VandenBroeck and Keller, Equation (6)

Average Accuracy: 68.2% → 91.5%

Time: 12.0s

Precision: binary64

Cost: 32969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+35} \lor \neg \left(\pi \cdot \ell \leq 4000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+35) (not (<= (* PI l) 4000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (tan (* PI l)) F) F))))

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) <= -2e+35) || !((((double) M_PI) * l) <= 4000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((tan((((double) M_PI) * l)) / F) / F);
	}
	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) <= -2e+35) || !((Math.PI * l) <= 4000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.tan((Math.PI * l)) / F) / F);
	}
	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) <= -2e+35) or not ((math.pi * l) <= 4000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.tan((math.pi * l)) / F) / F)
	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) <= -2e+35) || !(Float64(pi * l) <= 4000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(tan(Float64(pi * l)) / F) / F));
	end
	return tmp
end

MATLAB

function tmp = code(F, l)
	tmp = (pi * l) - ((1.0 / (F * F)) * tan((pi * l)));
end

↓

function tmp_2 = code(F, l)
	tmp = 0.0;
	if (((pi * l) <= -2e+35) || ~(((pi * l) <= 4000000000000.0)))
		tmp = pi * l;
	else
		tmp = (pi * l) - ((tan((pi * l)) / F) / F);
	end
	tmp_2 = 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[Or[LessEqual[N[(Pi * l), $MachinePrecision], -2e+35], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 4000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1.9999999999999999e35 or 4e12 < (*.f64 (PI.f64) l)

   1. Initial program 63.5%

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

   2. Simplified 63.5%

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

      [Start] 63.5	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 63.5	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 63.5	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Taylor expanded in l around 0 50.0%

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

   4. Simplified 50.0%

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

      [Start] 50.0	\[ \pi \cdot \ell - \frac{\ell \cdot \pi}{{F}^{2}} \]
      *-commutative [<=] 50.0	\[ \pi \cdot \ell - \frac{\color{blue}{\pi \cdot \ell}}{{F}^{2}} \]
      unpow2 [=>] 50.0	\[ \pi \cdot \ell - \frac{\pi \cdot \ell}{\color{blue}{F \cdot F}} \]
      times-frac [=>] 50.0	\[ \pi \cdot \ell - \color{blue}{\frac{\pi}{F} \cdot \frac{\ell}{F}} \]

   5. Taylor expanded in F around inf 99.2%

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

   if -1.9999999999999999e35 < (*.f64 (PI.f64) l) < 4e12

   1. Initial program 72.4%

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

   2. Applied egg-rr 84.6%

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

      [Start] 72.4	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 73.0	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-un-lft-identity [<=] 73.0	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]
      associate-/r* [=>] 84.6	\[ \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+35} \lor \neg \left(\pi \cdot \ell \leq 4000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	91.9%
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -0.0001 \lor \neg \left(\pi \cdot \ell \leq 2000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 2
Accuracy	72.5%
Cost	7640

\[\begin{array}{l} t_0 := \left(\pi \cdot \ell + 1\right) + -1\\ t_1 := \frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{-46}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-95}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq -1.65 \cdot 10^{-218}:\\ \;\;\;\;\frac{\ell}{F} \cdot \frac{\pi}{-F}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-123}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 3
Accuracy	73.3%
Cost	7376

\[\begin{array}{l} t_0 := \pi \cdot \left(-\frac{\ell}{F \cdot F}\right)\\ \mathbf{if}\;F \leq -8.5 \cdot 10^{-47}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{-122}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 4
Accuracy	73.2%
Cost	7376

\[\begin{array}{l} t_0 := \frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{if}\;F \leq -2.1 \cdot 10^{-46}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-125}:\\ \;\;\;\;\left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{elif}\;F \leq 0.000102:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5
Accuracy	91.9%
Cost	7177

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.2 \cdot 10^{-5} \lor \neg \left(\ell \leq 950000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \frac{\frac{\ell}{F}}{F}\right)\\ \end{array} \]

Alternative 6
Accuracy	73.4%
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023153 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))