```
#!/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 kind 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
http://grouper.ieee.org/groups/754/
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 struct
import sys
version_info = (1, 1)
if sys.version_info < (2, 7):
raise ImportError("Module anyfloat requires python 2.7 or newer.")
DEFAULT_SIZE = (11, 52)
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:
significand = 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 e == -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 main():
from math import exp
val = exp(2)
print ("exp(2) =", 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()
""" my output -->
exp(2) = 7.38905609893
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.
"""
```

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