VandenBroeck and Keller, Equation (6)

Average Error: 16.4 → 0.5

Time: 12.8s

Precision: binary64

Cost: 45769

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+14} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(\pi \cdot \ell\right)\right)\right)}{F}}{F}\\ \end{array} \]

FPCore

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

↓

(FPCore (F l)
 :precision binary64
 (if (or (<= (* PI l) -2e+14) (not (<= (* PI l) 10000000000000.0)))
   (* PI l)
   (- (* PI l) (/ (/ (log1p (expm1 (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) <= -2e+14) || !((((double) M_PI) * l) <= 10000000000000.0)) {
		tmp = ((double) M_PI) * l;
	} else {
		tmp = (((double) M_PI) * l) - ((log1p(expm1(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) <= -2e+14) || !((Math.PI * l) <= 10000000000000.0)) {
		tmp = Math.PI * l;
	} else {
		tmp = (Math.PI * l) - ((Math.log1p(Math.expm1(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) <= -2e+14) or not ((math.pi * l) <= 10000000000000.0):
		tmp = math.pi * l
	else:
		tmp = (math.pi * l) - ((math.log1p(math.expm1(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) <= -2e+14) || !(Float64(pi * l) <= 10000000000000.0))
		tmp = Float64(pi * l);
	else
		tmp = Float64(Float64(pi * l) - Float64(Float64(log1p(expm1(tan(Float64(pi * l)))) / F) / F));
	end
	return 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], -2e+14], N[Not[LessEqual[N[(Pi * l), $MachinePrecision], 10000000000000.0]], $MachinePrecision]], N[(Pi * l), $MachinePrecision], N[(N[(Pi * l), $MachinePrecision] - N[(N[(N[Log[1 + N[(Exp[N[Tan[N[(Pi * l), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) l) < -2e14 or 1e13 < (*.f64 (PI.f64) l)

   1. Initial program 23.7

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

   2. Simplified 23.7

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

      [Start] 23.7	\[ \pi \cdot \ell - \frac{1}{F \cdot F} \cdot \tan \left(\pi \cdot \ell\right) \]
      associate-*l/ [=>] 23.7	\[ \pi \cdot \ell - \color{blue}{\frac{1 \cdot \tan \left(\pi \cdot \ell\right)}{F \cdot F}} \]
      *-lft-identity [=>] 23.7	\[ \pi \cdot \ell - \frac{\color{blue}{\tan \left(\pi \cdot \ell\right)}}{F \cdot F} \]

   3. Taylor expanded in l around inf 0.4

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

   if -2e14 < (*.f64 (PI.f64) l) < 1e13

   1. Initial program 9.2

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

   2. Applied egg-rr 0.5

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

   3. Applied egg-rr 0.6

      \[\leadsto \pi \cdot \ell - \frac{\frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(\pi \cdot \ell\right)\right)\right)}}{F}}{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+14} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan \left(\pi \cdot \ell\right)\right)\right)}{F}}{F}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	32969

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+14} \lor \neg \left(\pi \cdot \ell \leq 10000000000000\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\tan \left(\pi \cdot \ell\right)}{F}}{F}\\ \end{array} \]

Alternative 2
Error	0.9
Cost	26569

\[\begin{array}{l} \mathbf{if}\;\pi \cdot \ell \leq -2 \cdot 10^{+14} \lor \neg \left(\pi \cdot \ell \leq 5 \cdot 10^{-15}\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\frac{\pi \cdot \ell}{F}}{F}\\ \end{array} \]

Alternative 3
Error	4.8
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\ell \leq -600000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \pi \cdot \frac{\ell}{F \cdot F}\\ \end{array} \]

Alternative 4
Error	0.8
Cost	13641

\[\begin{array}{l} \mathbf{if}\;\ell \leq -600000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\pi \cdot \ell - \frac{\pi}{F} \cdot \frac{\ell}{F}\\ \end{array} \]

Alternative 5
Error	5.1
Cost	13513

\[\begin{array}{l} \mathbf{if}\;\ell \leq -600000000000 \lor \neg \left(\ell \leq 0.5\right):\\ \;\;\;\;\pi \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\pi - \frac{\pi}{F \cdot F}\right)\\ \end{array} \]

Alternative 6
Error	13.6
Cost	6528

\[\pi \cdot \ell \]

Reproduce

herbie shell --seed 2023011 
(FPCore (F l)
  :name "VandenBroeck and Keller, Equation (6)"
  :precision binary64
  (- (* PI l) (* (/ 1.0 (* F F)) (tan (* PI l)))))