VandenBroeck and Keller, Equation (6)

Average Error: 16.1 → 0.6

Time: 10.9s

Precision: binary64

Cost: 32969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -4e+18) (not (<= (* PI l) 1e+14)))
   (* 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) <= -4e+18) || !((((double) M_PI) * l) <= 1e+14)) {
		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) <= -4e+18) || !((Math.PI * l) <= 1e+14)) {
		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) <= -4e+18) or not ((math.pi * l) <= 1e+14):
		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) <= -4e+18) || !(Float64(pi * l) <= 1e+14))
		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) <= -4e+18) || ~(((pi * l) <= 1e+14)))
		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], -4e+18], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 1e+14]], $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) < -4e18 or 1e14 < (*.f64 (PI.f64) l)

   1. Initial program 22.6

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

   2. Simplified 22.6

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

      [Start] 22.6	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 22.6	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 22.6	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Taylor expanded in l around inf 0.3

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

   if -4e18 < (*.f64 (PI.f64) l) < 1e14

   1. Initial program 9.8

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

   2. Applied egg-rr 1.0

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -4 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 10^{+14}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \]

Alternative 2
Error	13.4
Cost	13704

\[\begin{array}{l} t_0 := \left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{if}\;F \cdot F \leq 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-247}:\\ \;\;\;\;\frac{\pi \cdot \left(-\ell\right)}{{F}^{2}}\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 3
Error	0.9
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+15} \lor \neg \left(\ell \leq 30000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 4
Error	5.4
Cost	13513

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.5 \cdot 10^{+15} \lor \neg \left(\ell \leq 30000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 5
Error	13.4
Cost	7888

\[\begin{array}{l} t_0 := \frac{-\ell}{\frac{F \cdot F}{\pi}}\\ t_1 := \left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{if}\;F \cdot F \leq 7 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 7.2 \cdot 10^{-234}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 2.2 \cdot 10^{-97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-30}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 6
Error	13.5
Cost	7888

\[\begin{array}{l} t_0 := \left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{if}\;F \cdot F \leq 7.5 \cdot 10^{-292}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 2.7 \cdot 10^{-243}:\\ \;\;\;\;\frac{\pi}{F \cdot F} \cdot \left(-\ell\right)\\ \mathbf{elif}\;F \cdot F \leq 1.15 \cdot 10^{-96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 6.2 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7
Error	13.4
Cost	7888

\[\begin{array}{l} t_0 := \left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{if}\;F \cdot F \leq 10^{-296}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-247}:\\ \;\;\;\;\frac{-\ell}{\frac{F}{\frac{\pi}{F}}}\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-31}:\\ \;\;\;\;\frac{-\ell}{\frac{F \cdot F}{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 8
Error	13.7
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023059 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))