VandenBroeck and Keller, Equation (6)

Average Error: 73.9% → 99.1%

Time: 18.9s

Precision: binary64

Cost: 45961.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 2\right):\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F \cdot \cos \left(\pi \cdot \ell\right)}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -5e+16) (not (<= (* PI l) 2.0)))
   (+ (* PI l) (* (* PI l) 0.0))
   (- (* PI l) (/ (/ (sin (* PI l)) F) (* F (cos (* 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+16) || !((((double) M_PI) * l) <= 2.0)) {
		tmp = (((double) M_PI) * l) + ((((double) M_PI) * l) * 0.0);
	} else {
		tmp = (((double) M_PI) * l) - ((sin((((double) M_PI) * l)) / F) / (F * cos((((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+16) || !((Math.PI * l) <= 2.0)) {
		tmp = (Math.PI * l) + ((Math.PI * l) * 0.0);
	} else {
		tmp = (Math.PI * l) - ((Math.sin((Math.PI * l)) / F) / (F * Math.cos((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+16) or not ((math.pi * l) <= 2.0):
		tmp = (math.pi * l) + ((math.pi * l) * 0.0)
	else:
		tmp = (math.pi * l) - ((math.sin((math.pi * l)) / F) / (F * math.cos((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+16) || !(Float64(pi * l) <= 2.0))
		tmp = Float64(Float64(pi * l) + Float64(Float64(pi * l) * 0.0));
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(sin(Float64(pi * l)) / F) / Float64(F * cos(Float64(pi * l)))));
	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) <= -5e+16) || ~(((pi * l) <= 2.0)))
		tmp = (pi * l) + ((pi * l) * 0.0);
	else
		tmp = (pi * l) - ((sin((pi * l)) / F) / (F * cos((pi * l))));
	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], -5e+16], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2.0]], $MachinePrecision]], N[(N[(Pi * l), $MachinePrecision] + N[(N[(Pi * l), $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Sin[N[(Pi * l), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / N[(F * N[Cos[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -5e16 or 2 < (*.f64 (PI.f64) l)

   1. Initial program 63.4

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

   2. Applied egg-rr 63.4

      \[\leadsto \pi \cdot \ell - \color{blue}{\left(e^{\mathsf{log1p}\left({F}^{-2}\right)} - 1\right)} \cdot \tan \left(\pi \cdot \ell\right) \]

   3. Taylor expanded in F around inf 99.0

      \[\leadsto \pi \cdot \ell - \left(\color{blue}{1} - 1\right) \cdot \tan \left(\pi \cdot \ell\right) \]

   4. Taylor expanded in l around 0 99.0

      \[\leadsto \pi \cdot \ell - \left(1 - 1\right) \cdot \color{blue}{\left(\ell \cdot \pi\right)} \]

   if -5e16 < (*.f64 (PI.f64) l) < 2

   1. Initial program 85.3

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

   2. Applied egg-rr 99.2

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

4. Final simplification 99.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 2\right):\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\sin \left(\pi \cdot \ell\right)}{F}}{F \cdot \cos \left(\pi \cdot \ell\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	99.1%
Cost	33097.00

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 2\right):\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

Alternative 2
Error	99.1%
Cost	32969.00

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 2\right):\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 3
Error	98.8%
Cost	26569.00

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -5 \cdot 10^{+16} \lor \neg \left(\pi \cdot \ell \leq 2\right):\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\ell}{F}}{\frac{F}{\pi}}\\ \end{array} \]

Alternative 4
Error	98.8%
Cost	13641.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq -4.6 \cdot 10^{+14} \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\ell}{F} \cdot \frac{\pi}{F}\\ \end{array} \]

Alternative 5
Error	98.8%
Cost	13641.00

\[\begin{array}{l} \mathbf{if}\;\ell \leq -8 \cdot 10^{+14} \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot 0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F \cdot \frac{F}{\ell}}\\ \end{array} \]

Alternative 6
Error	78.8%
Cost	13248.00

\[\pi \cdot \ell + \left(\pi \cdot \ell\right) \cdot 0 \]

Reproduce

herbie shell --seed 2023093 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))