28.18 Machine Epsilon

In any floating-point system, three attributes are particularly important to know: base (the number that the exponent specifies a power of), precision (number of digits in the significand), and range (difference between most positive and most negative values). The allocation of bits between exponent and significand decides the answers to those questions.

A measure of the precision is the answer to the question: what is the smallest number that can be added to 1.0 such that the sum differs from 1.0? That number is called the machine epsilon.

We could define the needed machine-epsilon constants for float, double, and long double like this:

static const float  epsf = 0x1p-23;  /* about 1.192e-07 */
static const double eps  = 0x1p-52;  /* about 2.220e-16 */
static const long double epsl = 0x1p-63;  /* about 1.084e-19 */

Instead of the hexadecimal constants, we could also have used the Standard C macros, FLT_EPSILON, DBL_EPSILON, and LDBL_EPSILON.

It is useful to be able to compute the machine epsilons at run time, and we can easily generalize the operation by replacing the constant 1.0 with a user-supplied value:

double
macheps (double x)
{ /* Return machine epsilon for x,  */
      such that x + macheps (x) > x.  */
  static const double base = 2.0;
  double eps;

  if (isnan (x))
      eps = x;
  else
    {
      eps = (x == 0.0) ? 1.0 : x;

      while ((x + eps / base) != x)
          eps /= base;          /* Always exact!  */
    }

  return (eps);
}

If we call that function with arguments from 0 to 10, as well as Infinity and NaN, and print the returned values in hexadecimal, we get output like this:

macheps (  0) = 0x1.0000000000000p-1074
macheps (  1) = 0x1.0000000000000p-52
macheps (  2) = 0x1.0000000000000p-51
macheps (  3) = 0x1.8000000000000p-52
macheps (  4) = 0x1.0000000000000p-50
macheps (  5) = 0x1.4000000000000p-51
macheps (  6) = 0x1.8000000000000p-51
macheps (  7) = 0x1.c000000000000p-51
macheps (  8) = 0x1.0000000000000p-49
macheps (  9) = 0x1.2000000000000p-50
macheps ( 10) = 0x1.4000000000000p-50
macheps (Inf) = infinity
macheps (NaN) = nan

Notice that macheps has a special test for a NaN to prevent an infinite loop.

Our code made another test for a zero argument to avoid getting a zero return. The returned value in that case is the smallest representable floating-point number, here the subnormal value 2**(-1074), which is about 4.941e-324.

No special test is needed for an Infinity, because the eps-reduction loop then terminates at the first iteration.

Our macheps function here assumes binary floating point; some architectures may differ.

The C library includes some related functions that can also be used to determine machine epsilons at run time:

#include <math.h>           /* Include for these prototypes. */

double      nextafter  (double x, double y);
float       nextafterf (float x, float y);
long double nextafterl (long double x, long double y);

These return the machine number nearest x in the direction of y. For example, nextafter (1.0, 2.0) produces the same result as 1.0 + macheps (1.0) and 1.0 + DBL_EPSILON. See FP Bit Twiddling in The GNU C Library Reference Manual.

It is important to know that the machine epsilon is not symmetric about all numbers. At the boundaries where normalization changes the exponent, the epsilon below x is smaller than that just above x by a factor 1 / base. For example, macheps (1.0) returns +0x1p-52, whereas macheps (-1.0) returns +0x1p-53. Some authors distinguish those cases by calling them the positive and negative, or big and small, machine epsilons. You can produce their values like this:

eps_neg = 1.0 - nextafter (1.0, -1.0);
eps_pos = nextafter (1.0, +2.0) - 1.0;

If x is a variable, such that you do not know its value at compile time, then you can substitute literal y values with either -inf() or +inf(), like this:

eps_neg = x - nextafter (x, -inf ());
eps_pos = nextafter (x, +inf() - x);

In such cases, if x is Infinity, then the nextafter functions return y if x equals y. Our two assignments then produce +0x1.fffffffffffffp+1023 (about 1.798e+308) for eps_neg and Infinity for eps_pos. Thus, the call nextafter (INFINITY, -INFINITY) can be used to find the largest representable finite number, and with the call nextafter (0.0, 1.0), the smallest representable number (here, 0x1p-1074 (about 4.491e-324), a number that we saw before as the output from macheps (0.0)).