Today I learned what precision Python floats have.

Python floats are IEEE 754 double-precision binary floating-point numbers, commonly referred to as “doubles”, and take up 64 bits. Of those:

- 1 is for the sign of the number;
- 11 are for the exponent; and
- 52 are for the fraction.

We can verify experimentally that Python floats use 52 bits to store the fraction.
The number `1 << 53`

is an exact integer:

```
>>> n = 1 << 53
>>> n
9007199254740992
```

In binary, this number is a `1`

followed by 53 `0`

:

```
>>> bin(n)
'0b100000000000000000000000000000000000000000000000000000'
>>> bin(n)[2:]
'100000000000000000000000000000000000000000000000000000'
>>> bin(n)[2:].count("0")
53
```

Now, if you convert `n`

to a float and add `1`

, nothing happens;
whereas if you add `2`

, you get the correct value:

```
>>> float(n)
9007199254740992.0
>>> float(n) + 1
9007199254740992.0
>>> float(n) + 2
9007199254740994.0
```

Why?

Well, `n + 1`

in binary starts and ends with `1`

and has 52 zeroes in the middle:

```
>>> bin(n + 1)
'0b100000000000000000000000000000000000000000000000000001'
```

Represented in scientific notation (in binary), this number would have 53 digits after the decimal point:

\[ 1.00000000000000000000000000000000000000000000000000001_2 \times 2^{53}\]

However, doubles only have 52 digits after the decimal point, so the final `1`

is dropped and the number becomes

\[ 1.0000000000000000000000000000000000000000000000000000_2 \times 2^{53}\]

Which is exactly the same number.

However, if we add `2`

instead of just `1`

, the final result is

\[ 1.00000000000000000000000000000000000000000000000000010_2 \times 2^{53}\]

which looks like

\[ 1.0000000000000000000000000000000000000000000000000001_2 \times 2^{53}\]

if we only use 52 digits after the decimal point, which is the same as the correct result.

The fact that `n`

, `n + 2`

, `n + 4`

, ... give the correct results,
and `n + 1`

, `n + 3`

, `n + 5`

, ... don't,
together with the fact that this phenomenon started at `n = 1 << 53`

and not `n = 1 << 52`

shows that Python floats use 52 bits to store the fraction of a number.

That's it for now! Stay tuned and I'll see you around!

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- “Double-precision floating-point format”, Wikipedia, https://en.wikipedia.org/wiki/Double-precision_floating-point_format [last visited 14-09-2022];