Write a function, ulps$($x$,$y$),$ that takes two floating point parameters, x and y$,$ and returns

the number of ulps $($floating$-$point intervals$)$ between x and y$.$ You will of course need to

take into account that x and y may have different exponents of the floating$-$point base in

their representation. Also, do not assume that x $<=$ y $($handle either case$).$ Your function

may assume that its parameter will be the floating$-$point type. Do not use logarithms to

find the exponent of your floating$-$point numbers $$ use repeated multiplication$/$division

following the pattern discussed in class. Your function should handle negative numbers

and numbers less than $1.$

Your function will return infinity if the input parameters have the following properties:

$$ Opposite in signs, or

$$ Either one of them is zero, or

$$ Either one of them is either positive infinity or negative infinity.

If the input parameters are both negative, convert them to be positive numbers by taking

the absolute value. Your algorithm only needs to work with two positive floating$-$point

numbers.

The following code segment shows how to use math and sys modules to get the base,

infinity $($inf$),$ machine epsilon $($eps$),$ and mantissa digits $($prec$)$ in Python.

import sys

import math

base $=$ sys$.$float$\_$info.radix

eps $=$ sys$.$float$\_$info.epsilon

prec $=$ sys$.$float$\_$info.mant$\_$dig

inf $=$ math.inf

Algorithm analysis:

$1.$ Check the input parameters for special conditions.

$2.$ Find the exponents for both input parameters in the machine base $($base$).$

For example: Find exp such that $()<=<()+1.$

$3.$ Examine the exp for both parameter:

a$.$ If they are the same: count the intervals between them. $($Remember that

the spacing is $$ for each interval $[,+1])$

b$.$ If they differ by one: add the intervals from the smaller number to $+1$

to the intervals from $+1$ to the larger number.

c$.$ If they differ by more than one: In addition to the number of intervals in

part b$,$ add the numbers of intervals in the exponent ranges in between the

exponents of these two numbers.

Verify your algorithm with the following inputs with my answers which are listed after

each call:print$($ulps$(-1\mathrm{.}0,-1\mathrm{.}0000000000000003))//1$

print$($ulps$(1\mathrm{.}0,1\mathrm{.}0000000000000003))//1$

print$($ulps$(1\mathrm{.}0,1\mathrm{.}0000000000000004))//2$

print$($ulps$(1\mathrm{.}0,1\mathrm{.}0000000000000005))//2$

print$($ulps$(1\mathrm{.}0,1\mathrm{.}0000000000000006))//3$

print$($ulps$(0\mathrm{.}9999999999999999,1\mathrm{.}0))//1$

print$($ulps$(0\mathrm{.}4999999999999995,2\mathrm{.}0))//9007199254741001$

print$($ulps$(0\mathrm{.}5000000000000005,2\mathrm{.}0))//9007199254740987$

print$($ulps$(0\mathrm{.}5,2\mathrm{.}0))//9007199254740992$

print$($ulps$(1\mathrm{.}0,2\mathrm{.}0))//4503599627370496$

print$(2\mathrm{.}0**52)//4503599627370496\mathrm{.}0$

print$($ulps$(-1\mathrm{.}0,1\mathrm{.}0)//$inf

print$($ulps$(-1\mathrm{.}0,0\mathrm{.}0)//$inf

print$($ulps$(0\mathrm{.}0,1\mathrm{.}0)//$inf

print$($ulps$(5\mathrm{.}0,$ math.inf$)//$inf

print$($ulps$(15\mathrm{.}0,100\mathrm{.}0))//1210342399855820$