Float-IEEE754-didactic

Educational python module to parse floats and inspect the IEEE 754 representation internals.

Installation

pip install floatedu

Use case

It might be used to get a glimpse of how the IEEE 754 model works. This means it's useful almost exclusively for educational purposes.

Brief example

Create 3 floats using the float32 specification and print them as native floats:

from floatedu import Float32

f_1 = Float32("0 01111111 00000000000000000000000")
f_num = Float32("0 10001001 00110100100111010011101")
f_inf = Float32("0 11111111 00000000000000000000000")

print(f_1, f_num, f_inf)

# 1.0 1234.4566650390625 inf

Print number details as per general formula:

print(repr(f_num))

# {'value': float('1234.4566650390625'),
#  'kind': 'normal', 'k': 8, 'p': 24, 'bias': 127,
#  'bits': '0_10001001_00110100100111010011101',
#  'sign': 1, 'exponent': 137, 'fraction': 0.20552408695220947,
#  'significand': 1.2055240869522095,
# }

Everything is accessible as a property:

print(f_num.kind, f_num.sign, f_num.exponent, f_num.fraction)

# normal 1 137 0.20552408695220947

Float is a subclass of list and can be updated live:

print(f_num.sign_bit)
print(f_num.exponent_bits)
print(f_num.fraction_bits)
f_num.sign_bit[0] = 1 #  Make number negative
print(f_num)

# [0]
# [1, 0, 0, 0, 1, 0, 0, 1]
# [0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1]
# -1234.4566650390625

Formats and equations

IEEE 754 defines various types of binary floats. This module implements standard and, in addition, a couple of non-standard types.

Bits are laid out from left to right following the scheme sign . exponent . fraction. p value includes the implicit bit (not actually stored).

For example the BFloat16 is stored as:

Kind

A number could be Zero, Infinity, Not a number, Normal or Subnormal. The first three cases are already values. Normal and Subnormal numbers are computed according to an equation.

To determine how to compute the value is sufficient to test exp and fraction against zero (all zeroes) and -1 (all ones, two's complement):

Zero and infinity

For Zero and Infinity the value is trivial. If the leftmost bit is 0 then the value is positive (+0 or +inf). Otherwise it is negative (-0 or -inf).

Not a number

In case of Not a number no extra steps is required if not returning an appropriate "non-value".

Normal

Normal numbers values can be computed by the equation (general formula and float32 formula):

An another way to think about this formula is to consider the stored number as a fixed point binary number with sign bit.

In this case, the integer part would be the exponent and the fractional part (plus 1) would be the significand. I.e.

Implementation

Every implemented float type is available as a class:

from floatedu import *
[Float, Float8, Float16, BFloat16, Float64, Float32, Float128, Float256]

The actual implementation class is Float and it couldn't be instantiated directly.

It must be subclassed providing p, k, and bias values as class properties (see floatedu/Float.py).