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< E, is not an equivalence relation because a ~ b and b ~ c does not imply that a ~ c.

Instruction Sets

It is quite common for an algorithm to require a short burst of higher precision in order to produce accurate results. One example occurs in the quadratic formula ( )/2a. As discussed in the section Proof of Theorem 4, when b2 4ac, rounding error can contaminate up to half the digits in the roots computed with the quadratic formula. By performing the subcalculation of b2 - 4ac in double precision, half the double precision bits of the root are lost, which means that all the single precision bits are preserved.

The computation of b2 - 4ac in double precision when each of the quantities a, b, and c are in single precision is easy if there is a multiplication instruction that takes two single precision numbers and produces a double precision result. In order to produce the exactly rounded product of two p-digit numbers, a multiplier needs to generate the entire 2p bits of product, although it may throw bits away as it proceeds. Thus, hardware to compute a double precision product from single precision operands will normally be only a little more expensive than a single precision multiplier, and much cheaper than a double precision multiplier. Despite this, modern instruction sets tend to provide only instructions that produce a result of the same precision as the operands.23

If an instruction that combines two single precision operands to produce a double precision product was only useful for the quadratic formula, it wouldn't be worth adding to an instruction set. However, this instruction has many other uses. Consider the problem of solving a system of linear equations,

a11x1 + a12x2 +&#; &#; &#; + a1nxn= b1
a21x1 + a22x2 +&#; &#; &#; + a2nxn= b2
&#; &#; &#;
an1x1 + an2x2 +&#; &#; &#;+ annxn= bn

which can be written in matrix form as Ax = b, where


Suppose that a solution x(1) is computed by some method, perhaps Gaussian elimination. There is a simple way to improve the accuracy of the result called iterative improvement. First compute

(12) = Ax(1) - b

and then solve the system

(13) Ay =

Note that if x(1) is an exact solution, then is the zero vector, as is y. In general, the computation of and y will incur rounding error, so Ay    Ax(1) - b = A(x(1) - x), where x is the (unknown) true solution. Then  x(1) - x, so an improved estimate for the solution is

(14) x(2) = x(1) - y

The three steps (12), infinity box best range check error, (13), and (14) can be repeated, replacing x(1) with x(2), and x(2) with x(3). This argument that x(i + 1) is more accurate than x(i) is only informal. For more information, see [Golub and Van Loan ].

When performing iterative improvement, is a vector whose elements are the difference of nearby inexact floating-point numbers, and so can suffer from catastrophic cancellation. Thus iterative improvement is not very useful unless Ax(1) - b is computed in double precision, infinity box best range check error. Once again, this is a case of computing the product of two single precision numbers (A and x(1)), where the full double precision result is needed.

To summarize, instructions that multiply two floating-point numbers and return a product with twice the precision of the operands make a useful addition to a floating-point instruction set. Some of the implications of this for compilers are discussed in the next section.

Languages and Compilers

The interaction of compilers and floating-point is discussed in Farnum [], and much of the discussion in this section is taken from that paper.

Ambiguity

Ideally, a language definition should define the semantics of the language precisely enough to prove statements about programs. While this is usually true for the integer part of a language, language definitions often have a large grey area when it comes to floating-point. Perhaps this is due to the fact that many language designers believe that nothing can be proven about floating-point, since it entails rounding error, infinity box best range check error. If so, the previous sections have demonstrated the fallacy in this reasoning. This section discusses some common grey areas in language definitions, including suggestions about how to deal with them.

Remarkably enough, some languages don't clearly specify that if is a floating-point variable (with say a value of ), then every occurrence of (say) must have the same value. For example Ada, which is based on Brown's model, seems to imply that floating-point infinity box best range check error only has to satisfy Brown's axioms, and thus expressions can have one of many possible values. Thinking about floating-point in this fuzzy way stands in sharp contrast to the IEEE model, where the result of each floating-point operation is precisely defined. In the IEEE model, we can prove that evaluates to (Theorem 7). In Brown's model, we cannot.

Another ambiguity in most language definitions concerns what happens on overflow, underflow and other exceptions. The IEEE standard precisely specifies the behavior of exceptions, and so languages that use the standard as a model can avoid any ambiguity on this point.

Another grey area concerns the interpretation of parentheses. Due to roundoff errors, the associative laws of algebra do not necessarily hold for floating-point numbers. For example, the expression has a totally different answer than when x = 1030, y = 30 and z = 1 (it is 1 in the former case, 0 in the latter). The importance of preserving parentheses cannot be overemphasized. The algorithms presented in theorems 3, 4 and 6 all depend on it. For example, in Theorem 6, the formula infinity box best range check error = mx - (mx - x) would reduce to xh = x if it weren't for parentheses, thereby destroying the entire algorithm. A language definition that does not require parentheses to be honored is useless for floating-point calculations.

Subexpression evaluation is imprecisely defined in many languages. Suppose that is double critikal error 84, but and are single precision. Then in the expression    is the product performed in single or double precision? Another example: in where and are integers, is the division an integer operation or a floating-point one? There are two ways to deal with this problem, neither of which is completely satisfactory. The first is to require that all variables in an expression have the same type. This is the simplest solution, but has some drawbacks. First of all, languages like Pascal that have subrange types allow mixing subrange variables with integer variables, so it is somewhat bizarre to prohibit mixing single and double precision variables. Another problem concerns constants. In the expressioninfinity box best range check error, most languages interpret to be a single precision constant. Now suppose the programmer decides to change the declaration of all the floating-point variables from single to double precision. If is still treated as a single precision constant, then there will be a compile time error. The programmer will have to hunt down and change every floating-point constant, infinity box best range check error.

The second approach is to allow mixed expressions, in which case rules for subexpression evaluation must be provided. There are a number of guiding examples. The original definition of C required that every floating-point expression be computed in double precision [Kernighan and Ritchie ]. This leads to anomalies like the example at the beginning of this section. The expression is computed in double precision, but if is a single-precision variable, the quotient is rounded to infinity box best range check error precision for storage. Since 3/7 is a repeating binary fraction, infinity box best range check error, its computed value in double precision is different from its stored value in single precision. Thus the comparison q = 3/7 fails. This suggests that computing every expression in the highest precision available is not a good rule.

Another guiding example is inner products. If the inner product has thousands of terms, the rounding error in the sum can become substantial. One way to reduce this rounding error is to accumulate the sums in double precision (this will be discussed in more detail in the section Optimizers). If is a double precision variable, and and are single precision arrays, then the inner product loop will look like