Why are the bit strings representing 1.0 and 2.0 so different?

I recently started using Julia and I came upon the bits function, which returns the bit-string representation of its numeric argument. For example:

julia> bits(1.0)
"0011111111110000000000000000000000000000000000000000000000000000"

However, while playing with this function, I was surprised to discover that bits returns very different bit strings for 1.0 and 2.0:

julia> bits(1.0)
"0011111111110000000000000000000000000000000000000000000000000000"

julia> bits(2.0)
"0100000000000000000000000000000000000000000000000000000000000000"

I would have expected those two values to be similar...

What is the meaning of those bits? I vaguely recall something about bits encoding the exponent (from my numerical-analysis class), but I really do not remember it well and I did not manage to find a good description online...

Solution

To understand why the ASCIIString values of bits(1.0) and bits(2.0) are "so different", you need to know a bit (!) about IEEE-754 (binary) floating-point numbers. Each such double-precision number is stored as a 64-bit word, broken down into three parts:

the sign bit (0 for nonnegative numbers, 1 for nonpositive numbers), followed by
the biased exponent (11 bits), followed by
the significand (52 bits).

The value of a normalized number (such as 1.0 and 2.0) can be obtained by using the following formula:

(-1)^sign_bit x 1.significand x 2^(biased_exponent - bias)

(For double-precision floating-point numbers, the bias has a value of 2^10 - 1 = 1023)

Now,

1.0 = +1.000... x 2^(1023 - bias)

and 1023 corresponds to (0)1111111111 in base 2, so the corresponding bit string is
```
0 01111111111 0000000000000000000000000000000000000000000000000000
```
2.0 = +1.000... x 2^(1024 - bias)

and 1024 corresponds to 10000000000 in base 2, so the corresponding bit string is
```
0 10000000000 0000000000000000000000000000000000000000000000000000
```

3.0 = +1.100... x 2^(1024 - bias)

so the corresponding bit string is

0 10000000000 1000000000000000000000000000000000000000000000000000

etc.

In summary, you can obtain the bit string of 2.0 by incrementing the biased-exponent part in the bit string of 1.0, which is a power of 2, minus 1. Incrementing such a number causes all the bits of its binary representation to change, in the same way that incrementing the number 9999 (in decimal representation) causes all the digits to change.