Confusing math problem

[Chris Angelico]

On Fri, Feb 22, 2013 at 7:49 AM, Dave Angel <davea at davea.name> wrote:
> However, if I do:
>
> print 3**60, "\n", int(math.pow(3,60)), "\n", pow(3,60)
>
>
> I get:
>
> 42391158275216203514294433201
> 42391158275216203520420085760
> 42391158275216203514294433201
>
>
> and the middle one is the one that's wrong.

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.

> You can tell by casting out 9's.  The middle one gets 1
> instead of zero, showing that it's NOT divisible by 3.

Which I thought so cool and magical and awesome, until I started
exploring other bases and found that you could cast out F's in hex and
7's in octal... and you can cast out 1's in binary to find out if it's
a multiple of 1, too.

ChrisA