Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.6% → 96.6%

Time: 21.3s

Precision: binary64

Cost: 26692

• 35.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

Math

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

↓

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+102}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{2}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \end{array} \]

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 (<= (* l l) 1e+102)
   (* (/ l (tan k)) (/ (* (/ l (sin k)) (/ 2.0 k)) (* k t)))
   (* 2.0 (/ (cos k) (/ (* t (pow (sin k) 2.0)) (pow (/ l 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 ((l * l) <= 1e+102) {
		tmp = (l / tan(k)) * (((l / sin(k)) * (2.0 / k)) / (k * t));
	} else {
		tmp = 2.0 * (cos(k) / ((t * pow(sin(k), 2.0)) / pow((l / 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 ((l * l) <= 1d+102) then
        tmp = (l / tan(k)) * (((l / sin(k)) * (2.0d0 / k)) / (k * t))
    else
        tmp = 2.0d0 * (cos(k) / ((t * (sin(k) ** 2.0d0)) / ((l / 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 ((l * l) <= 1e+102) {
		tmp = (l / Math.tan(k)) * (((l / Math.sin(k)) * (2.0 / k)) / (k * t));
	} else {
		tmp = 2.0 * (Math.cos(k) / ((t * Math.pow(Math.sin(k), 2.0)) / Math.pow((l / 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 (l * l) <= 1e+102:
		tmp = (l / math.tan(k)) * (((l / math.sin(k)) * (2.0 / k)) / (k * t))
	else:
		tmp = 2.0 * (math.cos(k) / ((t * math.pow(math.sin(k), 2.0)) / math.pow((l / 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 (Float64(l * l) <= 1e+102)
		tmp = Float64(Float64(l / tan(k)) * Float64(Float64(Float64(l / sin(k)) * Float64(2.0 / k)) / Float64(k * t)));
	else
		tmp = Float64(2.0 * Float64(cos(k) / Float64(Float64(t * (sin(k) ^ 2.0)) / (Float64(l / 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 ((l * l) <= 1e+102)
		tmp = (l / tan(k)) * (((l / sin(k)) * (2.0 / k)) / (k * t));
	else
		tmp = 2.0 * (cos(k) / ((t * (sin(k) ^ 2.0)) / ((l / 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[LessEqual[N[(l * l), $MachinePrecision], 1e+102], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(2.0 / k), $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[Cos[k], $MachinePrecision] / N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[(l / k), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 9.99999999999999977e101

   1. Initial program 36.9%

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

   2. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]
      Step-by-step derivation

      [Start] 36.9%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 36.9%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 36.9%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 36.9%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 36.2%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 36.2%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 37.2%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 37.2%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 49.7%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 49.7%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 49.7%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 57.3%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 91.3%

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

   4. Simplified 91.3%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      Step-by-step derivation

      [Start] 91.3%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 91.3%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
      Step-by-step derivation

      [Start] 91.3%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l/ [=>] 91.3%	\[ \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]
      associate-*l* [=>] 95.9%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   6. Simplified 97.4%

      \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k}}{k \cdot t}\right)} \]
      Step-by-step derivation

      [Start] 95.9%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)} \]
      *-commutative [=>] 95.9%	\[ \frac{\color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot 2}}{k \cdot \left(k \cdot t\right)} \]
      associate-*r/ [<=] 95.9%	\[ \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}} \]
      associate-*r* [=>] 91.3%	\[ \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
      unpow2 [<=] 91.3%	\[ \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{\color{blue}{{k}^{2}} \cdot t} \]
      associate-/r* [=>] 91.5%	\[ \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \]
      unpow2 [=>] 91.5%	\[ \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \]
      *-commutative [=>] 91.5%	\[ \color{blue}{\left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\frac{2}{k \cdot k}}{t} \]
      associate-*l* [=>] 92.7%	\[ \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k \cdot k}}{t}\right)} \]
      associate-/r* [=>] 92.8%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{2}{k}}{k}}}{t}\right) \]
      associate-/l/ [=>] 97.4%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \color{blue}{\frac{\frac{2}{k}}{t \cdot k}}\right) \]
      *-commutative [<=] 97.4%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k}}{\color{blue}{k \cdot t}}\right) \]

   7. Applied egg-rr 99.1%

      \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{\frac{\ell}{\sin k} \cdot \frac{2}{k}}{k \cdot t}} \]
      Step-by-step derivation

      [Start] 97.4%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k}}{k \cdot t}\right) \]
      associate-*r/ [=>] 99.1%	\[ \frac{\ell}{\tan k} \cdot \color{blue}{\frac{\frac{\ell}{\sin k} \cdot \frac{2}{k}}{k \cdot t}} \]

   if 9.99999999999999977e101 < (*.f64 l l)

   1. Initial program 36.0%

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

   2. Simplified 40.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]
      Step-by-step derivation

      [Start] 36.0%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 36.0%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 36.0%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 36.0%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 36.0%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 36.0%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 35.9%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 35.9%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 40.4%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 40.4%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 40.4%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 40.4%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 70.1%

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

   4. Simplified 70.1%

      \[\leadsto \color{blue}{\frac{2}{\left(k \cdot k\right) \cdot t}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      Step-by-step derivation

      [Start] 70.1%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 70.1%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 74.5%

      \[\leadsto \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}} \]
      Step-by-step derivation

      [Start] 70.1%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l/ [=>] 70.1%	\[ \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]
      associate-*l* [=>] 74.5%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   6. Simplified 83.3%

      \[\leadsto \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k}}{k \cdot t}\right)} \]
      Step-by-step derivation

      [Start] 74.5%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)} \]
      *-commutative [=>] 74.5%	\[ \frac{\color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot 2}}{k \cdot \left(k \cdot t\right)} \]
      associate-*r/ [<=] 74.5%	\[ \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{k \cdot \left(k \cdot t\right)}} \]
      associate-*r* [=>] 70.1%	\[ \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{\color{blue}{\left(k \cdot k\right) \cdot t}} \]
      unpow2 [<=] 70.1%	\[ \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{2}{\color{blue}{{k}^{2}} \cdot t} \]
      associate-/r* [=>] 70.1%	\[ \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \color{blue}{\frac{\frac{2}{{k}^{2}}}{t}} \]
      unpow2 [=>] 70.1%	\[ \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \cdot \frac{\frac{2}{\color{blue}{k \cdot k}}}{t} \]
      *-commutative [=>] 70.1%	\[ \color{blue}{\left(\frac{\ell}{\tan k} \cdot \frac{\ell}{\sin k}\right)} \cdot \frac{\frac{2}{k \cdot k}}{t} \]
      associate-*l* [=>] 77.6%	\[ \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k \cdot k}}{t}\right)} \]
      associate-/r* [=>] 78.5%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\color{blue}{\frac{\frac{2}{k}}{k}}}{t}\right) \]
      associate-/l/ [=>] 83.3%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \color{blue}{\frac{\frac{2}{k}}{t \cdot k}}\right) \]
      *-commutative [<=] 83.3%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k}}{\color{blue}{k \cdot t}}\right) \]

   7. Applied egg-rr 91.4%

      \[\leadsto \frac{\ell}{\tan k} \cdot \color{blue}{\frac{\frac{\ell}{\sin k} \cdot \frac{2}{k}}{k \cdot t}} \]
      Step-by-step derivation

      [Start] 83.3%	\[ \frac{\ell}{\tan k} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\frac{2}{k}}{k \cdot t}\right) \]
      associate-*r/ [=>] 91.4%	\[ \frac{\ell}{\tan k} \cdot \color{blue}{\frac{\frac{\ell}{\sin k} \cdot \frac{2}{k}}{k \cdot t}} \]

   8. Taylor expanded in l around 0 70.1%

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

   9. Simplified 97.2%

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

      [Start] 70.1%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      unpow2 [=>] 70.1%	\[ 2 \cdot \frac{\cos k \cdot \color{blue}{\left(\ell \cdot \ell\right)}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      associate-/l* [=>] 70.1%	\[ 2 \cdot \color{blue}{\frac{\cos k}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\ell \cdot \ell}}} \]
      unpow2 [=>] 70.1%	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)}{\ell \cdot \ell}} \]
      *-commutative [=>] 70.1%	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{\left({\sin k}^{2} \cdot t\right) \cdot \left(k \cdot k\right)}}{\ell \cdot \ell}} \]
      associate-/l* [=>] 72.0%	\[ 2 \cdot \frac{\cos k}{\color{blue}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell \cdot \ell}{k \cdot k}}}} \]
      *-commutative [=>] 72.0%	\[ 2 \cdot \frac{\cos k}{\frac{\color{blue}{t \cdot {\sin k}^{2}}}{\frac{\ell \cdot \ell}{k \cdot k}}} \]
      times-frac [=>] 97.2%	\[ 2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\color{blue}{\frac{\ell}{k} \cdot \frac{\ell}{k}}}} \]
      unpow2 [<=] 97.2%	\[ 2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{\color{blue}{{\left(\frac{\ell}{k}\right)}^{2}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+102}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{2}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	96.6%
Cost	26692

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+102}:\\ \;\;\;\;\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{2}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k}{\frac{t \cdot {\sin k}^{2}}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 2
Accuracy	81.9%
Cost	14020

\[\begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-303}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{k} \cdot \frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(t_1 \cdot \frac{\ell}{\sin k}\right)\\ \end{array} \]

Alternative 3
Accuracy	84.8%
Cost	14020

\[\begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-303}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{k} \cdot \frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(2 \cdot \frac{\ell}{\left(k \cdot k\right) \cdot \left(\sin k \cdot t\right)}\right)\\ \end{array} \]

Alternative 4
Accuracy	93.8%
Cost	14020

\[\begin{array}{l} t_1 := \frac{\ell}{\tan k}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-303}:\\ \;\;\;\;t_1 \cdot \frac{\frac{2}{k} \cdot \frac{\ell}{k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\frac{\ell}{k \cdot t} \cdot \frac{\frac{2}{k}}{\sin k}\right)\\ \end{array} \]

Alternative 5
Accuracy	86.2%
Cost	13760

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

Alternative 6
Accuracy	89.7%
Cost	13760

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

Alternative 7
Accuracy	94.8%
Cost	13760

\[\frac{\ell}{\tan k} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{2}{k}}{k \cdot t} \]

Alternative 8
Accuracy	72.5%
Cost	7360

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

Alternative 9
Accuracy	73.4%
Cost	7360

\[\frac{\ell}{\tan k} \cdot \frac{\frac{2}{k} \cdot \frac{\ell}{k}}{k \cdot t} \]

Alternative 10
Accuracy	71.8%
Cost	7300

\[\begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{2 \cdot {\left(\frac{\ell}{k}\right)}^{2}}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 11
Accuracy	70.4%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 12
Accuracy	70.4%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\frac{k \cdot k}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 13
Accuracy	71.7%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;k \leq 3.5 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{2}{k}}{k \cdot t} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{k} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{t}\right)\\ \end{array} \]

Alternative 14
Accuracy	34.6%
Cost	704

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

Alternative 15
Accuracy	35.1%
Cost	704

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

Alternative 16
Accuracy	35.4%
Cost	704

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

Reproduce

herbie shell --seed 2023167 
(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))))