Toniolo and Linder, Equation (10+)

Average Accuracy: 48.8% → 89.4%

Time: 36.3s

Precision: binary64

Cost: 20937

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-79} \lor \neg \left(t \leq 1.95 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{-\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{t \cdot \left(k \cdot {\sin k}^{2}\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (t l k)
 :precision binary64
 (if (or (<= t -6.8e-79) (not (<= t 1.95e-146)))
   (/
    2.0
    (/
     (* (* t (* (/ t l) (sin k))) (* (tan k) (- -2.0 (pow (/ k t) 2.0))))
     (/ (- l) t)))
   (* l (/ (* (/ (cos k) k) (* 2.0 l)) (* t (* k (pow (sin k) 2.0)))))))

C

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

↓

double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.8e-79) || !(t <= 1.95e-146)) {
		tmp = 2.0 / (((t * ((t / l) * sin(k))) * (tan(k) * (-2.0 - pow((k / t), 2.0)))) / (-l / t));
	} else {
		tmp = l * (((cos(k) / k) * (2.0 * l)) / (t * (k * pow(sin(k), 2.0))));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((t <= (-6.8d-79)) .or. (.not. (t <= 1.95d-146))) then
        tmp = 2.0d0 / (((t * ((t / l) * sin(k))) * (tan(k) * ((-2.0d0) - ((k / t) ** 2.0d0)))) / (-l / t))
    else
        tmp = l * (((cos(k) / k) * (2.0d0 * l)) / (t * (k * (sin(k) ** 2.0d0))))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double t, double l, double k) {
	double tmp;
	if ((t <= -6.8e-79) || !(t <= 1.95e-146)) {
		tmp = 2.0 / (((t * ((t / l) * Math.sin(k))) * (Math.tan(k) * (-2.0 - Math.pow((k / t), 2.0)))) / (-l / t));
	} else {
		tmp = l * (((Math.cos(k) / k) * (2.0 * l)) / (t * (k * Math.pow(Math.sin(k), 2.0))));
	}
	return tmp;
}

Python

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

↓

def code(t, l, k):
	tmp = 0
	if (t <= -6.8e-79) or not (t <= 1.95e-146):
		tmp = 2.0 / (((t * ((t / l) * math.sin(k))) * (math.tan(k) * (-2.0 - math.pow((k / t), 2.0)))) / (-l / t))
	else:
		tmp = l * (((math.cos(k) / k) * (2.0 * l)) / (t * (k * math.pow(math.sin(k), 2.0))))
	return tmp

Julia

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

↓

function code(t, l, k)
	tmp = 0.0
	if ((t <= -6.8e-79) || !(t <= 1.95e-146))
		tmp = Float64(2.0 / Float64(Float64(Float64(t * Float64(Float64(t / l) * sin(k))) * Float64(tan(k) * Float64(-2.0 - (Float64(k / t) ^ 2.0)))) / Float64(Float64(-l) / t)));
	else
		tmp = Float64(l * Float64(Float64(Float64(cos(k) / k) * Float64(2.0 * l)) / Float64(t * Float64(k * (sin(k) ^ 2.0)))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((t <= -6.8e-79) || ~((t <= 1.95e-146)))
		tmp = 2.0 / (((t * ((t / l) * sin(k))) * (tan(k) * (-2.0 - ((k / t) ^ 2.0)))) / (-l / t));
	else
		tmp = l * (((cos(k) / k) * (2.0 * l)) / (t * (k * (sin(k) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[t_, l_, k_] := If[Or[LessEqual[t, -6.8e-79], N[Not[LessEqual[t, 1.95e-146]], $MachinePrecision]], N[(2.0 / N[(N[(N[(t * N[(N[(t / l), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(-2.0 - N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[((-l) / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * N[(2.0 * l), $MachinePrecision]), $MachinePrecision] / N[(t * N[(k * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.79999999999999951e-79 or 1.95000000000000001e-146 < t

   1. Initial program 61.6%

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

   2. Simplified 61.7%

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

      [Start] 61.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*l* [=>] 61.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)\right)}} \]
      +-commutative [=>] 61.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\right)} \]

   3. Applied egg-rr 72.4%

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

      [Start] 61.7	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      unpow3 [=>] 61.7	\[ \frac{2}{\left(\frac{\color{blue}{\left(t \cdot t\right) \cdot t}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      times-frac [=>] 72.4	\[ \frac{2}{\left(\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right)} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   4. Applied egg-rr 86.0%

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

      [Start] 72.4	\[ \frac{2}{\left(\left(\frac{t \cdot t}{\ell} \cdot \frac{t}{\ell}\right) \cdot \sin k\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l* [=>] 76.5	\[ \frac{2}{\color{blue}{\left(\frac{t \cdot t}{\ell} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-/l* [=>] 84.3	\[ \frac{2}{\left(\color{blue}{\frac{t}{\frac{\ell}{t}}} \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l/ [=>] 86.0	\[ \frac{2}{\color{blue}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]

   5. Applied egg-rr 90.1%

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

      [Start] 86.0	\[ \frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      frac-2neg [=>] 86.0	\[ \frac{2}{\color{blue}{\frac{-t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{-\frac{\ell}{t}}} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)} \]
      associate-*l/ [=>] 90.1	\[ \frac{2}{\color{blue}{\frac{\left(-t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{-\frac{\ell}{t}}}} \]
      *-commutative [=>] 90.1	\[ \frac{2}{\frac{\left(-\color{blue}{\left(\frac{t}{\ell} \cdot \sin k\right) \cdot t}\right) \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{-\frac{\ell}{t}}} \]
      distribute-rgt-neg-in [=>] 90.1	\[ \frac{2}{\frac{\color{blue}{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(-t\right)\right)} \cdot \left(\tan k \cdot \left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)\right)}{-\frac{\ell}{t}}} \]
      associate-+r+ [=>] 90.1	\[ \frac{2}{\frac{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(-t\right)\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}{-\frac{\ell}{t}}} \]
      metadata-eval [=>] 90.1	\[ \frac{2}{\frac{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(-t\right)\right) \cdot \left(\tan k \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{-\frac{\ell}{t}}} \]
      distribute-neg-frac [=>] 90.1	\[ \frac{2}{\frac{\left(\left(\frac{t}{\ell} \cdot \sin k\right) \cdot \left(-t\right)\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\color{blue}{\frac{-\ell}{t}}}} \]

   if -6.79999999999999951e-79 < t < 1.95000000000000001e-146

   1. Initial program 4.0%

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

   2. Simplified 3.8%

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

      [Start] 4.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      *-commutative [=>] 4.0	\[ \frac{2}{\color{blue}{\left(\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/r/ [<=] 4.0	\[ \frac{2}{\left(\tan k \cdot \color{blue}{\frac{{t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}}\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-*r/ [=>] 3.7	\[ \frac{2}{\color{blue}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell \cdot \ell}{\sin k}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      associate-/l* [=>] 3.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\color{blue}{\frac{\ell}{\frac{\sin k}{\ell}}}} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      +-commutative [=>] 3.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}} \]
      associate-+r+ [=>] 3.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \color{blue}{\left(\left(1 + 1\right) + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      metadata-eval [=>] 3.8	\[ \frac{2}{\frac{\tan k \cdot {t}^{3}}{\frac{\ell}{\frac{\sin k}{\ell}}} \cdot \left(\color{blue}{2} + {\left(\frac{k}{t}\right)}^{2}\right)} \]

   3. Taylor expanded in k around inf 56.3%

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

   4. Simplified 66.3%

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

      [Start] 56.3	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      times-frac [=>] 53.0	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}}} \]
      unpow2 [=>] 53.0	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{\cos k} \cdot \frac{{\sin k}^{2} \cdot t}{{\ell}^{2}}} \]
      *-commutative [=>] 53.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{\color{blue}{t \cdot {\sin k}^{2}}}{{\ell}^{2}}} \]
      unpow2 [=>] 53.0	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \frac{t \cdot {\sin k}^{2}}{\color{blue}{\ell \cdot \ell}}} \]
      times-frac [=>] 66.3	\[ \frac{2}{\frac{k \cdot k}{\cos k} \cdot \color{blue}{\left(\frac{t}{\ell} \cdot \frac{{\sin k}^{2}}{\ell}\right)}} \]

   5. Taylor expanded in k around inf 56.3%

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

   6. Simplified 63.6%

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

      [Start] 56.3	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [<=] 56.3	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]
      times-frac [=>] 52.8	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{t \cdot {\sin k}^{2}}\right)} \]
      unpow2 [=>] 52.8	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{t \cdot {\sin k}^{2}}\right) \]
      unpow2 [=>] 52.8	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{t \cdot {\sin k}^{2}}\right) \]
      associate-*r/ [=>] 53.3	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \ell\right)}{t \cdot {\sin k}^{2}}} \]
      *-commutative [<=] 53.3	\[ 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{k \cdot k}}}{t \cdot {\sin k}^{2}} \]
      associate-*r/ [=>] 53.3	\[ \color{blue}{\frac{2 \cdot \left(\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{k \cdot k}\right)}{t \cdot {\sin k}^{2}}} \]
      *-commutative [=>] 53.3	\[ \frac{2 \cdot \color{blue}{\left(\frac{\cos k}{k \cdot k} \cdot \left(\ell \cdot \ell\right)\right)}}{t \cdot {\sin k}^{2}} \]
      associate-*l* [<=] 53.3	\[ \frac{\color{blue}{\left(2 \cdot \frac{\cos k}{k \cdot k}\right) \cdot \left(\ell \cdot \ell\right)}}{t \cdot {\sin k}^{2}} \]
      associate-*l/ [<=] 56.2	\[ \color{blue}{\frac{2 \cdot \frac{\cos k}{k \cdot k}}{t \cdot {\sin k}^{2}} \cdot \left(\ell \cdot \ell\right)} \]
      associate-*r* [=>] 61.9	\[ \color{blue}{\left(\frac{2 \cdot \frac{\cos k}{k \cdot k}}{t \cdot {\sin k}^{2}} \cdot \ell\right) \cdot \ell} \]

   7. Applied egg-rr 87.1%

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

      [Start] 63.6	\[ \ell \cdot \left(\frac{\frac{\cos k}{k}}{k \cdot {\sin k}^{2}} \cdot \left(\frac{2}{t} \cdot \ell\right)\right) \]
      associate-*l/ [=>] 63.7	\[ \ell \cdot \left(\frac{\frac{\cos k}{k}}{k \cdot {\sin k}^{2}} \cdot \color{blue}{\frac{2 \cdot \ell}{t}}\right) \]
      frac-times [=>] 87.1	\[ \ell \cdot \color{blue}{\frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{\left(k \cdot {\sin k}^{2}\right) \cdot t}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.8 \cdot 10^{-79} \lor \neg \left(t \leq 1.95 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)\right) \cdot \left(\tan k \cdot \left(-2 - {\left(\frac{k}{t}\right)}^{2}\right)\right)}{\frac{-\ell}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{t \cdot \left(k \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.2%
Cost	21000

\[\begin{array}{l} t_1 := \frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}}\\ \mathbf{if}\;t \leq -2.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;\ell \cdot \frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{t \cdot \left(k \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)\right)}\\ \end{array} \]

Alternative 2
Accuracy	79.6%
Cost	20624

\[\begin{array}{l} t_1 := \frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ t_2 := 2 \cdot \frac{\ell}{\frac{t}{\ell} \cdot \frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k}}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-206}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {k}^{-2}\right)}{k}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	83.3%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -7.2 \cdot 10^{+181} \lor \neg \left(k \leq 2.7 \cdot 10^{+75}\right):\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{t \cdot {\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \end{array} \]

Alternative 4
Accuracy	80.9%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;t \leq -1.18 \cdot 10^{-139} \lor \neg \left(t \leq 1.46 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{{\sin k}^{2}} \cdot \frac{\cos k}{k \cdot k}\right)\\ \end{array} \]

Alternative 5
Accuracy	84.9%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-14} \lor \neg \left(t \leq 1.4 \cdot 10^{-124}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{\cos k}{k}}{k \cdot \left(\frac{t}{\ell} \cdot {\sin k}^{2}\right)}\\ \end{array} \]

Alternative 6
Accuracy	86.3%
Cost	20488

\[\begin{array}{l} t_1 := \frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)\right)}\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-146}:\\ \;\;\;\;\ell \cdot \frac{\frac{\cos k}{k} \cdot \left(2 \cdot \ell\right)}{t \cdot \left(k \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t_1 \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \end{array} \]

Alternative 7
Accuracy	80.0%
Cost	20361

\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{-140} \lor \neg \left(t \leq 1.85 \cdot 10^{-146}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\cos k}}\\ \end{array} \]

Alternative 8
Accuracy	78.0%
Cost	14793

\[\begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-130} \lor \neg \left(t \leq 4.8 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{k}{t \cdot \frac{t}{k}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {k}^{-2}\right)}{k}\\ \end{array} \]

Alternative 9
Accuracy	78.0%
Cost	14793

\[\begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{-130} \lor \neg \left(t \leq 1.05 \cdot 10^{-130}\right):\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\frac{t}{\ell} \cdot \sin k\right)}{\frac{\ell}{t}} \cdot \left(\tan k \cdot \left(1 + \left(1 + \frac{\frac{k}{t}}{\frac{t}{k}}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right) \cdot \left(\cos k \cdot {k}^{-2}\right)}{k}\\ \end{array} \]

Alternative 10
Accuracy	73.5%
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -9500000000 \lor \neg \left(k \leq 0.0077\right):\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \frac{\ell}{t}}{0.5 - 0.5 \cdot \cos \left(k + k\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \end{array} \]

Alternative 11
Accuracy	76.4%
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -0.00042 \lor \neg \left(k \leq 1.45\right):\\ \;\;\;\;\ell \cdot \left(\frac{\frac{\cos k}{k}}{k \cdot \left(0.5 - 0.5 \cdot \cos \left(k + k\right)\right)} \cdot \left(\ell \cdot \frac{2}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{t \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}{\frac{\frac{\ell}{k}}{t}}}\\ \end{array} \]

Alternative 12
Accuracy	66.9%
Cost	8073

\[\begin{array}{l} \mathbf{if}\;t \leq -8.2 \cdot 10^{-156} \lor \neg \left(t \leq 1.1 \cdot 10^{-135}\right):\\ \;\;\;\;\frac{2}{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\left(t \cdot \frac{k}{\ell}\right) \cdot \frac{t \cdot t}{\frac{\ell}{k}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k \cdot k}{\cos k} \cdot \left(\frac{t}{\ell} \cdot \frac{k \cdot k}{\ell}\right)}\\ \end{array} \]

Alternative 13
Accuracy	68.7%
Cost	7808

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

Alternative 14
Accuracy	53.1%
Cost	7305

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -1.15 \cdot 10^{+45} \lor \neg \left(t \leq 7.6 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{\ell}{k \cdot k} \cdot \frac{\ell}{{t}^{3}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 15
Accuracy	62.6%
Cost	7305

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -1.04 \cdot 10^{-48} \lor \neg \left(t \leq 2.4 \cdot 10^{-64}\right):\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell} \cdot \left(k \cdot {t}^{3}\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \end{array} \]

Alternative 16
Accuracy	61.5%
Cost	7304

\[\begin{array}{l} t_1 := k \cdot \frac{k}{\ell}\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{-98}:\\ \;\;\;\;\frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{{t}^{3}}\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-65}:\\ \;\;\;\;2 \cdot \frac{1}{t_1 \cdot \left(t \cdot t_1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{k}{\ell} \cdot \left(k \cdot {t}^{3}\right)}\\ \end{array} \]

Alternative 17
Accuracy	46.1%
Cost	1088

\[2 \cdot \left(\frac{\ell}{k \cdot k} \cdot \frac{1}{\frac{t}{\ell} \cdot \left(k \cdot k\right)}\right) \]

Alternative 18
Accuracy	46.3%
Cost	1088

\[2 \cdot \left(\frac{\ell \cdot \frac{\ell}{t}}{k \cdot k} \cdot \frac{1}{k \cdot k}\right) \]

Alternative 19
Accuracy	42.9%
Cost	960

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

Alternative 20
Accuracy	42.9%
Cost	960

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

Alternative 21
Accuracy	43.1%
Cost	960

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

Alternative 22
Accuracy	43.5%
Cost	960

\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot k}\\ 2 \cdot \frac{t_1 \cdot t_1}{t} \end{array} \]

Alternative 23
Accuracy	43.7%
Cost	960

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

Alternative 24
Accuracy	45.0%
Cost	960

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

Reproduce

herbie shell --seed 2023141 
(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))))