Toniolo and Linder, Equation (10-)

Average Accuracy: 25.2% → 99.3%

Time: 38.3s

Precision: binary64

Cost: 14025

Math

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

↓

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k}\\ \mathbf{if}\;k \leq -2.9 \cdot 10^{-135} \lor \neg \left(k \leq 1.7 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{\ell \cdot 2}{k \cdot \tan k} \cdot \frac{\frac{t_1}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{2 \cdot t_1}{k \cdot t}\\ \end{array} \]

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 (/ l (sin k))))
   (if (or (<= k -2.9e-135) (not (<= k 1.7e-155)))
     (* (/ (* l 2.0) (* k (tan k))) (/ (/ t_1 k) t))
     (* (/ (/ l k) (tan k)) (/ (* 2.0 t_1) (* 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 t_1 = l / sin(k);
	double tmp;
	if ((k <= -2.9e-135) || !(k <= 1.7e-155)) {
		tmp = ((l * 2.0) / (k * tan(k))) * ((t_1 / k) / t);
	} else {
		tmp = ((l / k) / tan(k)) * ((2.0 * t_1) / (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) :: t_1
    real(8) :: tmp
    t_1 = l / sin(k)
    if ((k <= (-2.9d-135)) .or. (.not. (k <= 1.7d-155))) then
        tmp = ((l * 2.0d0) / (k * tan(k))) * ((t_1 / k) / t)
    else
        tmp = ((l / k) / tan(k)) * ((2.0d0 * t_1) / (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 t_1 = l / Math.sin(k);
	double tmp;
	if ((k <= -2.9e-135) || !(k <= 1.7e-155)) {
		tmp = ((l * 2.0) / (k * Math.tan(k))) * ((t_1 / k) / t);
	} else {
		tmp = ((l / k) / Math.tan(k)) * ((2.0 * t_1) / (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):
	t_1 = l / math.sin(k)
	tmp = 0
	if (k <= -2.9e-135) or not (k <= 1.7e-155):
		tmp = ((l * 2.0) / (k * math.tan(k))) * ((t_1 / k) / t)
	else:
		tmp = ((l / k) / math.tan(k)) * ((2.0 * t_1) / (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)
	t_1 = Float64(l / sin(k))
	tmp = 0.0
	if ((k <= -2.9e-135) || !(k <= 1.7e-155))
		tmp = Float64(Float64(Float64(l * 2.0) / Float64(k * tan(k))) * Float64(Float64(t_1 / k) / t));
	else
		tmp = Float64(Float64(Float64(l / k) / tan(k)) * Float64(Float64(2.0 * t_1) / 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)
	t_1 = l / sin(k);
	tmp = 0.0;
	if ((k <= -2.9e-135) || ~((k <= 1.7e-155)))
		tmp = ((l * 2.0) / (k * tan(k))) * ((t_1 / k) / t);
	else
		tmp = ((l / k) / tan(k)) * ((2.0 * t_1) / (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_] := Block[{t$95$1 = N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[k, -2.9e-135], N[Not[LessEqual[k, 1.7e-155]], $MachinePrecision]], N[(N[(N[(l * 2.0), $MachinePrecision] / N[(k * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t$95$1 / k), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * t$95$1), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < -2.9000000000000002e-135 or 1.7e-155 < k

   1. Initial program 26.0%

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

   2. Simplified 39.9%

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

      [Start] 26.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 26.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 26.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 26.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 25.8	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 25.8	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 26.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 26.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 39.1	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 39.1	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 39.1	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 39.9	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 70.2%

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

   4. Simplified 73.8%

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

      [Start] 70.2	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 70.2	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l* [=>] 73.8	\[ \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 75.8%

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

      [Start] 73.8	\[ \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      *-commutative [=>] 73.8	\[ \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}} \]
      associate-*r/ [=>] 72.7	\[ \color{blue}{\frac{\frac{\ell}{\sin k} \cdot \ell}{\tan k}} \cdot \frac{2}{k \cdot \left(k \cdot t\right)} \]
      associate-/r* [=>] 73.0	\[ \frac{\frac{\ell}{\sin k} \cdot \ell}{\tan k} \cdot \color{blue}{\frac{\frac{2}{k}}{k \cdot t}} \]
      frac-times [=>] 75.8	\[ \color{blue}{\frac{\left(\frac{\ell}{\sin k} \cdot \ell\right) \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)}} \]
      clear-num [=>] 75.8	\[ \frac{\left(\color{blue}{\frac{1}{\frac{\sin k}{\ell}}} \cdot \ell\right) \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)} \]
      associate-*l/ [=>] 75.8	\[ \frac{\color{blue}{\frac{1 \cdot \ell}{\frac{\sin k}{\ell}}} \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)} \]
      *-un-lft-identity [<=] 75.8	\[ \frac{\frac{\color{blue}{\ell}}{\frac{\sin k}{\ell}} \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)} \]

   6. Applied egg-rr 84.6%

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

      [Start] 75.8	\[ \frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)} \]
      frac-times [=>] 84.6	\[ \frac{\color{blue}{\frac{\ell \cdot 2}{\frac{\sin k}{\ell} \cdot k}}}{\tan k \cdot \left(k \cdot t\right)} \]
      *-commutative [=>] 84.6	\[ \frac{\frac{\ell \cdot 2}{\color{blue}{k \cdot \frac{\sin k}{\ell}}}}{\tan k \cdot \left(k \cdot t\right)} \]

   7. Applied egg-rr 99.5%

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

      [Start] 84.6	\[ \frac{\frac{\ell \cdot 2}{k \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left(k \cdot t\right)} \]
      div-inv [=>] 84.5	\[ \frac{\color{blue}{\left(\ell \cdot 2\right) \cdot \frac{1}{k \cdot \frac{\sin k}{\ell}}}}{\tan k \cdot \left(k \cdot t\right)} \]
      associate-*r* [=>] 84.5	\[ \frac{\left(\ell \cdot 2\right) \cdot \frac{1}{k \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left(\tan k \cdot k\right) \cdot t}} \]
      times-frac [=>] 99.4	\[ \color{blue}{\frac{\ell \cdot 2}{\tan k \cdot k} \cdot \frac{\frac{1}{k \cdot \frac{\sin k}{\ell}}}{t}} \]
      *-commutative [=>] 99.4	\[ \frac{\ell \cdot 2}{\color{blue}{k \cdot \tan k}} \cdot \frac{\frac{1}{k \cdot \frac{\sin k}{\ell}}}{t} \]
      *-commutative [=>] 99.4	\[ \frac{\ell \cdot 2}{k \cdot \tan k} \cdot \frac{\frac{1}{\color{blue}{\frac{\sin k}{\ell} \cdot k}}}{t} \]
      associate-/r* [=>] 99.4	\[ \frac{\ell \cdot 2}{k \cdot \tan k} \cdot \frac{\color{blue}{\frac{\frac{1}{\frac{\sin k}{\ell}}}{k}}}{t} \]
      clear-num [<=] 99.5	\[ \frac{\ell \cdot 2}{k \cdot \tan k} \cdot \frac{\frac{\color{blue}{\frac{\ell}{\sin k}}}{k}}{t} \]

   if -2.9000000000000002e-135 < k < 1.7e-155

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 0.0	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 19.5%

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

   4. Simplified 76.0%

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

      [Start] 19.5	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 19.5	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l* [=>] 76.0	\[ \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 55.6%

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

      [Start] 76.0	\[ \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      *-commutative [=>] 76.0	\[ \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}} \]
      associate-*r/ [=>] 54.6	\[ \color{blue}{\frac{\frac{\ell}{\sin k} \cdot \ell}{\tan k}} \cdot \frac{2}{k \cdot \left(k \cdot t\right)} \]
      associate-/r* [=>] 54.6	\[ \frac{\frac{\ell}{\sin k} \cdot \ell}{\tan k} \cdot \color{blue}{\frac{\frac{2}{k}}{k \cdot t}} \]
      frac-times [=>] 55.6	\[ \color{blue}{\frac{\left(\frac{\ell}{\sin k} \cdot \ell\right) \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)}} \]
      clear-num [=>] 55.5	\[ \frac{\left(\color{blue}{\frac{1}{\frac{\sin k}{\ell}}} \cdot \ell\right) \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)} \]
      associate-*l/ [=>] 55.6	\[ \frac{\color{blue}{\frac{1 \cdot \ell}{\frac{\sin k}{\ell}}} \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)} \]
      *-un-lft-identity [<=] 55.6	\[ \frac{\frac{\color{blue}{\ell}}{\frac{\sin k}{\ell}} \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)} \]

   6. Applied egg-rr 78.1%

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

      [Start] 55.6	\[ \frac{\frac{\ell}{\frac{\sin k}{\ell}} \cdot \frac{2}{k}}{\tan k \cdot \left(k \cdot t\right)} \]
      frac-times [=>] 78.1	\[ \frac{\color{blue}{\frac{\ell \cdot 2}{\frac{\sin k}{\ell} \cdot k}}}{\tan k \cdot \left(k \cdot t\right)} \]
      *-commutative [=>] 78.1	\[ \frac{\frac{\ell \cdot 2}{\color{blue}{k \cdot \frac{\sin k}{\ell}}}}{\tan k \cdot \left(k \cdot t\right)} \]

   7. Applied egg-rr 94.7%

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

      [Start] 78.1	\[ \frac{\frac{\ell \cdot 2}{k \cdot \frac{\sin k}{\ell}}}{\tan k \cdot \left(k \cdot t\right)} \]
      times-frac [=>] 78.0	\[ \frac{\color{blue}{\frac{\ell}{k} \cdot \frac{2}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left(k \cdot t\right)} \]
      times-frac [=>] 94.8	\[ \color{blue}{\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\frac{2}{\frac{\sin k}{\ell}}}{k \cdot t}} \]
      div-inv [=>] 94.8	\[ \frac{\frac{\ell}{k}}{\tan k} \cdot \frac{\color{blue}{2 \cdot \frac{1}{\frac{\sin k}{\ell}}}}{k \cdot t} \]
      clear-num [<=] 94.7	\[ \frac{\frac{\ell}{k}}{\tan k} \cdot \frac{2 \cdot \color{blue}{\frac{\ell}{\sin k}}}{k \cdot t} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2.9 \cdot 10^{-135} \lor \neg \left(k \leq 1.7 \cdot 10^{-155}\right):\\ \;\;\;\;\frac{\ell \cdot 2}{k \cdot \tan k} \cdot \frac{\frac{\frac{\ell}{\sin k}}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{\tan k} \cdot \frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	90.2%
Cost	14420

\[\begin{array}{l} t_1 := 2 \cdot \left(\ell \cdot \frac{\frac{\frac{\ell}{k}}{\sin k}}{\tan k \cdot \left(k \cdot t\right)}\right)\\ t_2 := 2 \cdot {\left(\frac{\ell}{k}\right)}^{2}\\ \mathbf{if}\;k \leq -2.65 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq -4.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)}}\\ \mathbf{elif}\;k \leq -8.6 \cdot 10^{-141}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\left(k \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{elif}\;k \leq 4.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{\frac{t_2}{k \cdot t}}{k}\\ \mathbf{elif}\;k \leq 1.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{t_2}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	91.0%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;\ell \leq -3.4 \cdot 10^{-179} \lor \neg \left(\ell \leq -8 \cdot 10^{-288}\right):\\ \;\;\;\;\frac{\ell}{k \cdot \left(\tan k \cdot t\right)} \cdot \frac{2}{k \cdot \frac{\sin k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \left(\frac{\frac{\ell}{k}}{k} \cdot \frac{\ell}{t}\right)}}\\ \end{array} \]

Alternative 3
Accuracy	99.3%
Cost	14025

\[\begin{array}{l} \mathbf{if}\;k \leq -3.5 \cdot 10^{-141} \lor \neg \left(k \leq 3.6 \cdot 10^{-156}\right):\\ \;\;\;\;\frac{\ell \cdot 2}{k \cdot \tan k} \cdot \frac{\frac{\frac{\ell}{\sin k}}{k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot t}}{k}\\ \end{array} \]

Alternative 4
Accuracy	63.0%
Cost	7876

\[\begin{array}{l} \mathbf{if}\;t \leq -1.5 \cdot 10^{-95}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\left(k \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{k}{\frac{\cos k}{k} \cdot \left(\frac{\ell}{t} \cdot \left(\frac{\ell}{k \cdot k} + \ell \cdot 0.3333333333333333\right)\right)}}\\ \end{array} \]

Alternative 5
Accuracy	61.7%
Cost	7624

\[\begin{array}{l} t_1 := \frac{k}{\frac{\ell}{k}}\\ \mathbf{if}\;t \leq -9.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\left(k \cdot t\right) \cdot t_1}\\ \mathbf{elif}\;t \leq 3.8 \cdot 10^{+43}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot 2}{t_1}}{\tan k \cdot \left(k \cdot t\right)}\\ \end{array} \]

Alternative 6
Accuracy	61.4%
Cost	7556

\[\begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-248}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\left(k \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \left(-0.16666666666666666 + \frac{\frac{1}{k}}{k}\right)}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 7
Accuracy	62.0%
Cost	7432

\[\begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-289}:\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\left(k \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot k}}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{k \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 8
Accuracy	61.2%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;k \leq -2.7 \cdot 10^{-138} \lor \neg \left(k \leq 1.8 \cdot 10^{-141}\right):\\ \;\;\;\;\frac{2}{\frac{k \cdot \left(\left(k \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{k \cdot \frac{k}{\ell}}\\ \end{array} \]

Alternative 9
Accuracy	62.7%
Cost	1225

\[\begin{array}{l} \mathbf{if}\;t \leq -5.9 \cdot 10^{-292} \lor \neg \left(t \leq 4.3 \cdot 10^{-194}\right):\\ \;\;\;\;\frac{\ell \cdot \frac{2}{k}}{\left(k \cdot t\right) \cdot \frac{k}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{\ell}{\frac{t}{\ell}}}{k \cdot k}}{k \cdot k}\\ \end{array} \]

Alternative 10
Accuracy	59.0%
Cost	960

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

Alternative 11
Accuracy	62.4%
Cost	960

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

Alternative 12
Accuracy	46.9%
Cost	704

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

Reproduce

herbie shell --seed 2023151 
(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))))