A little diversion into floating point (im)precision, part 1

Question

Most mathematicians agree that:

e^πi + 1 = 0

However, most floating point implementations disagree. How well can we settle this dispute?

I'm keen to hear about different languages and implementations, and various methods to make the result as close to zero as possible. Be creative!

Answer 1

It's not that most floating point implementations disagree, it's just that they cannot get the accuracy necessary to get a 100% answer. And the correct answer is that they can't.

PI is an infinite series of digits that nobody has been able to denote by anything other than a symbolic representation, and e^X is the same, and thus the only way to get to 100% accuracy is to go symbolic.

Answer 2

Here's a short list of implementations and languages I've tried. It's sorted by closeness to zero:

• Scheme: (+ 1 (make-polar 1 (atan 0 -1)))
  • ⇒ 0.0+1.2246063538223773e-16i (Chez Scheme, MIT Scheme)
  • ⇒ 0.0+1.22460635382238e-16i (Guile)
  • ⇒ 0.0+1.22464679914735e-16i (Chicken with numbers egg)
  • ⇒ 0.0+1.2246467991473532e-16i (MzScheme, SISC, Gauche, Gambit)
  • ⇒ 0.0+1.2246467991473533e-16i (SCM)
• Common Lisp: (1+ (exp (complex 0 pi)))
  • ⇒ #C(0.0L0 -5.0165576136843360246L-20) (CLISP)
  • ⇒ #C(0.0d0 1.2246063538223773d-16) (CMUCL)
  • ⇒ #C(0.0d0 1.2246467991473532d-16) (SBCL)
• Perl: use Math::Complex; Math::Complex->emake(1, pi) + 1
  • ⇒ 1.22464679914735e-16i
• Python: from cmath import exp, pi; exp(complex(0, pi)) + 1
  • ⇒ 1.2246467991473532e-16j (CPython)
• Ruby: require 'complex'; Complex::polar(1, Math::PI) + 1
  • ⇒ Complex(0.0, 1.22464679914735e-16) (MRI)
  • ⇒ Complex(0.0, 1.2246467991473532e-16) (JRuby)
• R: complex(argument = pi) + 1
  • ⇒ 0+1.224606353822377e-16i