Toniolo and Linder, Equation (10-)

Average Accuracy: 25.2% → 97.4%

Time: 35.7s

Precision: binary64

Cost: 20489

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -2.2 \cdot 10^{-150} \lor \neg \left(k \leq 2.15 \cdot 10^{-13}\right):\\ \;\;\;\;2 \cdot \frac{{\sin k}^{-2}}{\left(\frac{k}{\ell \cdot \cos k} \cdot t\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \end{array} \]

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 (<= k -2.2e-150) (not (<= k 2.15e-13)))
   (* 2.0 (/ (pow (sin k) -2.0) (* (* (/ k (* l (cos k))) t) (/ k l))))
   (* 2.0 (* (/ (/ l k) k) (/ (/ l k) (* k t))))))

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 ((k <= -2.2e-150) || !(k <= 2.15e-13)) {
		tmp = 2.0 * (pow(sin(k), -2.0) / (((k / (l * cos(k))) * t) * (k / l)));
	} else {
		tmp = 2.0 * (((l / k) / k) * ((l / k) / (k * t)));
	}
	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 ((k <= (-2.2d-150)) .or. (.not. (k <= 2.15d-13))) then
        tmp = 2.0d0 * ((sin(k) ** (-2.0d0)) / (((k / (l * cos(k))) * t) * (k / l)))
    else
        tmp = 2.0d0 * (((l / k) / k) * ((l / k) / (k * t)))
    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 ((k <= -2.2e-150) || !(k <= 2.15e-13)) {
		tmp = 2.0 * (Math.pow(Math.sin(k), -2.0) / (((k / (l * Math.cos(k))) * t) * (k / l)));
	} else {
		tmp = 2.0 * (((l / k) / k) * ((l / k) / (k * t)));
	}
	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 (k <= -2.2e-150) or not (k <= 2.15e-13):
		tmp = 2.0 * (math.pow(math.sin(k), -2.0) / (((k / (l * math.cos(k))) * t) * (k / l)))
	else:
		tmp = 2.0 * (((l / k) / k) * ((l / k) / (k * t)))
	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 ((k <= -2.2e-150) || !(k <= 2.15e-13))
		tmp = Float64(2.0 * Float64((sin(k) ^ -2.0) / Float64(Float64(Float64(k / Float64(l * cos(k))) * t) * Float64(k / l))));
	else
		tmp = Float64(2.0 * Float64(Float64(Float64(l / k) / k) * Float64(Float64(l / k) / Float64(k * t))));
	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 ((k <= -2.2e-150) || ~((k <= 2.15e-13)))
		tmp = 2.0 * ((sin(k) ^ -2.0) / (((k / (l * cos(k))) * t) * (k / l)));
	else
		tmp = 2.0 * (((l / k) / k) * ((l / k) / (k * t)));
	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[k, -2.2e-150], N[Not[LessEqual[k, 2.15e-13]], $MachinePrecision]], N[(2.0 * N[(N[Power[N[Sin[k], $MachinePrecision], -2.0], $MachinePrecision] / N[(N[(N[(k / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / k), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -2.1999999999999999e-150 or 2.1499999999999999e-13 < k

   1. Initial program 28.4%

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

   2. Simplified 40.4%

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

      [Start] 28.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      +-rgt-identity [<=] 28.4	\[ \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0}} \]
      associate-*l* [=>] 28.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} + 0} \]
      mul0-rgt [<=] 10.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot 0}} \]
      distribute-lft-in [<=] 28.4	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0\right)}} \]
      +-rgt-identity [=>] 28.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      sub-neg [=>] 28.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}\right)} \]
      +-commutative [=>] 28.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + \left(-1\right)\right)\right)} \]
      associate-+l+ [=>] 40.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}\right)} \]
      metadata-eval [=>] 40.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \color{blue}{-1}\right)\right)\right)} \]
      metadata-eval [=>] 40.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]
      +-rgt-identity [=>] 40.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   3. Taylor expanded in t around 0 68.7%

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

   4. Simplified 75.8%

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

      [Start] 68.7	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 68.7	\[ 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 68.5	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 68.5	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{\ell \cdot \ell}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 68.5	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]
      times-frac [=>] 75.8	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{\sin k}^{2}}\right)}\right) \]

   5. Taylor expanded in k around inf 68.7%

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

   6. Simplified 86.7%

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

      [Start] 68.7	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 68.7	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 68.7	\[ 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{\left(k \cdot k\right) \cdot \left({\sin k}^{2} \cdot t\right)} \]
      *-commutative [<=] 68.7	\[ 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}} \]
      associate-*l* [=>] 73.0	\[ 2 \cdot \frac{\cos k \cdot \left(\ell \cdot \ell\right)}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}} \]
      associate-/r* [=>] 76.1	\[ 2 \cdot \color{blue}{\frac{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
      associate-*l/ [<=] 76.1	\[ 2 \cdot \frac{\color{blue}{\frac{\cos k}{k} \cdot \left(\ell \cdot \ell\right)}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]
      *-commutative [<=] 76.1	\[ 2 \cdot \frac{\color{blue}{\left(\ell \cdot \ell\right) \cdot \frac{\cos k}{k}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)} \]
      associate-/l* [=>] 73.0	\[ 2 \cdot \color{blue}{\frac{\ell \cdot \ell}{\frac{k \cdot \left(t \cdot {\sin k}^{2}\right)}{\frac{\cos k}{k}}}} \]
      associate-*l/ [<=] 68.7	\[ 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{\frac{k}{\frac{\cos k}{k}} \cdot \left(t \cdot {\sin k}^{2}\right)}} \]
      times-frac [=>] 79.5	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{\frac{k}{\frac{\cos k}{k}}} \cdot \frac{\ell}{t \cdot {\sin k}^{2}}\right)} \]
      *-commutative [=>] 79.5	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{t \cdot {\sin k}^{2}} \cdot \frac{\ell}{\frac{k}{\frac{\cos k}{k}}}\right)} \]

   7. Applied egg-rr 89.4%

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

      [Start] 86.7	\[ 2 \cdot \left(\left({\sin k}^{-2} \cdot \frac{\ell}{t}\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k}\right)\right) \]
      associate-*r* [=>] 90.1	\[ 2 \cdot \color{blue}{\left(\left(\left({\sin k}^{-2} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{k}\right) \cdot \frac{\cos k}{k}\right)} \]
      frac-2neg [=>] 90.1	\[ 2 \cdot \left(\left(\left({\sin k}^{-2} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{k}\right) \cdot \color{blue}{\frac{-\cos k}{-k}}\right) \]
      associate-*r/ [=>] 90.1	\[ 2 \cdot \color{blue}{\frac{\left(\left({\sin k}^{-2} \cdot \frac{\ell}{t}\right) \cdot \frac{\ell}{k}\right) \cdot \left(-\cos k\right)}{-k}} \]
      associate-*l* [=>] 89.4	\[ 2 \cdot \frac{\color{blue}{\left({\sin k}^{-2} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right)\right)} \cdot \left(-\cos k\right)}{-k} \]

   8. Simplified 91.2%

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

      [Start] 89.4	\[ 2 \cdot \frac{\left({\sin k}^{-2} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right)\right) \cdot \left(-\cos k\right)}{-k} \]
      associate-/l* [=>] 89.4	\[ 2 \cdot \color{blue}{\frac{{\sin k}^{-2} \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{k}\right)}{\frac{-k}{-\cos k}}} \]
      associate-/l* [=>] 89.0	\[ 2 \cdot \color{blue}{\frac{{\sin k}^{-2}}{\frac{\frac{-k}{-\cos k}}{\frac{\ell}{t} \cdot \frac{\ell}{k}}}} \]
      neg-mul-1 [=>] 89.0	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{\frac{\color{blue}{-1 \cdot k}}{-\cos k}}{\frac{\ell}{t} \cdot \frac{\ell}{k}}} \]
      neg-mul-1 [=>] 89.0	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{\frac{-1 \cdot k}{\color{blue}{-1 \cdot \cos k}}}{\frac{\ell}{t} \cdot \frac{\ell}{k}}} \]
      times-frac [=>] 89.0	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{\color{blue}{\frac{-1}{-1} \cdot \frac{k}{\cos k}}}{\frac{\ell}{t} \cdot \frac{\ell}{k}}} \]
      metadata-eval [=>] 89.0	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{\color{blue}{1} \cdot \frac{k}{\cos k}}{\frac{\ell}{t} \cdot \frac{\ell}{k}}} \]
      associate-*l/ [=>] 84.5	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{1 \cdot \frac{k}{\cos k}}{\color{blue}{\frac{\ell \cdot \frac{\ell}{k}}{t}}}} \]
      associate-/l* [=>] 91.2	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{1 \cdot \frac{k}{\cos k}}{\color{blue}{\frac{\ell}{\frac{t}{\frac{\ell}{k}}}}}} \]

   9. Applied egg-rr 97.8%

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

      [Start] 91.2	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{1 \cdot \frac{k}{\cos k}}{\frac{\ell}{\frac{t}{\frac{\ell}{k}}}}} \]
      associate-/r/ [=>] 97.7	\[ 2 \cdot \frac{{\sin k}^{-2}}{\color{blue}{\frac{1 \cdot \frac{k}{\cos k}}{\ell} \cdot \frac{t}{\frac{\ell}{k}}}} \]
      div-inv [=>] 97.7	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{1 \cdot \frac{k}{\cos k}}{\ell} \cdot \color{blue}{\left(t \cdot \frac{1}{\frac{\ell}{k}}\right)}} \]
      clear-num [<=] 97.7	\[ 2 \cdot \frac{{\sin k}^{-2}}{\frac{1 \cdot \frac{k}{\cos k}}{\ell} \cdot \left(t \cdot \color{blue}{\frac{k}{\ell}}\right)} \]
      associate-*r* [=>] 97.7	\[ 2 \cdot \frac{{\sin k}^{-2}}{\color{blue}{\left(\frac{1 \cdot \frac{k}{\cos k}}{\ell} \cdot t\right) \cdot \frac{k}{\ell}}} \]
      *-un-lft-identity [<=] 97.7	\[ 2 \cdot \frac{{\sin k}^{-2}}{\left(\frac{\color{blue}{\frac{k}{\cos k}}}{\ell} \cdot t\right) \cdot \frac{k}{\ell}} \]
      associate-/l/ [=>] 97.8	\[ 2 \cdot \frac{{\sin k}^{-2}}{\left(\color{blue}{\frac{k}{\ell \cdot \cos k}} \cdot t\right) \cdot \frac{k}{\ell}} \]

   if -2.1999999999999999e-150 < k < 2.1499999999999999e-13

   1. Initial program 2.5%

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

   2. Simplified 10.4%

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

      [Start] 2.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      +-rgt-identity [<=] 2.5	\[ \frac{2}{\color{blue}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0}} \]
      associate-*l* [=>] 2.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)} + 0} \]
      mul0-rgt [<=] 2.2	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right) + \color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot 0}} \]
      distribute-lft-in [<=] 2.6	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) + 0\right)}} \]
      +-rgt-identity [=>] 2.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \color{blue}{\left(\tan k \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      sub-neg [=>] 2.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + \left(-1\right)\right)}\right)} \]
      +-commutative [=>] 2.6	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left(\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} + \left(-1\right)\right)\right)} \]
      associate-+l+ [=>] 10.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \left(-1\right)\right)\right)}\right)} \]
      metadata-eval [=>] 10.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \left(1 + \color{blue}{-1}\right)\right)\right)} \]
      metadata-eval [=>] 10.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \left({\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}\right)\right)} \]
      +-rgt-identity [=>] 10.4	\[ \frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \left(\tan k \cdot \color{blue}{{\left(\frac{k}{t}\right)}^{2}}\right)} \]

   3. Taylor expanded in k around 0 30.9%

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   4. Simplified 41.1%

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

      [Start] 30.9	\[ 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      unpow2 [=>] 30.9	\[ 2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t} \]
      *-commutative [=>] 30.9	\[ 2 \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}} \]
      times-frac [=>] 41.1	\[ 2 \cdot \color{blue}{\left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)} \]

   5. Applied egg-rr 49.5%

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

      [Start] 41.1	\[ 2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right) \]
      associate-*r/ [=>] 36.2	\[ 2 \cdot \color{blue}{\frac{\frac{\ell}{t} \cdot \ell}{{k}^{4}}} \]
      metadata-eval [<=] 36.2	\[ 2 \cdot \frac{\frac{\ell}{t} \cdot \ell}{{k}^{\color{blue}{\left(2 \cdot 2\right)}}} \]
      pow-sqr [<=] 36.1	\[ 2 \cdot \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{{k}^{2} \cdot {k}^{2}}} \]
      pow-prod-down [=>] 36.1	\[ 2 \cdot \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{{\left(k \cdot k\right)}^{2}}} \]
      pow2 [<=] 36.1	\[ 2 \cdot \frac{\frac{\ell}{t} \cdot \ell}{\color{blue}{\left(k \cdot k\right) \cdot \left(k \cdot k\right)}} \]
      associate-/r* [=>] 49.5	\[ 2 \cdot \color{blue}{\frac{\frac{\frac{\ell}{t} \cdot \ell}{k \cdot k}}{k \cdot k}} \]
      *-commutative [=>] 49.5	\[ 2 \cdot \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{t}}}{k \cdot k}}{k \cdot k} \]

   6. Applied egg-rr 89.4%

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

      [Start] 49.5	\[ 2 \cdot \frac{\frac{\ell \cdot \frac{\ell}{t}}{k \cdot k}}{k \cdot k} \]
      times-frac [=>] 65.4	\[ 2 \cdot \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{\frac{\ell}{t}}{k}}}{k \cdot k} \]
      times-frac [=>] 74.3	\[ 2 \cdot \color{blue}{\left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\frac{\ell}{t}}{k}}{k}\right)} \]
      associate-/l/ [=>] 89.4	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\color{blue}{\frac{\ell}{k \cdot t}}}{k}\right) \]

   7. Taylor expanded in l around 0 69.6%

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

   8. Simplified 94.7%

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

      [Start] 69.6	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{{k}^{2} \cdot t}\right) \]
      unpow2 [=>] 69.6	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{\color{blue}{\left(k \cdot k\right)} \cdot t}\right) \]
      associate-*r* [<=] 84.2	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{\color{blue}{k \cdot \left(k \cdot t\right)}}\right) \]
      associate-/r* [=>] 94.7	\[ 2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \color{blue}{\frac{\frac{\ell}{k}}{k \cdot t}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.2 \cdot 10^{-150} \lor \neg \left(k \leq 2.15 \cdot 10^{-13}\right):\\ \;\;\;\;2 \cdot \frac{{\sin k}^{-2}}{\left(\frac{k}{\ell \cdot \cos k} \cdot t\right) \cdot \frac{k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	93.4%
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -5.6 \cdot 10^{-144} \lor \neg \left(k \leq 10^{-15}\right):\\ \;\;\;\;2 \cdot \left({\sin k}^{-2} \cdot \left(\ell \cdot \left(\frac{\cos k}{k} \cdot \frac{\frac{\ell}{k}}{t}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \end{array} \]

Alternative 2
Accuracy	93.7%
Cost	20488

\[\begin{array}{l} t_1 := \frac{\cos k}{k}\\ t_2 := {\sin k}^{-2}\\ \mathbf{if}\;k \leq -2.4 \cdot 10^{-34}:\\ \;\;\;\;2 \cdot \left(\left(\ell \cdot \left(t_2 \cdot t_1\right)\right) \cdot \frac{\ell}{k \cdot t}\right)\\ \mathbf{elif}\;k \leq 8.4 \cdot 10^{-16}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(t_2 \cdot \left(\ell \cdot \left(t_1 \cdot \frac{\frac{\ell}{k}}{t}\right)\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	80.2%
Cost	14288

\[\begin{array}{l} t_1 := \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{if}\;\ell \leq -9.2 \cdot 10^{+151}:\\ \;\;\;\;2 \cdot \left(\left(\frac{\ell}{k} \cdot \frac{\frac{1}{t}}{k}\right) \cdot \left({\sin k}^{-2} \cdot \ell\right)\right)\\ \mathbf{elif}\;\ell \leq -1.5 \cdot 10^{-170}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \leq 3 \cdot 10^{-176}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{elif}\;\ell \leq 1.75 \cdot 10^{+146}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \frac{\ell}{t}\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k}\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	80.3%
Cost	14280

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+291}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \frac{\ell}{t}\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k}\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	66.2%
Cost	8004

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{-187}:\\ \;\;\;\;2 \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(\left(0.3333333333333333 + \frac{1}{k \cdot k}\right) \cdot \frac{\ell}{t}\right) \cdot \left(\frac{\ell}{k} \cdot \frac{\cos k}{k}\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	60.7%
Cost	960

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

Alternative 7
Accuracy	63.7%
Cost	960

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

Alternative 8
Accuracy	46.9%
Cost	832

\[2 \cdot \frac{-0.16666666666666666 \cdot \frac{\ell \cdot \ell}{t}}{k \cdot k} \]

Alternative 9
Accuracy	48.0%
Cost	832

\[2 \cdot \frac{\frac{\ell}{\frac{t}{\ell}}}{\frac{k \cdot k}{-0.16666666666666666}} \]

Reproduce

herbie shell --seed 2023152 
(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))))