Toniolo and Linder, Equation (10-)

Percentage Accurate: 35.8% → 89.7%

Time: 26.9s

Precision: binary64

Cost: 20356

• 39.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot 2} \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{\frac{t \cdot {\sin k}^{2}}{\ell}}\right)\\ \end{array} \]

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 (<= t 1.5e+113)
   (/ (/ l (tan k)) (* (/ (* k (* t k)) (* l 2.0)) (sin k)))
   (* 2.0 (* (/ (cos k) (* k k)) (/ l (/ (* t (pow (sin k) 2.0)) l))))))

C

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

↓

double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.5e+113) {
		tmp = (l / tan(k)) / (((k * (t * k)) / (l * 2.0)) * sin(k));
	} else {
		tmp = 2.0 * ((cos(k) / (k * k)) * (l / ((t * pow(sin(k), 2.0)) / l)));
	}
	return tmp;
}

Fortran

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t <= 1.5d+113) then
        tmp = (l / tan(k)) / (((k * (t * k)) / (l * 2.0d0)) * sin(k))
    else
        tmp = 2.0d0 * ((cos(k) / (k * k)) * (l / ((t * (sin(k) ** 2.0d0)) / l)))
    end if
    code = tmp
end function

Java

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

↓

public static double code(double t, double l, double k) {
	double tmp;
	if (t <= 1.5e+113) {
		tmp = (l / Math.tan(k)) / (((k * (t * k)) / (l * 2.0)) * Math.sin(k));
	} else {
		tmp = 2.0 * ((Math.cos(k) / (k * k)) * (l / ((t * Math.pow(Math.sin(k), 2.0)) / l)));
	}
	return tmp;
}

Python

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

↓

def code(t, l, k):
	tmp = 0
	if t <= 1.5e+113:
		tmp = (l / math.tan(k)) / (((k * (t * k)) / (l * 2.0)) * math.sin(k))
	else:
		tmp = 2.0 * ((math.cos(k) / (k * k)) * (l / ((t * math.pow(math.sin(k), 2.0)) / l)))
	return tmp

Julia

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

↓

function code(t, l, k)
	tmp = 0.0
	if (t <= 1.5e+113)
		tmp = Float64(Float64(l / tan(k)) / Float64(Float64(Float64(k * Float64(t * k)) / Float64(l * 2.0)) * sin(k)));
	else
		tmp = Float64(2.0 * Float64(Float64(cos(k) / Float64(k * k)) * Float64(l / Float64(Float64(t * (sin(k) ^ 2.0)) / l))));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if (t <= 1.5e+113)
		tmp = (l / tan(k)) / (((k * (t * k)) / (l * 2.0)) * sin(k));
	else
		tmp = 2.0 * ((cos(k) / (k * k)) * (l / ((t * (sin(k) ^ 2.0)) / l)));
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[t_, l_, k_] := If[LessEqual[t, 1.5e+113], N[(N[(l / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision] / N[(l * 2.0), $MachinePrecision]), $MachinePrecision] * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision] * N[(l / N[(N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e113

   1. Initial program 39.6%

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

   2. Simplified 53.5%

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

      [Start] 39.6%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 39.6%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 39.6%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 39.6%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 39.1%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 39.1%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 39.6%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 39.6%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 45.8%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 45.8%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 45.8%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 53.5%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 83.3%

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

   4. Simplified 83.3%

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

      [Start] 83.3%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 83.3%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 87.8%

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

      [Start] 83.3%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l/ [=>] 83.3%	\[ \color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\left(k \cdot k\right) \cdot t}} \]
      associate-*l* [=>] 87.8%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{k \cdot \left(k \cdot t\right)}} \]

   6. Simplified 92.8%

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

      [Start] 87.8%	\[ \frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)} \]
      associate-*l/ [<=] 87.7%	\[ \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]
      associate-*r* [=>] 92.4%	\[ \color{blue}{\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\sin k}\right) \cdot \frac{\ell}{\tan k}} \]
      *-commutative [=>] 92.4%	\[ \color{blue}{\frac{\ell}{\tan k} \cdot \left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\sin k}\right)} \]
      associate-*l/ [=>] 92.8%	\[ \frac{\ell}{\tan k} \cdot \color{blue}{\frac{2 \cdot \frac{\ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
      associate-*r/ [=>] 92.8%	\[ \frac{\ell}{\tan k} \cdot \frac{\color{blue}{\frac{2 \cdot \ell}{\sin k}}}{k \cdot \left(k \cdot t\right)} \]

   7. Applied egg-rr 92.8%

      \[\leadsto \color{blue}{\left(\ell \cdot \frac{1}{\tan k}\right)} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \]
      Step-by-step derivation

      [Start] 92.8%	\[ \frac{\ell}{\tan k} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \]
      div-inv [=>] 92.8%	\[ \color{blue}{\left(\ell \cdot \frac{1}{\tan k}\right)} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \]

   8. Applied egg-rr 87.8%

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

      [Start] 92.8%	\[ \left(\ell \cdot \frac{1}{\tan k}\right) \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \]
      div-inv [<=] 92.8%	\[ \color{blue}{\frac{\ell}{\tan k}} \cdot \frac{\frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)} \]
      associate-*r/ [=>] 87.8%	\[ \color{blue}{\frac{\frac{\ell}{\tan k} \cdot \frac{2 \cdot \ell}{\sin k}}{k \cdot \left(k \cdot t\right)}} \]
      *-commutative [=>] 87.8%	\[ \frac{\frac{\ell}{\tan k} \cdot \frac{\color{blue}{\ell \cdot 2}}{\sin k}}{k \cdot \left(k \cdot t\right)} \]

   9. Simplified 92.8%

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

      [Start] 87.8%	\[ \frac{\frac{\ell}{\tan k} \cdot \frac{\ell \cdot 2}{\sin k}}{k \cdot \left(k \cdot t\right)} \]
      associate-/l* [=>] 92.8%	\[ \color{blue}{\frac{\frac{\ell}{\tan k}}{\frac{k \cdot \left(k \cdot t\right)}{\frac{\ell \cdot 2}{\sin k}}}} \]
      associate-/r/ [=>] 92.8%	\[ \frac{\frac{\ell}{\tan k}}{\color{blue}{\frac{k \cdot \left(k \cdot t\right)}{\ell \cdot 2} \cdot \sin k}} \]
      *-commutative [=>] 92.8%	\[ \frac{\frac{\ell}{\tan k}}{\frac{k \cdot \left(k \cdot t\right)}{\color{blue}{2 \cdot \ell}} \cdot \sin k} \]

   if 1.5e113 < t

   1. Initial program 5.9%

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

   2. Simplified 21.0%

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

      [Start] 5.9%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 5.9%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 5.9%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 5.9%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 5.9%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 5.9%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 5.9%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 5.9%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 21.0%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 21.0%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 21.0%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 21.0%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 65.4%

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

   4. Simplified 65.4%

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

      [Start] 65.4%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 65.4%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Taylor expanded in k around inf 64.9%

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

   6. Simplified 91.1%

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

      [Start] 64.9%	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 68.3%	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 68.3%	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 68.3%	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 68.3%	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]
      associate-/l* [=>] 91.1%	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \color{blue}{\frac{\ell}{\frac{t \cdot {\sin k}^{2}}{\ell}}}\right) \]

3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot 2} \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{\frac{t \cdot {\sin k}^{2}}{\ell}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	89.7%
Cost	20356

\[\begin{array}{l} \mathbf{if}\;t \leq 1.5 \cdot 10^{+113}:\\ \;\;\;\;\frac{\frac{\ell}{\tan k}}{\frac{k \cdot \left(t \cdot k\right)}{\ell \cdot 2} \cdot \sin k}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell}{\frac{t \cdot {\sin k}^{2}}{\ell}}\right)\\ \end{array} \]

Alternative 2
Accuracy	89.9%
Cost	13888

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

Alternative 3
Accuracy	89.7%
Cost	13760

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

Alternative 4
Accuracy	89.9%
Cost	13760

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

Alternative 5
Accuracy	89.9%
Cost	13760

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

Alternative 6
Accuracy	89.9%
Cost	13760

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

Alternative 7
Accuracy	74.6%
Cost	7360

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

Alternative 8
Accuracy	71.6%
Cost	960

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

Alternative 9
Accuracy	71.6%
Cost	960

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

Alternative 10
Accuracy	73.1%
Cost	960

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

Alternative 11
Accuracy	30.4%
Cost	704

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

Reproduce

herbie shell --seed 2023269 
(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))))