python - Strange behaviour with floats and string conversion

Question

I've typed this into python shell:

>>> 0.1*0.1
0.010000000000000002

I expected that 0.1*0.1 is not 0.01, because I know that 0.1 in base 10 is periodic in base 2.

>>> len(str(0.1*0.1))
4

I expected to get 20 as I've seen 20 characters above. Why do I get 4?

>>> str(0.1*0.1)
'0.01'

Ok, this explains why I len gives me 4, but why does str return '0.01'?

>>> repr(0.1*0.1)
'0.010000000000000002'

Why does str round but repr not? (I have read this answer, but I would like to know how they have decided when str rounds a float and when it doesn't)

>>> str(0.01) == str(0.0100000000001)
False
>>> str(0.01) == str(0.01000000000001)
True

So it seems to be a problem with the accuracy of floats. I thought Python would use IEEE 754 single precicion floats. So I've checked it like this:

#include <stdint.h>
#include <stdio.h> // printf

union myUnion {
    uint32_t i; // unsigned integer 32-bit type (on every machine)
    float f;    // a type you want to play with
};

int main() {
    union myUnion testVar;
    testVar.f = 0.01000000000001f;
    printf("%f
", testVar.f);

    testVar.f = 0.01000000000000002f;
    printf("%f
", testVar.f);

    testVar.f = 0.01f*0.01f;
    printf("%f
", testVar.f);
}

I got:

0.010000
0.010000
0.000100

Python gives me:

>>> 0.01000000000001
0.010000000000009999
>>> 0.01000000000000002
0.010000000000000019
>>> 0.01*0.01
0.0001

Why does Python give me these results?

(I use Python 2.6.5. If you know of differences in the Python versions, I would also be interested in them.)

Answer 1

The crucial requirement on repr is that it should round-trip; that is, eval(repr(f)) == f should give True in all cases.

In Python 2.x (before 2.7) repr works by doing a printf with format %.17g and discarding trailing zeroes. This is guaranteed correct (for 64-bit floats) by IEEE-754. Since 2.7 and 3.1, Python uses a more intelligent algorithm that can find shorter representations in some cases where %.17g gives unnecessary non-zero terminal digits or terminal nines. See What's new in 3.1? and issue 1580.

Even under Python 2.7, repr(0.1 * 0.1) gives "0.010000000000000002". This is because 0.1 * 0.1 == 0.01 is False under IEEE-754 parsing and arithmetic; that is, the nearest 64-bit floating-point value to 0.1, when multiplied by itself, yields a 64-bit floating-point value that is not the nearest 64-bit floating-point value to 0.01:

>>> 0.1.hex()
'0x1.999999999999ap-4'
>>> (0.1 * 0.1).hex()
'0x1.47ae147ae147cp-7'
>>> 0.01.hex()
'0x1.47ae147ae147bp-7'
                 ^ 1 ulp difference

The difference between repr and str (pre-2.7/3.1) is that str formats with 12 decimal places as opposed to 17, which is non-round-trippable but produces more readable results in many cases.