Confusing math problem

[Chris Angelico]

On Fri, Feb 22, 2013 at 8:59 AM, Peter Pearson <ppearson at nowhere.invalid> wrote:
> On Fri, 22 Feb 2013 08:23:27 +1100, Chris Angelico <rosuav at gmail.com> wrote:
>> In theory, a float should hold the nearest representable value to the
>> exact result. Considering that only one operation is being performed,
>> there should be no accumulation of error. The integer results show a
>> small number (618) of collisions, eg 2**16 and 4**8; why should some
>> of those NOT collide when done with floating point? My initial thought
>> was "Oh, this is comparing floats for equality", but after one single
>> operation, that should be not a problem.
>
> Does this help explain it?
>
>>>> print hex(int(math.pow(3,60))); print hex(3**60)
> 0x88f924eeceeda80000000000L
> 0x88f924eeceeda7fe92e1f5b1L
>

I understand how the inaccuracy works, but I'm expecting it to be as
consistent as Mr Grossmith's entertainments. It doesn't matter that
math.pow(3,60) != 3**60, but the number of collisions is different
when done with floats on the OP's Mac. Here's what I'm talking about:

>>> set((3**60,9**30,27**20))
{42391158275216203514294433201}
>>> set((math.pow(3,60),math.pow(9,30),math.pow(27,20)))
{4.23911582752162e+28}

Note how, in each case, calculating three powers that have the same
real-number result gives a one-element set. Three to the sixtieth
power can't be perfectly rendered with a 53-bit mantissa, but it's
rendered the same way whichever route is used to calculate it.

ChrisA