fprintf('One number is %f\n', 436598989898987.124132)

I am wondering if someone can explain why after I input the command below:

fprintf('One number is %f\n', 436598989898987.124132)

The result I received was:

One number is 436598989898987.120000

I am curious about the difference between the two numbers.

Thank you in advance, - Lisa

[Cedric]

Look at the eps documentation and then at these two cases:

 fprintf('One number is %f\n', 436598989898987.124132 + eps( 436598989898987 ))
 One number is 436598989898987.190000

 fprintf('One number is %f\n', 436598989898987.124132 + eps( 436598989898987 )/2)
 One number is 436598989898987.120000

If you want to see how the order of magnitude of the distance varies with the powers of 10

 plot( 0:20, log10( eps( 10.^(0:20) ))) ;
 grid ;

and you'll be able to find at 0 ( 10^0=1 ) the 2^-52 of the doc:

 log10( 2^-52 )
 ans =
  -15.6536

[Walter Roberson]

>> eps(436598989898987.124132)
ans =
                    0.0625

Your number is only stored internally to within the nearest 0.0625

We can tell from the output that you are using MS Windows; on OS-X or Linux you would have gotten

One number is 436598989898987.125000

The MS Windows libraries are poor at formatting values exactly.

Answer 1

[Mark Schwab]

The default data type for storing numbers in MATLAB is double. The double data type is a 64 bit data type that stores numbers in a representation similar to scientific notation. Within the 64 bits, one is reserved to specify sign (positive or negative). The next 11 bits are reserved for the exponent power and the remaining 52 bits are used for the fractional value. The number you are trying to store would require more than 52 bits to represent the fractional portion with exact precision so MATLAB uses floating point precision to round to a number it can represent in memory.

Please refer to the following documentation for more information: https://www.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html

[Stephen23]

https://www.mathworks.com/help/matlab/matlab_prog/floating-point-numbers.html