# -*- coding: utf-8 -*-
"""
Created on Sun Dec 20 15:58:19 2020

@author: neil
"""

# In numerical (computer) work there is always a limit to the precision of any calculation
# because real numbers can only be stored to some number of digits (in python the default
# is 15 digits, but can be extended if needed)
# Most common numerical errors propagate from simple subtraction!
# this is an artificial example, but shows that the order of subtractions
# can be very important if two numbers very close to each other are subtracted,
# the answer will be very small, but the error will be the same size as the original
# errors on the large numbers.  In the worst case, the error can be as large or larger than
# the answer!

# subtracting 2 numbers that are large, with only a tiny difference
# python only keeps approximatly 15 or so digits in any number
a = 1.23456789             # a 'small' number
b = 1.0e16                 # a 'big' number
c = -1.0e16
# we all know that in 'math' addition/subtraction is commutative, so:
ans1 = (a+b)+c                 # (a+b) is subtracted by c, 2 big numbers only slightly different
ans2 = a+(b+c)                 # (b+c) this subtraction is exactly 1, leaving ans2 equal to 'a'

# we know the answer should be the small number exactly, by basic arithmatic
print("#1 An example of truncation error enhancement by subtraction")
print("incorrect answer for a+b+c = {0:f}".format(ans1) )
print("correct answer for a+(b+c) = {0:f}".format(ans2) )
print("The error is the same order of magnitude as 'a'! namely {0:e}".format(ans1-a))

# a more typical example would usually be harder to spot, 
print("\n\r#2 A harder to spot example")  # the '\n\r' is a linefeed-cariage return (or newline)
x = 1e-10          # again a 'small' number, which goes into a quadratic equation with a 'big' number
y1 = 1 - (1+x)**2  # this leads to a significant error, but can be fixed by noticing
# that (1+x)**2 could be written as 1+2*x+x**2
y2 = (1 - 1) - 2*x - x**2  # get rid of the 2 big numbers!
print("the correct answer is {0:1.15e}".format(y2))
print("the incorrect is      {0:1.15e}".format(y1))
print("the difference in the 2 answers is {0:e}".format(y1-y2))