Garbage bits in the floating point representation of numbers in CC program

Questions

Possible Duplicate:
http://stackoverflow.com/questions/588004/is-javascripts-math-broken http://stackoverflow.com/questions/588004/is-javascripts-math-broken http://stackoverflow.com/questions/590822/dealing-with-accuracy-problems-in-floating-point-numbers http://stackoverflow.com/questions/590822/dealing-with-accuracy-problems-in-floating-point-numbers

Consider the code below and its output:

#include <iostream>
#include <iomanip>
#include <cstdlib>
#include <cmath>
#include <limits>
#include <vector>


int main(int argc, char *argv[])
{
  double xleft  = 0;
  double xright = 1.0;

  double dx = 0.1;
  std::cout << std::setprecision(36) << "dx is " << dx << std::endl; 

  int numgridpts = ((int) ceil (( xright - xleft )/dx))  + 1; 


  for (int i = 0; i < numgridpts  ; ++i)
    {
      std::cout << std::setprecision(36) << xleft + i*dx << std::endl;
    }

  return 0;
}

[~:ICgen/$ ./a.out                                                                                                                               
dx is 0.100000000000000005551115123125782702
0
0.100000000000000005551115123125782702
0.200000000000000011102230246251565404
0.300000000000000044408920985006261617
0.400000000000000022204460492503130808
0.5
0.600000000000000088817841970012523234
0.700000000000000066613381477509392425
0.800000000000000044408920985006261617
0.900000000000000022204460492503130808
1

My question is when I print out the numbers till a precision of 36 bits, why are the numbers, 0 , 0.5 and 1.0 represented exactly, wherars the other numbers seem to have some garbage numbers placed at the end?

Also if I add the floating point representations of 0.2 and 0.1 as shown in the output above, they dont seem to add up to the representation of 0.3, in the part of the garbage -bits.

I am using Linux Ubuntu 10.10 and the gcc compiler

Answers

These are not garbage numbers but as as accurate values of those numbers as possible within the floating point representation according to the IEEE 754 standard.

That standard uses binary numbers, not decimal numbers. Therefore, fractions like 1/2, 1/4, 1/8 (= 0.5, 0.25, 0.125, ...) and their multiples are represented exactly, whereas fractions like 1/3 or 1/10 are not.

If this was a decimal digit system, fractions like 1/10, 1/100, 1/1000... and their multiples would be the only numbers represented exactly with a finite number of digits.

In both systems, all integers are represented exactly if they are not extremely large.

Source

License : cc by-sa 3.0

http://stackoverflow.com/questions/10019127/garbage-bits-in-the-floating-point-representation-of-numbers-in-c-c-program

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