} \]
      associate-/l/ [<=] 93.0	\[ \frac{\ell}{k \cdot t} \cdot \frac{\color{blue}{\frac{\frac{1}{\frac{t}{\ell}}}{k}}}{t} \]
      clear-num [<=] 93.6	\[ \frac{\ell}{k \cdot t} \cdot \frac{\frac{\color{blue}{\frac{\ell}{t}}}{k}}{t} \]
      associate-/l/ [=>] 95.1	\[ \frac{\ell}{k \cdot t} \cdot \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{t} \]

   if 1.07999999999999998e39 < k

   1. Initial program 46.9%

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

   2. Simplified 46.9%

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

      [Start] 46.9	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 46.9	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 46.9	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 46.9	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 46.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 46.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 46.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 46.9	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 67.4%

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

   4. Simplified 85.3%

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

      [Start] 67.4	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [=>] 67.4	\[ 2 \cdot \frac{\color{blue}{{\ell}^{2} \cdot \cos k}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 64.7	\[ 2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 64.7	\[ 2 \cdot \left(\frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 64.7	\[ 2 \cdot \left(\frac{\ell \cdot \ell}{\color{blue}{k \cdot k}} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      times-frac [=>] 85.3	\[ 2 \cdot \left(\color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)} \cdot \frac{\cos k}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 85.3	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 91.9%

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

      [Start] 85.3	\[ 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right) \]
      associate-*l* [=>] 91.9	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right)} \]
      clear-num [=>] 91.9	\[ 2 \cdot \left(\color{blue}{\frac{1}{\frac{k}{\ell}}} \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\right) \]
      associate-*l/ [=>] 91.9	\[ 2 \cdot \color{blue}{\frac{1 \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)}{\frac{k}{\ell}}} \]
      *-un-lft-identity [<=] 91.9	\[ 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}}}{\frac{k}{\ell}} \]
      associate-/r* [=>] 91.9	\[ 2 \cdot \frac{\frac{\ell}{k} \cdot \color{blue}{\frac{\frac{\cos k}{t}}{{\sin k}^{2}}}}{\frac{k}{\ell}} \]
      associate-*r/ [=>] 91.9	\[ 2 \cdot \frac{\color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{{\sin k}^{2}}}}{\frac{k}{\ell}} \]
      associate-/l/ [=>] 91.9	\[ 2 \cdot \color{blue}{\frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot {\sin k}^{2}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.85 \cdot 10^{+58}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{\frac{\cos k}{t}}{{\sin k}^{2}}\right)\right) \cdot -2\\ \mathbf{elif}\;k \leq -5.8 \cdot 10^{-85}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{\tan k}}{t \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot \frac{t}{\ell}}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{k \cdot t}}{t}\\ \mathbf{elif}\;k \leq 1.08 \cdot 10^{+39}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{\tan k}}{t \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{{\sin k}^{2} \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	87.0%
Cost	21400

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ t_2 := \frac{\cos k}{t}\\ t_3 := {\sin k}^{2}\\ t_4 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_5 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -1.5 \cdot 10^{+25}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{t_2}{t_3}\right)\right) \cdot -2\\ \mathbf{elif}\;k \leq -2.6 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right) \cdot \left(\sin k \cdot \left(\tan k \cdot t_4\right)\right)}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_5 \cdot \frac{t_5}{t}\\ \mathbf{elif}\;k \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 3.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot t_4\right)}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_3 \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 2
Accuracy	86.7%
Cost	21136

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{{\sin k}^{2} \cdot \frac{k}{\ell}}\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{-61}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-32}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 1.06 \cdot 10^{+48}:\\ \;\;\;\;\frac{\frac{2}{\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}}{t \cdot t} \cdot \frac{\ell \cdot \ell}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	89.6%
Cost	21136

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{t}\\ t_3 := \frac{2 \cdot \ell}{\frac{t}{\ell} \cdot \left(t \cdot \left(t \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)\right)}\\ t_4 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -2.3 \cdot 10^{+26}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{t_2}{t_1}\right)\right) \cdot -2\\ \mathbf{elif}\;k \leq -2.05 \cdot 10^{-79}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-143}:\\ \;\;\;\;t_4 \cdot \frac{t_4}{t}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{+48}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_2}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 4
Accuracy	88.9%
Cost	21136

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\ell}{k \cdot t}\\ t_3 := t \cdot \left(t \cdot \left(\tan k \cdot \left(\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)\right)\\ t_4 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -2.5 \cdot 10^{+26}:\\ \;\;\;\;\left(\frac{\ell}{-k} \cdot \left(\frac{\ell}{k} \cdot \frac{t_4}{t_1}\right)\right) \cdot -2\\ \mathbf{elif}\;k \leq -9.5 \cdot 10^{-81}:\\ \;\;\;\;\frac{2 \cdot \ell}{\frac{t}{\ell} \cdot t_3}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-143}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 10^{+49}:\\ \;\;\;\;\frac{2 \cdot \left(\ell \cdot \frac{\ell}{t}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_4}{t_1 \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 5
Accuracy	85.9%
Cost	20752

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{{\sin k}^{2} \cdot \frac{k}{\ell}}\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -4.5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-32}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{2}{t \cdot t}}{t \cdot \left(\frac{\tan k}{\ell \cdot 0.5} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	68.8%
Cost	14812

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ t_2 := \frac{\ell}{k \cdot t}\\ t_3 := 2 \cdot \frac{t_2}{{\sin k}^{2} \cdot \frac{k}{\ell}}\\ t_4 := \frac{\sin k}{\ell}\\ \mathbf{if}\;k \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -1.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{0.5}{\frac{\tan k}{\ell}} \cdot \frac{\frac{2}{t \cdot t}}{t \cdot t_4}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -9.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}}{k \cdot t}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 1.65 \cdot 10^{-32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 4.7 \cdot 10^{+38}:\\ \;\;\;\;\frac{\frac{2}{t}}{\left(t \cdot t\right) \cdot \left(\frac{\tan k}{\ell \cdot 0.5} \cdot t_4\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 7
Accuracy	68.8%
Cost	14812

\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ t_2 := \frac{\ell}{k \cdot t}\\ t_3 := 2 \cdot \frac{t_2}{{\sin k}^{2} \cdot \frac{k}{\ell}}\\ t_4 := \frac{2}{t \cdot t}\\ t_5 := \frac{\sin k}{\ell}\\ \mathbf{if}\;k \leq -3.1 \cdot 10^{+26}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -1.35 \cdot 10^{-20}:\\ \;\;\;\;\frac{0.5}{\frac{\tan k}{\ell}} \cdot \frac{t_4}{t \cdot t_5}\\ \mathbf{elif}\;k \leq -4.5 \cdot 10^{-56}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -6.8 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot t\right)}}{k \cdot t}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;\frac{t_4}{t \cdot \left(\frac{\tan k}{\ell \cdot 0.5} \cdot t_5\right)}\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Accuracy	82.4%
Cost	14672

\[\begin{array}{l} t_1 := 2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\right)\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -2100000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 2.15 \cdot 10^{-32}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 7.5 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{2}{t \cdot t}}{t \cdot \left(\frac{\tan k}{\ell \cdot 0.5} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	86.2%
Cost	14672

\[\begin{array}{l} t_1 := 2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -25000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-30}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 7.6 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{2}{t \cdot t}}{t \cdot \left(\frac{\tan k}{\ell \cdot 0.5} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Accuracy	86.2%
Cost	14672

\[\begin{array}{l} t_1 := 0.5 - \frac{\cos \left(k + k\right)}{2}\\ t_2 := \frac{\ell}{k \cdot t}\\ t_3 := \frac{\cos k}{t}\\ \mathbf{if}\;k \leq -2400000:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot t_3}{\frac{k}{\ell} \cdot t_1}\\ \mathbf{elif}\;k \leq 5 \cdot 10^{-61}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-29}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq 4.8 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{2}{t \cdot t}}{t \cdot \left(\frac{\tan k}{\ell \cdot 0.5} \cdot \frac{\sin k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot \frac{t_1}{t_3}}\\ \end{array} \]

Alternative 11
Accuracy	72.0%
Cost	13961

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -0.9 \lor \neg \left(t \leq 4 \cdot 10^{-79}\right):\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_1}{{\sin k}^{2} \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 12
Accuracy	72.1%
Cost	7753

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-14} \lor \neg \left(t \leq 1.06 \cdot 10^{-81}\right):\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\cos k}{t}}{\frac{k}{\ell} \cdot \left(k \cdot k\right)}\\ \end{array} \]

Alternative 13
Accuracy	70.7%
Cost	7305

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-14} \lor \neg \left(t \leq 2.65 \cdot 10^{-79}\right):\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \end{array} \]

Alternative 14
Accuracy	70.3%
Cost	1225

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-14} \lor \neg \left(t \leq 10^{-79}\right):\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k}}{\frac{k}{\ell} \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \end{array} \]

Alternative 15
Accuracy	63.5%
Cost	1097

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-14} \lor \neg \left(t \leq 5 \cdot 10^{-81}\right):\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}}{t \cdot t}\\ \end{array} \]

Alternative 16
Accuracy	55.2%
Cost	832

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

Alternative 17
Accuracy	63.3%
Cost	832

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ t_1 \cdot \frac{t_1}{t} \end{array} \]

Reproduce

herbie shell --seed 2023140 
(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))))