Problems Of Floating Point Arithmetic

I purposely wrote a hub to explain CVS, CVD, MKS$ and MKD$ functions of QBasic so that those who have access to QBasic can make sense of what I am saying here. Other than that this article is not dependent of QBasic. I am merely using QBasic to make few points about problems of floating point arithmetic. Material here applies to all systems which use IEEE Standard for Floating-Point Arithmetic or anything similar to that.

First of all let me say something about the sub-title: Why Overflow and Underflow Are not the Only Problems. If you look at the first hub in this series of hubs I started with the problem of How to Calculate Binomial Probability. I showed that if we calculate n! as a first step then it takes a very small value of n to cause overflow error. I showed how to avoid such incident by rearranging terms in the multiplications and divisions that we do and furthermore doing those computations using integers only. That is, we carry our fractional numbers as a/b, where both a and b are integers. When calculations are done we carry out one final floating point division taking care that at this final juncture we do not cause underflow or overflow error. So underflow and overflow are not the main problem of floating point arithmetic.

Problem with floating point arithmetic is truncation which causes loss of precision. Especially when such truncated numbers are used again and again when the error caused by truncation adds up. It is to solve this problem that we must think of integer way of calculating.

Even simplest of decimal fractions which can be written as non-recurring fraction in decimal will become recurring fraction when converted to binary fraction. For example 0.1 is recurring fraction as a binary number. And, of course, even simplest of decimal fractions too are recurring. For example: one third. And 1/3 is recurring in binary too.

Even when overflow/underflow problem has not occurred the problem of truncation will invariably occur if an intermediate result was as a result of division and in binary system it was recurring fraction.

Let us visit the problem of calculation of binomial probability again. The formula is as follows:

P(k) = C(n, k) * p^k * (1 - p)^{(n - k)}

where C(n, k) is well known as number of combinations of selecting k objects out of n objects, k ≤ n.

The integer method that was described in that hub will work until we come to last step when we must divide the numerator by denominator when resulting value of P(k) is likely to be so small that there can be underflow error. How to prevent that from happening?

How to Prevent Underflow Error in Calculation of P(k)?

When it is time to evaluate fraction which will give us P(k) we simply change numerator by multiplying it with a power of 2 (or whatever radix that is used in the floating point representation of hardware/software). We select power large enough that would not cause underflow error. So final result of calculation is carried as a floating point number which we know is multiplied by a known power of radix used by target software. If we intend to use this method to calculate sum of P(k)'s (say, in a program which will calculate probability of upper or lower tail), then we should use same or biggest or smallest power of radix that will work for for first P(k) that we calculate. We use biggest if for later P(k)'s will gradually see increase in value, and therefore, possibility of exponent increasing. Using biggest power of radix to multiply to beef up the value of first P(k) will cause the exponent of calculated floating point number to at the lower edge just inside region which prevents underflow error. And later P(k)'s calculated are likely to increase then gradually this exponent too will increase, thereby remaining in acceptable region.

So, as you can see prevention of underflow in intermediate result is also not an unsolvable problem. But a problem now of adding newly computed P(k) to Sum of earlier P(k)'s will become apparent to smarter and alert minds. What happens when we calculate series of terms to be all summed up according algorithm suggested by following?

S(j) = S(j-1) + P(j)

As calculation proceeds S(j-1) becomes large compared to P(j). When two floating point number are added first a common exponent is decided upon, and then mantissas of both number are calculated with that common exponent and then these are added. (This is exponent equalization.) Here is a contrived example in radix 10 (decimal):

10⁺⁵ * 0.12345 + 10^-5 * 0.98765

Let us take +5 as common exponent. So we will have to change the second number so that its exponent is also +5:

(10¹⁰ * 10^-5) * (0.98765 / 10¹⁰) = 10⁵ * 0.000000000098765

Now that we have common exponent and we can add mantissas:

0.12345 + 0.000000000098765 = 0.123450000098765

When our floating point can store only 5 decimal digits "contribution" to sum is absolutely zero. And it will remain zero as long as accumulated sum remains large number. Say, in calculation of type: S(j) = S(j-1) + P(j). Here as soon as S(j) becomes one big fat sum it is quite futile waste of computational power to keep calculating P(j). How do solve this problem?

How to Prevent Loss of Precision When Adding A Large and A Small Floating Point Number?

One way to solve this problem is to accumulate new term in a new sum. As soon as provisional S(j) becomes too big we create another S(j) setting aside old S(j). By the by accumulated sum in new S(j) too will become large enough when it can be added to old accumulated sum. Final accumulated sum is just added to old sum. If it had enough precision even after exponent equalization it will contribute something to sum else there will be no change.

I hope I have made a good case of sticking with integer way of computing fractions. Integer way is the way of maintaining numerator and denominator all through computation until it becomes absolutely necessary to carry out division of numerator by denominator.

Our next stop should be developing a framework of floating point computation in this integer way. We should develop a systematic approach which can be implemented eaily in a portable manner. Hopefully that will be my next hub.