The gap between two adjacent IEEE754 DP numbers is more or less proportional to the magnitude of these numbers. The overwhelming majority of DP numbers do not exactly equal a power of 2. In this case, the gaps between such a DP number and its direct neighbours on both the smaller as well as the bigger sides have a magnitude of about 1.11E-16 times the smallest power of 2 that is larger than the number. For example, consider the DP number 1.5 (hex 3F F8 00 00 00 00 00 00). The smallest power of 2 which is bigger than 1.5 is 2^1=2. Thus, the biggest DP number being smaller than 1.5 is about 1.5 – 2⋅1.11E-16 = 1,499999999999999778 (exactly hex 3F F7 FF FF FF FF FF FF), and the smallest DP number being bigger than 1.5 is about 1.5 + 2⋅1.11E-16 = 1,500000000000000222 (exactly hex 3F F8 00 00 00 00 00 01).
At all exact powers of 2 the gap between adjacent DP numbers doubles in size. This means the question “does x equal 1.0” will yield the answer “true” for any real x in the range 1.0-5.55E-17 <= x < 1.0+1.11E-16, if rounding to the nearest is used in order to convert an arbitrary real number x into an IEEE754 DP number. Please note that in this special case, the interval limits are located asymmetrically around 1.0! That happens for all exact powers of 2. The size of the gap between two adjacent DP numbers in the interval from 1 to 2, i.e. 2.22E-16, is called machine epsilon.
The following figures show the discretisation effect occurring when using DP numbers. In particular, the region around 1 is explored using cell formulae in the spreadsheet application Microsoft®™ Excel®™ 2002 as well as using the VBA 6®™ built-in programming language. The FOSS add-in xNumbers is used to show how the results look like if the calculations are carried out with considerably higher numerical accuracy, i.e., 30 instead of the 17 decimal places in case of the IEEE754 DP numbers. The vertical scaling had to be shifted downwards by 1 in the figures, because XL cannot deal with axis limits of 1+/-2E-15, for example. The discretisation effect occurs in the first step of all shown calculations, i.e., when the relatively small quantity x is added to 1, and is thus not influenced by the final shift process.
There is a remarkable difference between the results obtained using XL directly on the one hand and those obtained by invoking the programming language VBA 6 from XL on the other hand. MS seems to have implemented some “politically correct mathematics” into XL here in order to maximise the interval for which the answer to “does x equal 1” yields true… The results of the XL cell formulae are clearly not IEEE754 compliant.
DP numbers are most dense around 0. Neglecting the special concept of “denormalised numbers”, the absolute value of the smallest numbers different from 0 is 2^(-1022), which is about 2.225E-308. The absolute value of the biggest DP number is 2^1023+2^1022+…+2^972+2^971 (hex
7F EF FF FF FF FF FF FF), which nearly equals 1.7976931348623157E+308. The adjacent DP number on its lower side, i.e. 1.7976931348623155E+308, is smaller by an amount of 2^971, which is about 2.00E+292. In the biggest dyad of the DP number system, the gap of 2.00E+292 between two adjacent DP numbers is huge in absolute terms, but still small in relative terms: 2.00E+292/1,79769E+308 = 1.11E-16, just as it was to be expected.
I have “translated” a routine from FORTRAN into a VBA array function that gives you all relevant information about the numerical characteristics of your computer. You may download the VBA6 code and use it if you follow the GPL v3 (included).
The size of the DP gap is usually small even in absolute terms for all numbers humans deal with in their every day or business life. However, it should be kept in mind that the DP gap around a number is roughly proportional to its magnitude. For example, in the range between 2^52 and 2^53, which equals about 4.5036E+15 to 9.0072E+15, only the integers (gap is 2^0=1) are exactly representable by DP numbers. In the next bigger dyad from 9.0072E+15 to 1.80144E+16, only the even integers (gap is 2^1=2) are exactly representable by DP numbers, and so on. Of course, this is only a consequence of limiting the mantissa of IEEE DP numbers 53 bits or roughly 17 significant decimal digits.
Even in science and engineering, numbers are very rarely known with an accuracy of more than 10 significant decimal digits. In fact, even most scientific measurements typically yield an accuracy of less than 5 significant decimal digits. This is why usually virtually nobody is bothered by neither the discreteness of the set of DP numbers in the continuum of the real numbers nor their limited accuracy of about 17 decimal digits. Consequently, people are quite surprised if even very simple calculations fail due to an effect called catastrophic cancellation. This will be the subject of a follow-up post.