VandenBroeck and Keller, Equation (6)

Average Error: 16.2 → 0.5

Time: 12.3s

Precision: binary64

Cost: 32969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 20000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+18) (not (<= (* PI l) 20000000000000.0)))
   (* 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) <= -1e+18) || !((((double) M_PI) * l) <= 20000000000000.0)) {
		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) <= -1e+18) || !((Math.PI * l) <= 20000000000000.0)) {
		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) <= -1e+18) or not ((math.pi * l) <= 20000000000000.0):
		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) <= -1e+18) || !(Float64(pi * l) <= 20000000000000.0))
		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) <= -1e+18) || ~(((pi * l) <= 20000000000000.0)))
		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], -1e+18], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 20000000000000.0]], $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) < -1e18 or 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 23.3

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

   2. Taylor expanded in l around 0 32.4

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

   3. Simplified 32.4

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

      [Start] 32.4	\[ \pi \cdot \ell - \frac{\ell \cdot \pi}{{F}^{2}} \]
      associate-/l* [=>] 32.4	\[ \pi \cdot \ell - \color{blue}{\frac{\ell}{\frac{{F}^{2}}{\pi}}} \]
      unpow2 [=>] 32.4	\[ \pi \cdot \ell - \frac{\ell}{\frac{\color{blue}{F \cdot F}}{\pi}} \]
      associate-/r/ [=>] 32.4	\[ \pi \cdot \ell - \color{blue}{\frac{\ell}{F \cdot F} \cdot \pi} \]

   4. Taylor expanded in F around inf 0.3

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

   if -1e18 < (*.f64 (PI.f64) l) < 2e13

   1. Initial program 8.8

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

   2. Applied egg-rr 0.8

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 20000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+18} \lor \neg \left(\pi \cdot \ell \leq 0.001\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{\frac{F}{\pi}}}{F}\\ \end{array} \]

Alternative 2
Error	4.5
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+16} \lor \neg \left(\ell \leq 5500\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 3
Error	4.6
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+16} \lor \neg \left(\ell \leq 5500\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{\frac{F \cdot F}{\pi}}\\ \end{array} \]

Alternative 4
Error	4.8
Cost	13513

\[\begin{array}{l} \mathbf{if}\;\ell \leq -5.7 \cdot 10^{+16} \lor \neg \left(\ell \leq 5500\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \left(\ell - \ell \cdot {F}^{-2}\right)\\ \end{array} \]

Alternative 5
Error	13.1
Cost	7376

\[\begin{array}{l} t_0 := \frac{\pi}{F} \cdot \frac{-\ell}{F}\\ \mathbf{if}\;F \leq -5.5 \cdot 10^{-94}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{-88}:\\ \;\;\;\;1 + \left(\pi \cdot \ell + -1\right)\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 6
Error	13.2
Cost	7376

\[\begin{array}{l} t_0 := \frac{-\pi}{F \cdot \frac{F}{\ell}}\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{-92}:\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{elif}\;F \leq -5 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-88}:\\ \;\;\;\;1 + \left(\pi \cdot \ell + -1\right)\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 7
Error	13.2
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023054 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))