Machine epsilon: conclusion

Following the discussion about the machine epsilon, I have modified the
text as follows. I thank all people that contributed. It was interesting, and I surely learned stuff I did not consider before.
--------------------------------------------------------------
The machine epsilon
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The standard defines for each floating point representation (float, double, long double) the difference between 1 and the least value greater than 1 that is representable in the given floating point type. In IEEE754 representation this number has an exponent value of the bias, and a fraction of 1. In hexadecimal notation this would be
0x1p-52
Another way to define this quantity in terms of the mathematical functions of the C library is:

DBL_EPSILON = nextafter(1.0,2.0) - 1.0;

For the different representations we have in the standard header <float.h>:

#define FLT_EPSILON 1.19209290e-07F // float
#define DBL_EPSILON 2.2204460492503131e-16 // double
#define LDBL_EPSILON 1.084202172485504434007452e-19L //long double
// qfloat epsilon truncated so that it fits in this page...
#define QFLT_EPSILON 1.09003771904865842969737513593110651 ... E-106

These definitions (except the qfloat part) are part of the C99 ANSI standard. For the standard types (float, double and long double) they should always exist in other compilers.

Here is a program that will find out the machine epsilon for a given floating point representation.

#include <stdio.h>
int main(void)
{
double float_radix=2.0;
double inverse_radix = 1.0/float_radix;
double machine_precision = 1.0;
double temp = 1.0 + machine_precision;

while (temp != 1.0) {
machine_precision *= inverse_radix;
temp = 1.0 + machine_precision ;
printf("%.17g\n",machine_precision);
}
return 0;
}
Exercises:
1: Explain why in the above program, the value of DBL_EPSILON is not the last number but the number before.
2: An alternative version of the above program is:
#include <stdio.h>
int main(void)
{
volatile double oneplus = 2, epsilon = 1;
while (1 + epsilon/2 > 1) {
epsilon /= 2;
oneplus = 1 + epsilon;
}
epsilon = oneplus - 1;
printf("DBL_EPSILON is %g\n", epsilon);
return 0;
}

Explain why this program prints

DBL_EPSILON is 0

in lcc-win32.
.

Relevant Pages

  • Re: Point between 2 points
    ... the distance from A to C plus the distance from B to C is ... (lowercase epsilon). ... floating-points arithmetics at all... ... post represented using some funky uncommon representation? ...
    (comp.lang.java.programmer)
  • Re: Floating point arithmetic (epsilon)
    ... suggesting that delta is large ... To find the epsilon of any given floating-point number, ... where b is the floating point representation base, ... FLOAT if OPERATION were applied to FLOAT and this number. ...
    (comp.lang.lisp)
  • Re: An strange point
    ... where epsilon() is a standard function returning the smallest number E ... Arjen- Tekst uit oorspronkelijk bericht niet weergeven - ...
    (comp.lang.fortran)
  • Re: Machine epsilon: conclusion
    ... The standard defines for each floating point representation the difference between 1 and the least ... // qfloat epsilon truncated so that it fits in this page... ... oneplus = 1 + epsilon; ...
    (comp.lang.c)
  • Re: The machine epsilon
    ... The machine epsilon is the smallest number that changes the result of an addition operation at the point where the representation of the numbers is the densest. ... encoding of the exponent, then I think this statement is wrong. ... An FP epsilon has to do with the precision of the fraction, ...
    (comp.lang.c)