1.2-1. == 10.2-10.

1.2-1. != 10.2-10.

0.2-0.

1.2-1.

10.2-10.

100.2-100.

1000.2-1000.

10000.2-10000.

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

# remember that in numpy, unless otherwise specified by the user, numbers are treated as floats
# it is good practice though to indicate what numbers you consider as floats by using the decimal point
ls = np.logspace(1.,25,25)
x = 0.2
# operators act elementwise on array
x_tilde = (ls+x)-ls
absolute_error = np.abs(x_tilde-x)

plt.plot(ls, absolute_error)
plt.xscale('log')
plt.yscale('log')
plt.xlabel('$O(x)$')
plt.ylabel('$O(\epsilon)$')
plt.title('Absolute Error')
plt.show()

relative_error = np.abs((x_tilde-x)/x)

plt.plot(ls, relative_error)
plt.xscale('log')
plt.yscale('log')
plt.xlabel('$O(x)$')
plt.ylabel('$O(\epsilon)$')
plt.title('Relative Error')
plt.ylim(None, 1.0E1)
plt.show()

1.2-1. == 1.2-1.

0*2**7 + 1*2**6 + 1*2**5 + 1*2**4 + 1*2**3 + 0*2**2 + 1*2**1 + 1*2**0

1*2**7 + 1*2**6 + 1*2**5 + 1*2**4 + 0*2**3 + 1*2**2 + 1*2**1 + 0*2**0

plt.plot(ls, relative_error)
plt.xscale('log')
plt.yscale('log')
plt.xlabel('$O(x)$')
plt.ylabel('$O(\epsilon)$')
plt.title('Relative Error')
plt.ylim(None, 1.0E1)
plt.show()

relative_error_2 = np.abs((x_tilde-x)/ls)

plt.plot(ls, relative_error_2)
plt.xscale('log')
plt.yscale('log')
plt.xlabel('$O(x)$')
plt.ylabel('$O(\epsilon)$')
plt.title('Absolute Error')
plt.ylim(None, 1.0E1)
plt.annotate('$\epsilon_{machine}$', xy=(1.1E1, 1.1E-16), xycoords='data', xytext=(1E6, 1E-10), textcoords='data', 
            arrowprops=dict(arrowstyle="->", connectionstyle="arc3"), fontsize=20 )
plt.show()

1E17*2.220446049250313E-16

1e17 - 0.2

1e17 - 2

1e17 - 20

1*16**4 + 14*16**3 + 0*16**2 + 7*16**1 + 8*16**0

1.2-1. == 10.2-10.

np.allclose(1.2-1., 10.2-10., rtol=1e-016, atol=1e-15)

v = 1.0E154
v**2 # largest representable number

v = 1.0E155
v**2 # overflow

1e-16 + 1 - 1e-16 == 1e-16 - 1e-16 + 1

def is_equal(a, b, rtol):
    return abs(a-b) < max(a,b)*rtol

is_equal(1.2-1., 10.2-10., 1.0E-14)

np.sqrt(1.0E-16 + 1.) - 1.

#       numbers       bits    bits/byte
bytes = 1E6**2    *    64   /     8
kbytes =  bytes/1024
mbytes = kbytes/1024
gbytes = mbytes/1024
tbytes = gbytes/1024

print bytes
print kbytes
print mbytes
print gbytes
print tbytes

epsilon_half_16 = 4.88e-04
epsilon_single_32 = 5.96e-08
epsilon_double_64 = 1.11e-16
order = 100.
print order*epsilon_half_16
print order*epsilon_single_32
print order*epsilon_double_64

#       numbers       bits    bits/byte
bytes = 1E6**2    *    16   /     8
kbytes =  bytes/1024
mbytes = kbytes/1024
gbytes = mbytes/1024
tbytes = gbytes/1024

print bytes
print kbytes
print mbytes
print gbytes
print tbytes