How To Represent 0.1 In Floating Point Arithmetic And Decimal

Question

I am trying to understand floating point arithmetic better and have seen a few links to 'What Every Computer Scientist Should Know About Floating Point Arithmetic'.

I still don't understand how a number like 0.1 or 0.5 is stored in floats and as decimals.

Can someone please explain how it is laid out is memory?

I know about the float being two parts (i.e., a number to the power of something).

Solution 1

I've always pointed people towards Harald Schmidt's online converter, along with the Wikipedia IEEE754-1985 article with its nice pictures.

For those two specific values, you get (for 0.1):

s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm    1/n
0 01111011 10011001100110011001101
           |  ||  ||  ||  ||  || +- 8388608
           |  ||  ||  ||  ||  |+--- 2097152
           |  ||  ||  ||  ||  +---- 1048576
           |  ||  ||  ||  |+-------  131072
           |  ||  ||  ||  +--------   65536
           |  ||  ||  |+-----------    8192
           |  ||  ||  +------------    4096
           |  ||  |+---------------     512
           |  ||  +----------------     256
           |  |+-------------------      32
           |  +--------------------      16
           +-----------------------       2

The sign is positive, that's pretty easy.

The exponent is 64+32+16+8+2+1 = 123 - 127 bias = -4, so the multiplier is 2-4 or 1/16.

The mantissa is chunky. It consists of 1 (the implicit base) plus (for all those bits with each being worth 1/(2n) as n starts at 1 and increases to the right), {1/2, 1/16, 1/32, 1/256, 1/512, 1/4096, 1/8192, 1/65536, 1/131072, 1/1048576, 1/2097152, 1/8388608}.

When you add all these up, you get 1.60000002384185791015625.

When you multiply that by the multiplier, you get 0.100000001490116119384765625, which is why they say you cannot represent 0.1 exactly as an IEEE754 float, and provides so much opportunity on SO for people answering "why doesn't 0.1 + 0.1 + 0.1 == 0.3?"-type questions :-)

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The 0.5 example is substantially easier. It's represented as:

s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm
0 01111110 00000000000000000000000

which means it's the implicit base, 1, plus no other additives (all the mantissa bits are zero).

The sign is again positive. The exponent is 64+32+16+8+4+2 = 126 - 127 bias = -1. Hence the multiplier is 2-1 which is 1/2 or 0.5.

So the final value is 1 multiplied by 0.5, or 0.5. Voila!

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I've sometimes found it easier to think of it in terms of decimal.

The number 1.345 is equivalent to

1 + 3/10   + 4/100 + 5/1000

or:

        -1       -2      -3
1 + 3*10   + 4*10  + 5*10

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Similarly, the IEEE754 representation for decimal 0.8125 is:

s eeeeeeee mmmmmmmmmmmmmmmmmmmmmmm
0 01111110 10100000000000000000000

With the implicit base of 1, that's equivalent to the binary:

         01111110-01111111
1.101 * 2

or:

                     -1
(1   + 1/2 + 1/8) * 2     (no 1/4 since that bit is 0)

which becomes:

(8/8 + 4/8 + 1/8) * 1/2

and then becomes:

13/8 * 1/2 = 0.8125

Solution 2

See the Wikipedia entry and the IEEE group, first.

Basically, there's a sign, a number, and an exponent. A number in one base cannot be represented finitely in another base if the source base has factors not present in the destination base. For instance, 1/3 cannot be represented as a finite decimal number, but is trivial to represent as a ternary (base-3) number: (0.1)₃.

So 0.5 has a finite binary representation, (0.1)₂, that is, 2^-1, but 0.1 has a repeating representation because 2 and 10 have a factor (5) not in common.