IEEE 754 floating point

Fractional binary and fixed point, the single-precision format (sign, exponent biased by 127, mantissa), double precision for comparison, special values, rounding errors and comparing floats.

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IEEE 754: what is it?
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The standard that fixes how floating-point numbers are encoded in machines, their special values and the rounding rules of calculations.
Almost all processors follow it: float and double in C and Java, and Python's float (in double precision) therefore follow the same formats and rounding rules from one machine to another.
What does this Python program print?

```python
total = 0
for _ in range(10):
total += 0.1
print(total == 1.0)
print(total)
`
- False puis 0.9999999999999999
- True puis 1.0
- False puis 1.0
- True puis 0.9999999999999999
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False puis 0.9999999999999999
0.1 is not exact in binary: ten additions accumulate the error and leave a gap of one last bit below 1. The comparison is therefore false, and print shows the gap.
A double-precision float takes … bits: 1 sign bit, … exponent bits and 52 mantissa bits.
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A double-precision float takes 64 bits: 1 sign bit, 11 exponent bits and 52 mantissa bits.
Single precision uses only 32 (1, 8 and 23). More mantissa bits give about 16 significant decimal digits instead of 7.
Floating-point number: what is it?
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A number coded by a sign, a mantissa and an exponent, of the form ± mantissa × 2 to the power of the exponent: the exponent sets where the point lies.
In single precision: 1 sign bit, 8 exponent bits, 23 mantissa bits. It covers very large or very small values, but with a limited number of significant digits.

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