import matplotlib as mpl
import matplotlib.pyplot as plt
%matplotlib inline
import numpy as np

Floating Point Numbers

Integer Range

We discussed in class that we can obtain the range of integers that can be represented exactly as

$$[1,2^{p}]$$

So let's try this using Python (where double precision gives p=53)

# This number is within the range
2**52

# We add one to get the next integer 
2**52 + 1

# All good so far, right? Let's check the integer:
2**53

# If I add one, what do you expect us to get?
2**53 + 1

Humm... not what you expected right? Why do you think this happened?

Integer type

# to find the maximum integer (using 64-bit signed integer)
# uses only 63 bits, since one is reserved for the sign
maxint = 0
for i in range(63):
     maxint += 2**i
maxint

What will happen when we run these code snippets?

A) it won't stop (infinite loop)

B) it will stop when b reaches underflow

C) it will stop when b reaches machine epsilon

D) none of the above

a = 10**4
b = 1.0
i=0
while (a + b > a):
    #print("{0:d}, {1:1.16f}". format(i, b)) 
    print(i,b)
    b = b / 2.0
    i+=1

A) it won't stop (infinite loop)

B) it will stop when b reaches underflow

C) it will stop when b reaches machine epsilon

D) none of the above

a = 1.0
while a > 0.0: 
    a = a / 2.0
    print(a)
    #print("% .16e"% (a)) 

a = 1.0
while a != np.inf:
    a = a * 2.0
    print(a)
    #print("% .16e"% (a)) 

Let's write the following number (not exacly OFL, but close)

float(2**1023)

# Try the UFL definition
2**(1024)*(1-2**(-53))

# Can we make this work?