VandenBroeck and Keller, Equation (6)

Average Error: 16.5 → 1.5

Time: 11.8s

Precision: binary64

Cost: 33097

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+27} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-48}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -1e+27) (not (<= (* PI l) 2e-48)))
   (* PI l)
   (+ (* PI l) (/ (/ -1.0 F) (/ F (tan (* 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) <= -1e+27) || !((((double) M_PI) * l) <= 2e-48)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) + ((-1.0 / F) / (F / tan((((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) <= -1e+27) || !((Math.PI * l) <= 2e-48)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) + ((-1.0 / F) / (F / Math.tan((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) <= -1e+27) or not ((math.pi * l) <= 2e-48):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) + ((-1.0 / F) / (F / math.tan((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) <= -1e+27) || !(Float64(pi * l) <= 2e-48))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) + Float64(Float64(-1.0 / F) / Float64(F / tan(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) <= -1e+27) || ~(((pi * l) <= 2e-48)))
		tmp = pi * l;
	else
		tmp = (pi * l) + ((-1.0 / F) / (F / tan((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], -1e+27], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 2e-48]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] + N[(N[(-1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -1e27 or 1.9999999999999999e-48 < (*.f64 (PI.f64) l)

   1. Initial program 22.1

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

   2. Simplified 22.1

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

      [Start] 22.1	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 22.1	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 22.1	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Taylor expanded in l around inf 1.9

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

   if -1e27 < (*.f64 (PI.f64) l) < 1.9999999999999999e-48

   1. Initial program 10.1

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

   2. Applied egg-rr 1.1

      \[\leadsto \pi \cdot \ell - \color{blue}{\frac{\frac{1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+27} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-48}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell + \frac{\frac{-1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.5
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+27} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-48}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2
Error	1.4
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+14} \lor \neg \left(\pi \cdot \ell \leq 2 \cdot 10^{-48}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\frac{\ell}{F}}{F}\\ \end{array} \]

Alternative 3
Error	5.5
Cost	13513

\[\begin{array}{l} \mathbf{if}\;\ell \leq -340000 \lor \neg \left(\ell \leq 7.5 \cdot 10^{-41}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 4
Error	13.3
Cost	7888

\[\begin{array}{l} t_0 := \frac{-\ell}{\frac{F \cdot F}{\pi}}\\ t_1 := \left(\pi \cdot \ell + 1\right) + -1\\ \mathbf{if}\;F \cdot F \leq 2 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;F \cdot F \leq 2 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \cdot F \leq 5 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell\\ \end{array} \]

Alternative 5
Error	13.2
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023073 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))