VandenBroeck and Keller, Equation (6)

Percentage Accurate: 57.5% → 97.6%

Time: 18.1s

Precision: binary64

Cost: 45712

• 44.4% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\log \left(\frac{1}{{\left(e^{\ell}\right)}^{\pi}}\right)\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

FPCore

(FPCore (F l)
 :precision binary64
 (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))

↓

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -5e+84)
   (log (/ 1.0 (pow (exp l) PI)))
   (if (<= (* PI l) -5e+26)
     (* PI l)
     (if (<= (* PI l) 200000000000.0)
       (- (* PI l) (/ (/ (tan (* PI l)) F) F))
       (if (<= (* PI l) 2e+75) (* PI l) (log1p (expm1 (* PI l))))))))

C

double code(double F, double l) {
	return (((double) M_PI) * l) - ((1.0 / (F * F)) * tan((((double) M_PI) * l)));
}

↓

double code(double F, double l) {
	double tmp;
	if ((((double) M_PI) * l) <= -5e+84) {
		tmp = log((1.0 / pow(exp(l), ((double) M_PI))));
	} else if ((((double) M_PI) * l) <= -5e+26) {
		tmp = ((double) M_PI) * l;
	} else if ((((double) M_PI) * l) <= 200000000000.0) {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	} else if ((((double) M_PI) * l) <= 2e+75) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = log1p(expm1((((double) M_PI) * l)));
	}
	return tmp;
}

Java

public static double code(double F, double l) {
	return (Math.PI * l) - ((1.0 / (F * F)) * Math.tan((Math.PI * l)));
}

↓

public static double code(double F, double l) {
	double tmp;
	if ((Math.PI * l) <= -5e+84) {
		tmp = Math.log((1.0 / Math.pow(Math.exp(l), Math.PI)));
	} else if ((Math.PI * l) <= -5e+26) {
		tmp = Math.PI * l;
	} else if ((Math.PI * l) <= 200000000000.0) {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	} else if ((Math.PI * l) <= 2e+75) {
		tmp = Math.PI * l;
	} else {
		tmp = Math.log1p(Math.expm1((Math.PI * l)));
	}
	return tmp;
}

Python

def code(F, l):
	return (math.pi * l) - ((1.0 / (F * F)) * math.tan((math.pi * l)))

↓

def code(F, l):
	tmp = 0
	if (math.pi * l) <= -5e+84:
		tmp = math.log((1.0 / math.pow(math.exp(l), math.pi)))
	elif (math.pi * l) <= -5e+26:
		tmp = math.pi * l
	elif (math.pi * l) <= 200000000000.0:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	elif (math.pi * l) <= 2e+75:
		tmp = math.pi * l
	else:
		tmp = math.log1p(math.expm1((math.pi * l)))
	return tmp

Julia

function code(F, l)
	return Float64(Float64(pi * l) - Float64(Float64(1.0 / Float64(F * F)) * tan(Float64(pi * l))))
end

↓

function code(F, l)
	tmp = 0.0
	if (Float64(pi * l) <= -5e+84)
		tmp = log(Float64(1.0 / (exp(l) ^ pi)));
	elseif (Float64(pi * l) <= -5e+26)
		tmp = Float64(pi * l);
	elseif (Float64(pi * l) <= 200000000000.0)
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	elseif (Float64(pi * l) <= 2e+75)
		tmp = Float64(pi * l);
	else
		tmp = log1p(expm1(Float64(pi * l)));
	end
	return tmp
end

Wolfram

code[F_, l_] := N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[F_, l_] := If[LessEqual[N[(Pi * l), $MachinePrecision], -5e+84], N[Log[N[(1.0 / N[Power[N[Exp[l], $MachinePrecision], Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], -5e+26], N[(Pi * l), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 200000000000.0], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 2e+75], N[(Pi * l), $MachinePrecision], N[Log[1 + N[(Exp[N[(Pi * l), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (PI.f64) l) < -5.0000000000000001e84

   1. Initial program 15.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]

   2. Taylor expanded in l around 0 50.7%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right)} \]

   3. Simplified 50.7%

      \[\leadsto \color{blue}{\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)} \]
      Step-by-step derivation

      [Start] 50.7%	\[ \ell \cdot \left(\pi - \frac{\pi}{{F}^{2}}\right) \]
      unpow2 [=>] 50.7%	\[ \ell \cdot \left(\pi - \frac{\pi}{\color{blue}{F \cdot F}}\right) \]

   4. Applied egg-rr 50.7%

      \[\leadsto \ell \cdot \color{blue}{\sqrt[3]{{\left(\pi - \pi \cdot {F}^{-2}\right)}^{3}}} \]
      Step-by-step derivation

      [Start] 50.7%	\[ \ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right) \]
      add-cbrt-cube [=>] 50.7%	\[ \ell \cdot \color{blue}{\sqrt[3]{\left(\left(\pi - \frac{\pi}{F \cdot F}\right) \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\right) \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)}} \]
      pow3 [=>] 50.7%	\[ \ell \cdot \sqrt[3]{\color{blue}{{\left(\pi - \frac{\pi}{F \cdot F}\right)}^{3}}} \]
      div-inv [=>] 50.7%	\[ \ell \cdot \sqrt[3]{{\left(\pi - \color{blue}{\pi \cdot \frac{1}{F \cdot F}}\right)}^{3}} \]
      pow2 [=>] 50.7%	\[ \ell \cdot \sqrt[3]{{\left(\pi - \pi \cdot \frac{1}{\color{blue}{{F}^{2}}}\right)}^{3}} \]
      pow-flip [=>] 50.7%	\[ \ell \cdot \sqrt[3]{{\left(\pi - \pi \cdot \color{blue}{{F}^{\left(-2\right)}}\right)}^{3}} \]
      metadata-eval [=>] 50.7%	\[ \ell \cdot \sqrt[3]{{\left(\pi - \pi \cdot {F}^{\color{blue}{-2}}\right)}^{3}} \]

   5. Taylor expanded in F around inf 0.2%

      \[\leadsto \ell \cdot \sqrt[3]{\color{blue}{{\pi}^{3}}} \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\log \left(\frac{1}{{\left(e^{\ell}\right)}^{\pi}}\right)} \]
      Step-by-step derivation

      [Start] 0.2%	\[ \ell \cdot \sqrt[3]{{\pi}^{3}} \]
      rem-cbrt-cube [=>] 0.2%	\[ \ell \cdot \color{blue}{\pi} \]
      add-sqr-sqrt [=>] 0.2%	\[ \ell \cdot \color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)} \]
      sqrt-unprod [=>] 0.2%	\[ \ell \cdot \color{blue}{\sqrt{\pi \cdot \pi}} \]
      sqr-neg [<=] 0.2%	\[ \ell \cdot \sqrt{\color{blue}{\left(-\pi\right) \cdot \left(-\pi\right)}} \]
      sqrt-unprod [<=] 0.0%	\[ \ell \cdot \color{blue}{\left(\sqrt{-\pi} \cdot \sqrt{-\pi}\right)} \]
      add-sqr-sqrt [<=] 6.1%	\[ \ell \cdot \color{blue}{\left(-\pi\right)} \]
      distribute-rgt-neg-in [<=] 6.1%	\[ \color{blue}{-\ell \cdot \pi} \]
      add-log-exp [=>] 100.0%	\[ -\color{blue}{\log \left(e^{\ell \cdot \pi}\right)} \]
      neg-log [=>] 100.0%	\[ \color{blue}{\log \left(\frac{1}{e^{\ell \cdot \pi}}\right)} \]
      exp-prod [=>] 100.0%	\[ \log \left(\frac{1}{\color{blue}{{\left(e^{\ell}\right)}^{\pi}}}\right) \]

   if -5.0000000000000001e84 < (*.f64 (PI.f64) l) < -5.0000000000000001e26 or 2e11 < (*.f64 (PI.f64) l) < 1.99999999999999985e75

   1. Initial program 44.8%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]

   2. Taylor expanded in l around 0 32.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]

   3. Simplified 32.7%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]
      Step-by-step derivation

      [Start] 32.7%	\[ \pi \cdot \ell - \frac{\ell \cdot \pi}{{F}^{2}} \]
      unpow2 [=>] 32.7%	\[ \pi \cdot \ell - \frac{\ell \cdot \pi}{\color{blue}{F \cdot F}} \]
      times-frac [=>] 32.7%	\[ \pi \cdot \ell - \color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]

   4. Taylor expanded in F around inf 85.5%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   if -5.0000000000000001e26 < (*.f64 (PI.f64) l) < 2e11

   1. Initial program 91.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]

   2. Applied egg-rr 99.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]
      Step-by-step derivation

      [Start] 91.4%	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 91.4%	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-un-lft-identity [<=] 91.4%	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]
      associate-/r* [=>] 99.6%	\[ \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

   if 1.99999999999999985e75 < (*.f64 (PI.f64) l)

   1. Initial program 12.4%

      \[\pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]

   2. Taylor expanded in l around 0 3.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell \cdot \pi}{{F}^{2}}} \]

   3. Simplified 3.6%

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]
      Step-by-step derivation

      [Start] 3.6%	\[ \pi \cdot \ell - \frac{\ell \cdot \pi}{{F}^{2}} \]
      unpow2 [=>] 3.6%	\[ \pi \cdot \ell - \frac{\ell \cdot \pi}{\color{blue}{F \cdot F}} \]
      times-frac [=>] 3.6%	\[ \pi \cdot \ell - \color{blue}{\frac{\ell}{F} \cdot \frac{\pi}{F}} \]

   4. Taylor expanded in F around inf 6.2%

      \[\leadsto \color{blue}{\ell \cdot \pi} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)} \]
      Step-by-step derivation

      [Start] 6.2%	\[ \ell \cdot \pi \]
      *-commutative [=>] 6.2%	\[ \color{blue}{\pi \cdot \ell} \]
      log1p-expm1-u [=>] 100.0%	\[ \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\log \left(\frac{1}{{\left(e^{\ell}\right)}^{\pi}}\right)\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	97.6%
Cost	45712

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\log \left(\frac{1}{{\left(e^{\ell}\right)}^{\pi}}\right)\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	92.9%
Cost	52308

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+160}:\\ \;\;\;\;\sqrt{{\left(\pi \cdot \ell\right)}^{2}}\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\left|\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\right|\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternative 3
Accuracy	86.3%
Cost	45712

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+84}:\\ \;\;\;\;\left|\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\right|\\ \mathbf{elif}\;\pi \cdot \ell \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternative 4
Accuracy	94.0%
Cost	39116

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+105}:\\ \;\;\;\;{\left({\left(\pi \cdot \ell\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternative 5
Accuracy	94.8%
Cost	39116

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+105}:\\ \;\;\;\;{\left({\left(\pi \cdot \ell\right)}^{3}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;\pi \cdot \ell \leq 200000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{elif}\;\pi \cdot \ell \leq 2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(\pi \cdot \ell\right)\right)\\ \end{array} \]

Alternative 6
Accuracy	70.8%
Cost	13776

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+76}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 106000000000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 + {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 7
Accuracy	71.2%
Cost	13776

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+76}:\\ \;\;\;\;\left|\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\right|\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 106000000000:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 + {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	76.9%
Cost	13776

\[\begin{array}{l} \mathbf{if}\;\ell \leq -1.05 \cdot 10^{+76}:\\ \;\;\;\;\left|\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\right|\\ \mathbf{elif}\;\ell \leq -3.3 \cdot 10^{+15}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;\ell \leq 106000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \mathbf{elif}\;\ell \leq 5.2 \cdot 10^{+75}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi \cdot \left(1 + {F}^{-2}\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	57.8%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\ell \cdot \frac{-\pi}{F \cdot F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 10
Accuracy	63.5%
Cost	7108

\[\begin{array}{l} \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-19}:\\ \;\;\;\;\frac{\pi}{F} \cdot \left(-\frac{\ell}{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 11
Accuracy	44.0%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+95}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\frac{F}{\frac{\pi}{F}}}\\ \end{array} \]

Alternative 12
Accuracy	44.0%
Cost	6916

\[\begin{array}{l} \mathbf{if}\;\ell \leq 3 \cdot 10^{+95}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\pi}{F}}{\frac{F}{\ell}}\\ \end{array} \]

Alternative 13
Accuracy	37.8%
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023255 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))