Picking apart a floating point number

In [3]:
#keep

# Never mind the details of this function...

def pretty_print_fp(x):
    print("---------------------------------------------")
    print("Floating point structure for %r" % x)
    print("---------------------------------------------")
    import struct
    s = struct.pack("d", x)

    def get_bit(i):
        byte_nr, bit_nr = divmod(i, 8)
        return int(bool(
            s[byte_nr] & (1 << bit_nr)
            ))

    def get_bits(lsb, count):
        return sum(get_bit(i+lsb)*2**i for i in range(count))

    # https://en.wikipedia.org/wiki/Double_precision_floating-point_format

    print("                                 1         2         3         4         5")
    print("indices              : 012345678901234567890123456789012345678901234567890123456789")
    print("Sign bit (1:negative):", get_bit(63))
    exponent = get_bits(52, 11)
    print("Exponent     (binary):", bin(exponent)[2:])
    print("Exponent    (shifted): %d" % (exponent - 1023))
    fraction = get_bits(0, 52)
    significand = fraction + 2**52
    print("Significand  (binary):", bin(significand)[2:])
    print("Significand (shifted):", repr(significand / (2**52)))
In [8]:
#keep
pretty_print_fp(2.0**1022)
---------------------------------------------
Floating point structure for 4.49423283715579e+307
---------------------------------------------
                                 1         2         3         4         5
indices              : 012345678901234567890123456789012345678901234567890123456789
Sign bit (1:negative): 0
Exponent     (binary): 11111111101
Exponent    (shifted): 1022
Significand  (binary): 10000000000000000000000000000000000000000000000000000
Significand (shifted): 1.0
In [9]:
pretty_print_fp(2.0**1024)
---------------------------------------------------------------------------
OverflowError                             Traceback (most recent call last)
<ipython-input-9-0a6a1392a22e> in <module>()
----> 1 pretty_print_fp(2.0**1024)

OverflowError: (34, 'Numerical result out of range')

Things to try:

  • Twiddle the sign bit
  • 1,2,4,8
  • 0.5,0.25
  • $2^{1023}$, $2^{1024}$
  • $2^{-1023}$, $2^{-1024}$
  • float("nan")
In [15]:
import numpy as np
In [6]:
np.binary_repr(1024)
Out[6]:
'10000000000'
In [7]:
bin(1024+1023)
Out[7]:
'0b11111111111'
In [ ]: