Machine Precision

Machine precision is the smallest number eps such that the difference between 1 and 1 + eps is nonzero, ie., it is the smallest difference between two numbers that the computer recognizes. On a 32 bit computer, single precision is 2-23 (approximately 10-7) while double precision is 2-52 (approximately10-16) . The following fortran program finds the machine precision (This is based on a usenet posting by Guido Pagnacco). 

      PROGRAM MAIN 
      IMPLICIT NONE
      DOUBLE PRECISION DEPS, DA
      REAL EPS, A
      INTEGER N, DN
 
      EPS = 1.0
      DEPS = 1.0D0
      N = 0
      DN = 0
 
* determination of epsilon for reals (single precision) *
 
1     EPS = EPS / 2.0
      A = EPS + 1.0
*      PRINT *, A
      IF (A .GT. 1.0) THEN
         N = N + 1
         GO TO 1
      ELSE
         EPS = ABS(2.0 * EPS)
      END IF
* determination of epsilon for double precision  *
 
2     DEPS = DEPS / 2.0D0
      DA = DEPS + 1.0D0
*      PRINT *, DA
      IF (DA .GT. 1.0D0) THEN
         DN = DN + 1
         GO TO 2
      ELSE
         DEPS = DABS(2.0D0 * DEPS)
      END IF
 
      IF(N .eq. DN)THEN
           PRINT*,'Your computer probably has a math coprocessor.'
           PRINT*,'Uncomment the two print statements and rerun.'
           STOP
      ENDIF
      PRINT*,'                              ',-N
      WRITE(*,10) eps
10    FORMAT(' Single precision =',e16.8,' or 2')
      PRINT*,'                              ',-DN
      WRITE(*,11) deps
11    FORMAT(' Double precision =',e16.8,' or 2')
      STOP
      END

The above program finds the largest number N such that the difference between 1 and 1 + 2-N is nonzero which gives the machine precision. Computers also have a secondary processor for floating point operations (sometimes called Floating Point Unit or FPU). Let me now quote Pagnacco.

A note of caution about my routine: many compilers will optimize it to run completely inside the math coprocessor of your computer (if it has one). But coprocessors usually perform their internal computations in what is sometimes called long double precision format (the number is 80 bit instead of 64 for the double and 32 for the real). This results in the routine giving out the same value for both the epsilon in real and double precisions, usually a value of 2-64 if the machine works by IEEE standards. To avoid that it is necessary to force the routine to exit from the coprocessor every cycle. A dirty solution is the one I implemented: print something.

If you are using the gcc compiler you can find the machine precision using the following code.

#include<stdio.h>
#include<float.h>
int main(){
    printf("Real precision        = %e\n", FLT_EPSILON);
    printf("Double precision      = %e\n", DBL_EPSILON);
}

Machine precision must be taken into consideration while writing numerical programs. The choice between single and double precision must be made judiciously. For example, while performing turbulent Navier-Stokes computations, the grid is highly clustered near solid walls. The difference between neighbouring grid points can be very small which requires using double precision. The choice of equations can also be important and can be made intelligently. For example, if you want to compute 1 - cos(x) for a very small value of x then it is better to use the identical expression 2*sin2(x/2) since the former expression is susceptible to round-off error due to machine precision.

Note that precision does not represent the smallest number that can be stored in a computer.