what is the difference between 1.0 and 1.1

[James Quinton]

given this code:

double num1=1.0;
double num2=1.1;
System.out.println(num1==1.0f);
System.out.println(num2==1.1f);

It prints out "true" and "false"
why? Looks like a fraction makes a difference. How?

[Franz Fountain]

A double has higher precision than a float. The number 1 can be represented perfectly in binary (in just 1 bit). So there is no problem with comparing it. The number 0.1 must not have a perfect representation in binary. (I've never been inspired to take a close look at binary fractions so I can't provide any details.) Here the higher precision of the double gives a more accurate representation than the float. Essentially there is a rounding error that is the difference.

I tried this and when expanding 1.1 to 12 decimal places I got:
double = 1.100000000000, float = 1.100000023842

I don't know how many digits are really accurate on a double or a float. But anyway you can see there is a difference after 8 decimal places.

[Franz Fountain]

Hi James,

I got interested in this problem and wrote a program to see what's going on. Here's the code.

        double x=0;
        final double num=0.1;
        double tmp=0;
        StringBuilder s0_1 = new StringBuilder("0.");
        String sTmp = "";
        for(int i=1; i <= 80; i++) {
            tmp = 1/java.lang.Math.pow(2, i);
            if(x + tmp <= num) {
                x += tmp;
                s0_1.append(1);
                // use sTmp to make it easy to figure out the number of decimal places
                sTmp = String.valueOf(x);
                out.printf("%-24s %2d    %-60s %2d\n", sTmp, sTmp.length()-2, s0_1, s0_1.length()-2);
            } else {
                s0_1.append("0");
            }
            if(x == num) {
                break;
            }
        }
        
        out.println("\n");
        float xf=0;
        final float numf=0.1f;
        float tmpf=0;
        StringBuilder s0_1f = new StringBuilder("0.");
        for(int i=1; i <= 80; i++) {
            tmpf = (float)(1/java.lang.Math.pow(2, i));
            if(xf + tmpf <= numf) {
                xf += tmpf;
                s0_1f.append(1);
                // use sTmp to make it easy to figure out the number of fraction digits
                // subtract 2 to get rid of "0." chars and just count fraction digits
                sTmp = String.valueOf(xf);
                out.printf("%-24s %2d    %-60s %2d\n", sTmp, sTmp.length()-2, s0_1f, s0_1f.length()-2);
            } else {
                s0_1f.append("0");
            }
            if(xf == numf) {
                break;
            }
        }

And here's the output:
0.0625 4 0.0001 4
0.09375 5 0.00011 5
0.09765625 8 0.00011001 8
0.099609375 9 0.000110011 9
0.099853515625 12 0.000110011001 12
0.0999755859375 13 0.0001100110011 13
0.0999908447265625 16 0.0001100110011001 16
0.09999847412109375 17 0.00011001100110011 17
0.09999942779541016 17 0.00011001100110011001 20
0.09999990463256836 17 0.000110011001100110011 21
0.09999996423721313 17 0.000110011001100110011001 24
0.09999999403953552 17 0.0001100110011001100110011 25
0.09999999776482582 17 0.0001100110011001100110011001 28
0.09999999962747097 17 0.00011001100110011001100110011 29
0.09999999986030161 17 0.00011001100110011001100110011001 32
0.09999999997671694 17 0.000110011001100110011001100110011 33
0.09999999999126885 17 0.000110011001100110011001100110011001 36
0.09999999999854481 17 0.0001100110011001100110011001100110011 37
0.0999999999994543 16 0.0001100110011001100110011001100110011001 40
0.09999999999990905 17 0.00011001100110011001100110011001100110011 41
0.0999999999999659 16 0.00011001100110011001100110011001100110011001 44
0.09999999999999432 17 0.000110011001100110011001100110011001100110011 45
0.09999999999999787 17 0.000110011001100110011001100110011001100110011001 48
0.09999999999999964 17 0.0001100110011001100110011001100110011001100110011 49
0.09999999999999987 17 0.0001100110011001100110011001100110011001100110011001 52
0.09999999999999998 17 0.00011001100110011001100110011001100110011001100110011 53
0.1 1 0.0001100110011001100110011001100110011001100110011001101 55


0.0625 4 0.0001 4
0.09375 5 0.00011 5
0.09765625 8 0.00011001 8
0.099609375 9 0.000110011 9
0.099853516 9 0.000110011001 12
0.099975586 9 0.0001100110011 13
0.099990845 9 0.0001100110011001 16
0.099998474 9 0.00011001100110011 17
0.09999943 8 0.00011001100110011001 20
0.099999905 9 0.000110011001100110011 21
0.099999964 9 0.000110011001100110011001 24
0.099999994 9 0.0001100110011001100110011 25
0.1 1 0.000110011001100110011001101 27

I hope this looks OK on the screen. The lines are pretty wide.
:roll:

[Franz Fountain]

Oh darn the output got mangled
I'll try putting it in a code window. If that doesn't work you'll just have to run it yourself.

    
0.0625                    4    0.0001                                                        4
0.09375                   5    0.00011                                                       5
0.09765625                8    0.00011001                                                    8
0.099609375               9    0.000110011                                                   9
0.099853515625           12    0.000110011001                                               12
0.0999755859375          13    0.0001100110011                                              13
0.0999908447265625       16    0.0001100110011001                                           16
0.09999847412109375      17    0.00011001100110011                                          17
0.09999942779541016      17    0.00011001100110011001                                       20
0.09999990463256836      17    0.000110011001100110011                                      21
0.09999996423721313      17    0.000110011001100110011001                                   24
0.09999999403953552      17    0.0001100110011001100110011                                  25
0.09999999776482582      17    0.0001100110011001100110011001                               28
0.09999999962747097      17    0.00011001100110011001100110011                              29
0.09999999986030161      17    0.00011001100110011001100110011001                           32
0.09999999997671694      17    0.000110011001100110011001100110011                          33
0.09999999999126885      17    0.000110011001100110011001100110011001                       36
0.09999999999854481      17    0.0001100110011001100110011001100110011                      37
0.0999999999994543       16    0.0001100110011001100110011001100110011001                   40
0.09999999999990905      17    0.00011001100110011001100110011001100110011                  41
0.0999999999999659       16    0.00011001100110011001100110011001100110011001               44
0.09999999999999432      17    0.000110011001100110011001100110011001100110011              45
0.09999999999999787      17    0.000110011001100110011001100110011001100110011001           48
0.09999999999999964      17    0.0001100110011001100110011001100110011001100110011          49
0.09999999999999987      17    0.0001100110011001100110011001100110011001100110011001       52
0.09999999999999998      17    0.00011001100110011001100110011001100110011001100110011      53
0.1                       1    0.0001100110011001100110011001100110011001100110011001101    55
 

0.0625                    4    0.0001                                                        4
0.09375                   5    0.00011                                                       5
0.09765625                8    0.00011001                                                    8
0.099609375               9    0.000110011                                                   9
0.099853516               9    0.000110011001                                               12
0.099975586               9    0.0001100110011                                              13
0.099990845               9    0.0001100110011001                                           16
0.099998474               9    0.00011001100110011                                          17
0.09999943                8    0.00011001100110011001                                       20
0.099999905               9    0.000110011001100110011                                      21
0.099999964               9    0.000110011001100110011001                                   24
0.099999994               9    0.0001100110011001100110011                                  25
0.1                       1    0.000110011001100110011001101                                27