Float multiplication give Wrong Answer (python)

Question

we know that multiplication and division are inverse each other, so in python Suppose i have a number 454546456756765675454.00 and i want to divided the number with 32 lets define a variable for example

value = 454546456756765675454.00/32

so the output will be, 1.4204576773648927e+19 or 14204576773648926720.000000, now i want to multiply the output with 32 so if i multiply 14204576773648926720.000000 * 32 then the output give me 454546456756765655040.00 not 454546456756765675454.00 why this happend? i am not good at math, but my question is why float multiply give me wrong answer ( i also try decimal module but its not work for me or maybe i dont know how to use decimal module to get exact answer)

You need to consider the limitations of floating point numbers. When you change from float to integers there will be some loss, particularly for very large or very small numbers. — Jason Chia, May 04 '20 at 11:07
Also check this: https://stackoverflow.com/questions/588004/is-floating-point-math-broken — chtz, May 04 '20 at 11:25

score 1 · Accepted Answer · answered May 04 '20 at 11:25

Floating points are stored as binary fractions. Some number cannot be precisely written in base 2 form. So, their approximated value is store.
Now if this approximation had an error of +0.0001 for some number, and if this number is multiplied by 10000, then we our result will shift by value of 0.0001*10000 = 1.

It is same in pretty much all programming languages.

For operations where precision is very important, decimal module should be preferred.

i also try decimal module but its not work for me or maybe i dont know how to use decimal module to get exact answer

Your example, using decimal module, will look something like:

import decimal

value = decimal.Decimal(454546456756765675454)
vd = value/decimal.Decimal(32)
vm = vd*32

diff = vm - value
assert diff == decimal.Decimal(0)
# assert diff == 0.0

score 0 · Answer 2 · answered May 04 '20 at 14:18

Within wide range limits, multiplication and division by powers of two, including 32, are exact in binary floating point. Conversion of a decimal is inexact. 454546456756765655040 is the closest IEEE 754 64-bit binary number to 454546456756765675454. The division and multiplication by 32 made no difference.

More generally, division and multiplication by the same number can result in rounding error in finite width decimal/binary etc. fractions unless all the prime factors of the divisor are also prime factors of the radix being used to represent fractions. In both binary and decimal fractions, division and multiplication by 3 can cause rounding error, because 3 is a factor of neither 2 nor 10.

Division and multiplication by 32 can be exact, given enough significand width, in both decimal and binary because two, the only prime factor of 32, is a factor of both 10 and 2.

Float multiplication give Wrong Answer (python)

2 Answers2