“Between One and One Plus Epsilon” is set to the music of “Just Around the Riverbend” by Alan Menken with lyrics by Stephen Schwartz. “Just Around the Riverbend” is from the 1995 Disney film Pocahontas. Alternative lyrics are by Susanne Bradley, with thanks to Chen Greif for providing cancellation-related suggestions.
What I love most about reals is
You’ll never list every single case
There’s always digits more than you’ll be showing1
Computers, I guess, can’t work with that
There’s only so much space
Using bits, we lose our chance of ever knowing
What’s halfway to epsilon2
Between one and one plus epsilon3
Bits binary, standard words4 of zero/one
For all your needs: doubles, ints, and floats
e, pi, root(3)
None of these are as they seem
Rounding errors intervene with code, mess up my code
I see it in these random spikes5
That clutter my solution plot
Okay, perhaps my method makes them larger6
But just how can I compute things
As precisely as I ought
When I keep losing digits off my numbers?
What’s halfway to epsilon
Between one and one plus epsilon
Bits binary, standard words of zero/one
I triple E7 shows us how it’s done
But I can’t subtract a one
From one plus O of epsilon
Cancellation,8 that’s just wrong
- This refers vaguely to Georg Cantor’s proof that the real numbers are uncountably infinite – see, e.g., Gray 1994. The proof is by contradiction: we assume that we have a sequence containing all numbers in an interval $[a,b]$. Then Cantor’s diagonal argument shows how we can construct a new number in $[a,b]$ that’s not in the sequence by selecting digits from numbers in the sequence and then adding new digits at the end (hence “always digits more than you’ll be showing”).
- When we say epsilon we are, in this context, referring to machine epsilon, and not an arbitrary small constant. Unfortunately, the scansion of “machine epsilon” doesn’t work well with this song, so we’ve had to make do with the shorthand. (Another option would have been to try to make “unit roundoff” work, but that sounds less poetic than “epsilon.” It also would have denied us our repeatedly used semi-rhyme between “epsilon” and “one,” and that would just be ignorant.)
- Machine epsilon (which we’ll denote here by $\epsilon$) is the smallest number such that $1 + \epsilon$ is greater than $1$ in machine representation. Hence, anything between $1$ and $1 + \epsilon$ will be (inaccurately) represented on a computer as being equal to $1$ – so, this is a great, unknown space, much like “just around the riverbend” in this song’s original, non-parody lyrics. For more information on machine epsilon (including an experiment to compute it yourself), see the Wikipedia article.
- Some sources (e.g., Ascher and Greif 2011) use “IEEE standard word” to describe the standard 64-bit representation of a number, which consists of a sign bit, 11-bit exponent, and 52-bit fraction. Also see Overton 2001 for details and examples.
- Roundoff errors tend to be non-smooth and random-looking (though they are, of course, not actually random). That’s quite different from truncation errors, which are due to errors in the modeling and discretization of a problem (and tend to be smoother and more predictable).
- Sometimes roundoff errors aren’t due to the problem itself, but rather to an unstable algorithm. See chapter 1 of Ascher and Greif 2011 again for a discussion of the difference between ill-conditioning (when the problem is badly behaved) and instability (when your algorithm makes roundoff errors blow up, even though the problem itself might be perfectly well-behaved).
- Refers to the Institute of Electrical and Electronics Engineers, originators of the IEEE 754: the standard for floating-point computation used today in most floating point units.
- Subtracting two nearby numbers – that is, performing computations like $(1 + O(\epsilon)) - 1$ – can lead to unacceptably high rounding errors because we lose many significant digits. For example, suppose we have two numbers with twelve significant digits, but the first nine are the same. If we subtract one from the other, we’re left with just three significant digits, and this is irreversible! For more information on so-called “catastrophic cancellation,” including some ways to avoid it, consult a numerical analysis text (for example, chapter 2 of Ascher and Greif 2011).
- Shout-out to the textbook of Elman, Silvester, and Wathen 2014 for coining the truly excellent phrase “algebraic suicide” on p. 131. (It was actually “linear algebraic suicide,” but close enough.)
- Several types of numerical method rely on some kind of “step” (either in space or in time), the size of which can affect the stability of the method. It’s most commonly of concern in the solution of ordinary and partial differential equations – see, e.g., Kraaijevanger et. al. 1987. ^