VandenBroeck and Keller, Equation (6)

Average Error: 16.5 → 0.7

Time: 21.1s

Precision: binary64

Cost: 33096

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\frac{0.6666666666666666}{\ell}} \cdot 0.5 + 0.5 \cdot \left(\pi \cdot \left(\ell \cdot 0.5\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (<= (* PI l) -1e+27)
   (* l PI)
   (if (<= (* PI l) 20000000000000.0)
     (- (* PI l) (/ (/ 1.0 F) (/ F (tan (* PI l)))))
     (+ (* (/ PI (/ 0.6666666666666666 l)) 0.5) (* 0.5 (* PI (* l 0.5)))))))

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) {
		tmp = l * ((double) M_PI);
	} else if ((((double) M_PI) * l) <= 20000000000000.0) {
		tmp = (((double) M_PI) * l) - ((1.0 / F) / (F / tan((((double) M_PI) * l))));
	} else {
		tmp = ((((double) M_PI) / (0.6666666666666666 / l)) * 0.5) + (0.5 * (((double) M_PI) * (l * 0.5)));
	}
	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) {
		tmp = l * Math.PI;
	} else if ((Math.PI * l) <= 20000000000000.0) {
		tmp = (Math.PI * l) - ((1.0 / F) / (F / Math.tan((Math.PI * l))));
	} else {
		tmp = ((Math.PI / (0.6666666666666666 / l)) * 0.5) + (0.5 * (Math.PI * (l * 0.5)));
	}
	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:
		tmp = l * math.pi
	elif (math.pi * l) <= 20000000000000.0:
		tmp = (math.pi * l) - ((1.0 / F) / (F / math.tan((math.pi * l))))
	else:
		tmp = ((math.pi / (0.6666666666666666 / l)) * 0.5) + (0.5 * (math.pi * (l * 0.5)))
	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)
		tmp = Float64(l * pi);
	elseif (Float64(pi * l) <= 20000000000000.0)
		tmp = Float64(Float64(pi * l) - Float64(Float64(1.0 / F) / Float64(F / tan(Float64(pi * l)))));
	else
		tmp = Float64(Float64(Float64(pi / Float64(0.6666666666666666 / l)) * 0.5) + Float64(0.5 * Float64(pi * Float64(l * 0.5))));
	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)
		tmp = l * pi;
	elseif ((pi * l) <= 20000000000000.0)
		tmp = (pi * l) - ((1.0 / F) / (F / tan((pi * l))));
	else
		tmp = ((pi / (0.6666666666666666 / l)) * 0.5) + (0.5 * (pi * (l * 0.5)));
	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[LessEqual[N[(Pi * l), $MachinePrecision], -1e+27], N[(l * Pi), $MachinePrecision], If[LessEqual[N[(Pi * l), $MachinePrecision], 20000000000000.0], N[(N[(Pi * l), $MachinePrecision] - N[(N[(1.0 / F), $MachinePrecision] / N[(F / N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(Pi / N[(0.6666666666666666 / l), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + N[(0.5 * N[(Pi * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (PI.f64) l) < -1e27

   1. Initial program 23.0

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

   2. Simplified 23.0

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

      [Start] 23.0	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      rational.json-simplify-50 [=>] 23.0	\[ \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]
      rational.json-simplify-27 [<=] 23.0	\[ \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{\frac{F \cdot F}{1}}} \]
      rational.json-simplify-26 [=>] 23.0	\[ \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Taylor expanded in l around inf 0.2

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

   if -1e27 < (*.f64 (PI.f64) l) < 2e13

   1. Initial program 9.5

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

   2. Applied egg-rr 1.1

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

   if 2e13 < (*.f64 (PI.f64) l)

   1. Initial program 24.2

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

   2. Simplified 24.2

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

      [Start] 24.2	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      rational.json-simplify-50 [=>] 24.2	\[ \pi \cdot \ell - \color{blue}{\tan \left(\pi \cdot \ell\right) \cdot \frac{1}{F \cdot F}} \]
      rational.json-simplify-27 [<=] 24.2	\[ \pi \cdot \ell - \color{blue}{\frac{\tan \left(\pi \cdot \ell\right)}{\frac{F \cdot F}{1}}} \]
      rational.json-simplify-26 [=>] 24.2	\[ \pi \cdot \ell - \frac{\tan \left(\pi \cdot \ell\right)}{\color{blue}{F \cdot F}} \]

   3. Taylor expanded in l around inf 0.4

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

   4. Applied egg-rr 0.9

      \[\leadsto \color{blue}{\frac{\ell \cdot 0.5 - 0.5 \cdot \left(\ell \cdot 3\right)}{\frac{-1}{\pi}}} \]

   5. Applied egg-rr 0.8

      \[\leadsto \color{blue}{\pi \cdot \left(\ell \cdot -0.5 - \frac{1}{\frac{-0.6666666666666666}{\ell}}\right)} \]

   6. Applied egg-rr 0.5

      \[\leadsto \color{blue}{\frac{\pi}{\frac{0.6666666666666666}{\ell}} \cdot 0.5 + 0.5 \cdot \left(\pi \cdot \left(\ell \cdot 0.5\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{1}{F}}{\frac{F}{\tan \left(\pi \cdot \ell\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\frac{0.6666666666666666}{\ell}} \cdot 0.5 + 0.5 \cdot \left(\pi \cdot \left(\ell \cdot 0.5\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.7
Cost	32968

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+27}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\frac{0.6666666666666666}{\ell}} \cdot 0.5 + 0.5 \cdot \left(\pi \cdot \left(\ell \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 2
Error	0.8
Cost	26824

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\pi \cdot \ell \leq 20000000000000:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{\pi}{\frac{0.6666666666666666}{\ell}} \cdot 0.5 + 0.5 \cdot \left(\pi \cdot \left(\ell \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3
Error	1.3
Cost	26568

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\pi \cdot \ell \leq 4 \cdot 10^{-40}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\frac{\ell}{F}}{F}\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{1}{\frac{1}{\ell}}\\ \end{array} \]

Alternative 4
Error	5.2
Cost	26440

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -1 \cdot 10^{+14}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;\pi \cdot \ell \leq 4 \cdot 10^{-40}:\\ \;\;\;\;\left(\ell - \frac{\ell}{{F}^{2}}\right) \cdot \pi\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \frac{1}{\frac{1}{\ell}}\\ \end{array} \]

Alternative 5
Error	13.3
Cost	13512

\[\begin{array}{l} \mathbf{if}\;F \leq 6 \cdot 10^{-161}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;F \leq 3.05 \cdot 10^{-27}:\\ \;\;\;\;-1 \cdot \frac{\pi}{\frac{{F}^{2}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]

Alternative 6
Error	13.3
Cost	13448

\[\begin{array}{l} \mathbf{if}\;F \leq 2.1 \cdot 10^{-151}:\\ \;\;\;\;\ell \cdot \pi\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-31}:\\ \;\;\;\;\ell \cdot \frac{\pi}{-{F}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \pi\\ \end{array} \]

Alternative 7
Error	13.2
Cost	6528

\[\ell \cdot \pi \]

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)))))