# newbie seeks inaccurate arithmetic insight

Tim Peters tim.one at home.com
Tue May 1 01:35:10 CEST 2001

```[/F]
> > try using "%.17g" (show all significant digits) instead of "%f"
> (fudge it)

[Stanley Krute]
> Aha !  Python defaults to Truth !

Nope:  repr(float) gives a closer *approximation* to machine truth than does
str(float), and just *enough* of the truth so that eval(repr(x)) == x.  In
general, you cannot rely on eval(str(x)) == x (in fact, it's unusual to get
the same float back if you go thru str() conversion first).  If you wanted
the full truth, then e.g. you would get this instead:

>>> 0.1
0.1000000000000000055511151231257827021181583404541015625
>>>

That very long decimal number is the exact value of the binary approximation
to 0.1 stored in the machine by your HW -- and assuming that your platform C
library does best-possible conversion of the string "0.1" to an IEEE-754
double precision number to begin with.

> ...
> The inaccuracies exist 15 or so places to the right of the decimal
> point.

In general, since there 53 bits of precision in an IEEE-double, the tiniest
errors creep into the 53rd significant bit.  An error of 1 in 2**53 is an
error of 1 in 10**x, for *some* value of x.  A little math reveals x =
log10(2**53) = 53 * log10(2)

>>> import math
>>> 53 * math.log10(2)
15.954589770191003
>>>

So when viewed as decimal again, even the tiniest error will show up in the
15th or 16th significant decimal digit.  Since that's what you've already
observed, you have reason to believe it <wink>.