Play with the IEEE 754 format in python

anyfloat.py

#!/usr/bin/env python
# -*-coding: utf8-*-
# Title: anyfloat.py
# Author: Gribouillis for the python forum at www.daniweb.com
# Created: 2012-05-02 06:46:42.708131 (isoformat date)
# License: Public Domain
# Use this code freely.

"""Conversion module between floating point representations.

This module handles floating point values encoded with the IEEE 754 format. The following
is assumed about this format:

    * Real numbers are represented in 3 fields of bits:
    
        b bbbbbbbb bbbbbbbbbbbbbbbbbbbbbbb  (b = 0 or 1)
       
        + the first field is the _sign_ field (width = 1 bit)
        + the second field is the _exponent_ field (width = w bits)
        + the third field is the _significand_ field (width = p bits)
       
        Typical values are (w, p) = (8, 23) for single precision floating point numbers
        and (w, p) = (11, 52) for double precision floating point numbers. The number
        p is called the precision for this representation. The exponent and the significand
        of a real number are the integers represented by the exponent and significand fields
        in binary notation. For example, the number
       
        0 00000101 00000000000000000010000
       
        has exponent = 5 and significand = 16.
       
    * Apart from special values, there are two kinds of numbers:

        - _normal_ numbers have an exponent between 00000001 and 11111110 in our example,
        more precisely: 1 <= exponent < 2^w - 1. They represent the real value
            val = +/- 1.significand x 2^(exponent - bias)
        The bias is 2^(w-1) - 1. The sign bit gives the +/-  part which is - if the sign bit is set.

        - _subnormal_ numbers have an exponent = 0, they represent the real value
            val = +/- 0.significand x 2^(1 - bias)
            
    * Special values have exponent = 2^w - 1 (11111111 in our example) they are
        - +/- infinity, which have significand = 0
        - NaN which have a non-zero significand and a zero sign bit.

More information about the IEEE floating point number format can be found at
https://www.ieee.org

This module offers a class "anyfloat" which is a named tuple with 3 fields: sign, exponent, significand,
but without restrictions on the size of the exponent and the significand.

An instance of anyfloat represents the real value

    val = +/- 0.significand x 2^(1 + exponent)

    * The sign field has value 1 for a negative value and 0 otherwise.
    * The exponent is the logarithm of |val| in base 2 (except for special values)
    * The significand is >= 0 except for special values:
        + significand = -1 is used to represent +/- infinity. In this case, exponent = 0
        + significand = -2 is used to represent NaN value. In this case, exponent = 0
        
Methods are provided to convert ieee numbers or python floats to/from anyfloat:

    anyfloat.to_ieee(size) converts to an integer which binary representation is the ieee754
        representation of this anyfloat with given size = (w, p).
        
    anyfloat.from_ieee(n, size) creates an anyfloat from an integer containing the ieee754
        representation of a value.
        
    anyfloat.__float__() converts to a python float.
    
    anyfloat.from_float(val) creates an anyfloat from a python floating point number.
    
Numbers can be converted from one ieee754 format to another, for example:

    >>> n = int("00000010100000000000000000010000", 2) # a single precision ieee number.
    >>> af = anyfloat.from_ieee(n, (8, 23))
    >>> n = af.to_ieee(af, (11, 52)) # convert to the double precision ieee number.

These conversions may occasionaly transform non zero values into +/-0 or +/- infinity
if the value is too small or too big for the target format.
"""

from __future__ import print_function
from collections import namedtuple
from fractions import Fraction
from math import isnan
import re
import struct
import sys

__version__ = '2023.12.03'

if sys.version_info < (2, 7):
    raise ImportError("Module anyfloat requires python 2.7 or newer.")

# A set of popular sizes
BINARY8   = ( 3,   4)  # Quarter precision (hypothetical)
BINARY16  = ( 5,  10)  # Half precision
BINARY32  = ( 8,  23)  # Single precision
BINARY64  = (11,  52)  # Double precision
BINARY128 = (15, 112)  # Quadruple precision
BINARY256 = (19, 236)  # Octuple precision

DEFAULT_SIZE = BINARY64

def trunc_round(n, k):
    rshift = n.bit_length() - 1 - k
    if rshift >= 0:
        n >>= (rshift)
    else:
        n <<= (-rshift)
    return (n + 1) >> 1
    
def more_bin_digits(n, k):
    return bool(n >> k)
    
def unset_high_bit(n):
    assert n > 0
    return n ^ (1 << (n.bit_length() - 1))
    
def fbin(n, nbits):
    assert (0 <= n)
    assert not (n >> nbits)
    return "{val:0>{width}}".format(val = bin(n)[2:], width = nbits)

_anyfloat = namedtuple("anyfloat", "sign exponent significand")

class anyfloat(_anyfloat):
    __slots__ = ()
    _b32 = 1 << 32
    _b64 = 1 << 64
    
    def __new__(cls, sign, exponent, significand):
        assert sign in (0, 1)
        if significand > 0:
            significand //= significand & -significand
        return _anyfloat.__new__(cls, sign, exponent, significand)

    @staticmethod
    def _encode(log2, mantissa, a, b):
        A = ~(~0 << a)
        AA = A >> 1
        if mantissa <= 0:
            return ( (A, 0) if (mantissa == -1) else (A, 1 << (b-1)) ) if mantissa else (0, 0)
        elif log2 <= - AA:
            nbits = b + log2 + AA
            rounded = trunc_round(mantissa, nbits) if (nbits >= 0) else 0
            return (1, 0) if more_bin_digits(rounded, b) else (0, rounded)
        elif log2 <= AA:
            rounded = trunc_round(mantissa, b + 1)
            return (( (log2 + 1 + AA, 0) if (log2 < AA) else (A, 0) )
            if more_bin_digits(rounded, b+1) else (log2 + AA, unset_high_bit(rounded)) )
        else:
            return (A, 0)

    @staticmethod
    def _decode(exponent, significand, a, b):
        A = ~(~0 << a)
        AA = A >> 1
        assert 0 <= exponent <= A
        assert 0 <= significand < (1 << b)
        if exponent == A:
            return (0, -2 if significand else -1)
        elif exponent: # normal case
            return (exponent - AA, significand|(1 << b))
        else: # subnormal case
            if significand:
                return (significand.bit_length() - AA - b, significand)
            else:
                return (0, 0)

    def __float__(self):
        return self.int64_to_float(self.to_ieee())
        
    @classmethod
    def from_float(cls, x):
        """Create an anyfloat instance from a python float (64 bits double precision number)."""
        return cls.from_ieee(cls.float_to_int64(x))
        
    @classmethod
    def from_ieee(cls, n, size = DEFAULT_SIZE):
        """Create an anyfloat from an ieee754 integer.
        
        Create an anyfloat from an integer which binary representation is the ieee754
        format of a floating point number. The argument 'size' is a tuple (w, p)
        containing the width of the exponent part and the significand part in
        this ieee754 format."""
        w, p = size
        r = n >> p
        significand = (r << p) ^ n
        sign = int(r >> w)
        if not sign in (0, 1):
            raise ValueError(("Integer value out of range for ieee754 format", n, size))
        exponent = (sign << w) ^ r
        e, s = cls._decode(exponent, significand, w, p)
        if s == -2:
            sign = 0
        return cls(sign, e, s)

    def ieee_parts(self, size = DEFAULT_SIZE):
        w, p = size
        e, s = self._encode(self.exponent, self.significand, w, p)
        sign = 0 if (e + 1) >> w else self.sign
        return sign, e, s

    def to_ieee(self, size = DEFAULT_SIZE):
        """Convert to an ieee754 integer.
        
        Convert self to an integer which binary representation is the ieee754 format corresponding
        to the 'size' argument (read the documentation of from_ieee() for the meaning of the size
        argument.
        """
        sign, e, s = self.ieee_parts(size)
        return (((sign << size[0]) | e) << size[1]) | s
        
    @classmethod
    def int64_to_float(cls, n):
        """Convert a 64 bits integer to a python float.
        
        This class method converts an integer representing a 64 bits floating point
        number in the ieee754 double precision format to this floating point number."""
        
        if not (0 <= n < cls._b64):
            raise ValueError(("Integer value out of range for 64 bits ieee754 format", n))
        u, v = divmod(n, cls._b32)
        return struct.unpack(">d", struct.pack(">LL", u, v))[0]
        
    @classmethod
    def float_to_int64(cls, x):
        """Convert a python float to a 64 bits integer.
        
        This class method converts a float to an integer representing this
        float in the 64 bits ieee754 double precision format."""
        
        u, v = struct.unpack(">LL", struct.pack(">d", x))
        return (u << 32) | v
        
    def bin(self, size = DEFAULT_SIZE, sep=' '):
        """Return a binary representation of self.
        
        The returned string contains only the characters '0' and '1' and shows the
        ieee754 representation of the real number corresponding to self whith the given
        size = (w, p).
        """
        if sep:
            sign, e, s = self.ieee_parts(size)
            return sep.join((fbin(sign, 1), fbin(e, size[0]), fbin(s, size[1])))
        else:
            return fbin(self.to_ieee(size), sum(size) + 1)
        
    def to_fraction(self):
        s = self.significand
        b = s.bit_length()
        k = self.exponent + 1 - b
        if k < 0:
            d = Fraction(s, 2 ** (-k))
        else:
            d = Fraction(s * 2**k, 1)
        return -d if self.sign else d
    
    def trim(self, w=None, p=None):
        """Return the nearest anyfloat fully representable in size (w, p)
        
        This method can be called with two integer arguments
        or a single argument which is a tuple of two positive integers
        representing a size.
        
            >>> a = anyfloat.from_float(math.pi)
            >>> b = a.trim(3, 4)
            >>> print(b)
            anyfloat(sign=0, exponent=1, significand=25)
            >>> print(float(b))
            3.125
        """
        size = (w or DEFAULT_SIZE) if (p is None) else (w, p)
        return self.from_ieee(self.to_ieee(size), size)
    
    def hex(self):
        """Return a hexadecimal representation of self.
        
        The meaning of the result is similar to the result of float.hex()

        XXX not sufficiently tested
        """
        if self.significand <= 0:
            if not self.significand:
                return '0x0.0p+0'
            elif self.significand == -1:
                return ('-' if self.sign else '') + 'inf'
            else:
                return 'nan'
        s = hex(self.significand)[2:]
        expo = self.exponent - 3 + (4 - int(s[0], 16).bit_length())
        return '{}0x{}.{}p{}{}'.format(
            '-' if self.sign else '', s[0], s[1:],
            '+' if expo >= 0 else '', expo
            )
    
    @classmethod
    def from_hex(cls, string):
        """Convert to a hexadecimal representation.
        
        Similar to float.fromhex() for anyfloats.
        
        XXX not sufficiently tested
        """
        m = re.match(r'^([-+]?)(?:0x)?((?:\d|[a-fA-F])+)(?:[.]((?:\d|[a-fA-F])*))?(?:p([-+]?)(\d*))?$', string)
        if not m:
            raise ValueError('Invalid hex string', repr(string))
        sign, ent, frac, se, e = m.groups()
        e = int(e) if e else 0
        if se == '-':
            e = -e
        sign = 1 if sign == '-' else 0
        if int(ent, 16):
            frac = ent + frac
            e += 4 * len(ent)
        frac = frac.rstrip('0')
        z = len(frac)
        frac = frac.lstrip('0')
        z -= len(frac)
        e -= 4 * z
        if frac:
            e -= (4 - int(frac[0], 16).bit_length())
            significand = int(frac, 16)
        else:
            significand = 0
        return anyfloat(sign, e - 1, significand)

    def _bit_length(self):
        return self.significand.bit_length()
    
    def _shift(self):
        return 1 + self.exponent - self._bit_length()
    
    def _signed_significand(self):
        return -self.significand if self.sign else self.significand


    def __add__(self, other):
        """XXX not sufficiently tested
        """
        if self.significand < 0 or other.significand < 0:
            raise ValueError
        a, b = self._shift(), other._shift()
        if a < b:
            self, other = other, self
            a, b = b, a
        s = (self._signed_significand()
                << (a-b)) + other._signed_significand()
        if not s:
            return anyfloat(0, 0, 0)
        if s < 0:
            sign, s = True, -s
        else:
            sign = False
        return anyfloat(sign, b - 1 + s.bit_length(), s)
    
    def __neg__(self):
        """XXX not sufficiently tested
        """
        if self.significand > 0 or self.significand == -1:
            return anyfloat(not self.sign, self.exponent, self.significand)
        else:
            return self
        
    def __sub__(self, other):
        """XXX not sufficiently tested
        """
        return self + (-other)

def main():
    from math import exp
    val = exp(2)
    print ("exp(2) =", repr(val))
    af = anyfloat.from_float(val)
    print (af)
    print (af.bin(), "(64 bits float)")
    print (" " * 2, af.bin(size = (8,23)), "(32 bits)")
    print (" " * 7, af.bin(size = (3, 4)), "(8 bits)")
    print ("conversion to float works:", float(af) == val)
    
if __name__ == "__main__":
    main()
    
""" output -->
exp(2) = 7.38905609893065
anyfloat(sign=0, exponent=2, significand=4159668786720471)
0 10000000001 1101100011100110010010111000110101001101110110101110 (64 bits float)
   0 10000001 11011000111001100100110 (32 bits)
        0 101 1110 (8 bits)
conversion to float works: True

Notice that during the conversion to shorter formats, for example 64 bits to 32 bits,
the significand is not only truncated to 23 bits, it is also rounded to the nearest value
depending on the value of the next bit.
"""

Line 189 should read

if s == -2:

Line 189 should read

if significand == -2:

Hi Chris and thank you very much for your comment. Actually, it should read if s == -2 to handle the case where the _decode() function return (0, -2) at line 156. You make me realize that this code needs a serious refactoring and a series of unit tests. I will add them when I find the time to do so. In any event, this code can be used to explore floating points numbers at the Python console as I did many times. Don't hesitate to comment again if you have other ideas!

Thanks. I changed my comment before I realized you had replied. I've found your code helpful to write test scripts for my system that is converting between SP floats and fixed point formats.

Hi. Thanks for sharing the code, very useful.

I tried to use the code to directly create a custom float type (specifying my own size), but could not find a way to do so.
However, I checked that the following does the trick:

from anyfloat import anyfloat
import math
af = anyfloat.from_float(math.pi)
temp_int = anyfloat.to_ieee(af, size=(3,4))
af2 = anyfloat.from_ieee(temp_int, size=(3,4))

print("full-precision: {} \ncustom-precision: {}".format(float(af), float(af2)))

Output:

full-precision: 3.141592653589793 
custom-precision: 3.125

What do you think if that behavior could be part of the from_float method?

Something like:

@classmethod
    def from_float(cls, x, size = DEFAULT_SIZE):
        """Create an anyfloat instance from a python float (64 bits double precision number)."""
        return cls.from_ieee(cls.to_ieee(cls.from_ieee(cls.float_to_int64(x)), size), size)

Instead of the previous:

@classmethod
    def from_float(cls, x):
        """Create an anyfloat instance from a python float (64 bits double precision number)."""
        return cls.from_ieee(cls.float_to_int64(x))

This makes the previous code cleaner:

from anyfloat import anyfloat
import math

af = anyfloat.from_float(math.pi, size=(3,4))
print("full-precision: {} \ncustom-precision: {}".format(math.pi, float(af)))

Haven't tested more cases though...

@pedroexenberger
Hi and thank you very much for your comment.
I updated the class with a new method trim() to compute the nearest anyfloat that is fully representable in a given size. I think it is more natural because an instance of anyfloat does not carry any size information. You can now use

af = anyfloat.from_float(math.pi).trim(3, 4)
# or
af = anyfloat.from_float(math.pi).trim((3, 4))

This operation is a projection, in the sense that af.trim(size).trim(size) == af.trim(size) should always be true.

Great, thank you

Here's a set of popular sizes:

BINARY8   = ( 3,   4)  # Quarter precision (hypothetical)
BINARY16  = ( 5,  10)  # Half precision
BINARY32  = ( 8,  23)  # Single precision
BINARY64  = (11,  52)  # Double precision
BINARY128 = (15, 112)  # Quadruple precision
BINARY256 = (19, 236)  # Octuple precision

DEFAULT_SIZE = BINARY64

@vshymanskyy Hi Volodymyr and thank you very much for your contribution! I added this set in the above code, and best whishes for Ukraine in 2024!

@Gribouillis thanks!

I needed to work with 80-bit extended (x87) floats, so I added these 2 functions.
This works by converting to/from BINARY128, but it may also be implemented more directly (I'm good with this for now).

    @classmethod
    def from_x87(cls, n):
        ieee  = (n >> 64) << 112
        ieee |= (n & 0x7fffffffffffffff) << 49
        return cls.from_ieee(ieee, BINARY128)

    def to_x87(self):
        ieee = self.to_ieee(BINARY128)
        result  = ((ieee >> 112) & 0xffff) << 64
        result |= ((ieee >> 49) & 0xffffffffffffffff)
        result |= (1 if ieee != 0 else 0) << 63
        return result

Moreover, if anyone needs to load/save an IBM double-double format, this can be done (approximately) like this:

def load_ibmlongdouble(val):
    hi = val >> 64
    lo = val & 0xffffffffffffffff
    return anyfloat.from_ieee(hi, BINARY64) + anyfloat.from_ieee(lo, BINARY64)

def save_ibmlongdouble(val):
    hi = val.trim(BINARY64)
    lo = val - hi
    return hi.to_ieee(BINARY64) << 64 | lo.to_ieee(BINARY64)