The IEEE standard has a number of flags and modes. As discussed above, there is one status flag for each of the five exceptions: underflow, overflow, division by zero, invalid operation and inexact. It is strongly recommended that there be an enable mode bit for each of the five exceptions. This section gives some simple examples of how these modes and flags can be put to good use. A more sophisticated example is discussed in the section Binary to Decimal Conversion.
Consider writing a subroutine to compute x n , where n is an integer. In the second expression these are exact i. Unfortunately, these is a slight snag in this strategy.
If PositivePower x, -n underflows, then either the underflow trap handler will be called, or else the underflow status flag will be set. This is incorrect, because if x - n underflows, then x n will either overflow or be in range. It simply turns off the overflow and underflow trap enable bits and saves the overflow and underflow status bits. If neither the overflow nor underflow status bit is set, it restores them together with the trap enable bits.
Another example of the use of flags occurs when computing arccos via the formula. The solution to this problem is straightforward. Simply save the value of the divide by zero flag before computing arccos, and then restore its old value after the computation.
The design of almost every aspect of a computer system requires knowledge about floating-point. Computer architectures usually have floating-point instructions, compilers must generate those floating-point instructions, and the operating system must decide what to do when exception conditions are raised for those floating-point instructions. Computer system designers rarely get guidance from numerical analysis texts, which are typically aimed at users and writers of software, not at computer designers.
As an example of how plausible design decisions can lead to unexpected behavior, consider the following BASIC program. This example will be analyzed in the next section.
Incidentally, some people think that the solution to such anomalies is never to compare floating-point numbers for equality, but instead to consider them equal if they are within some error bound E. This is hardly a cure-all because it raises as many questions as it answers. What should the value of E be? It is quite common for an algorithm to require a short burst of higher precision in order to produce accurate results.
As discussed in the section Proof of Theorem 4 , when b 2 4 ac , rounding error can contaminate up to half the digits in the roots computed with the quadratic formula. By performing the subcalculation of b 2 - 4 ac 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 b 2 - 4 ac 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 2 p 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.
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,.
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. Note that if x 1 is an exact solution, then is the zero vector, as is y. Then y x 1 - x , so an improved estimate for the solution is. The three steps 12 , 13 , and 14 can be repeated, replacing x 1 with x 2 , and x 2 with x 3. 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. 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. The interaction of compilers and floating-point is discussed in Farnum [], and much of the discussion in this section is taken from that paper.
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. 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 x is a floating-point variable with say a value of 3.
For example Ada, which is based on Brown's model, seems to imply that floating-point arithmetic 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 3. 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. The importance of preserving parentheses cannot be overemphasized. The algorithms presented in theorems 3, 4 and 6 all depend on it.
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 ds is double precision, but x and y are single precision. 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 expression 0. Now suppose the programmer decides to change the declaration of all the floating-point variables from single to double precision. The programmer will have to hunt down and change every floating-point constant. 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 3. 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 the multiplication is done in single precision, than much of the advantage of double precision accumulation is lost, because the product is truncated to single precision just before being added to a double precision variable. A rule that covers both of the previous two examples is to compute an expression in the highest precision of any variable that occurs in that expression.
However, this rule is too simplistic to cover all cases cleanly. A more sophisticated subexpression evaluation rule is as follows.
First assign each operation a tentative precision, which is the maximum of the precisions of its operands. This assignment has to be carried out from the leaves to the root of the expression tree.
Then perform a second pass from the root to the leaves. In this pass, assign to each operation the maximum of the tentative precision and the precision expected by the parent. Farnum [] presents evidence that this algorithm in not difficult to implement. The disadvantage of this rule is that the evaluation of a subexpression depends on the expression in which it is embedded.
This can have some annoying consequences. For example, suppose you are debugging a program and want to know the value of a subexpression. You cannot simply type the subexpression to the debugger and ask it to be evaluated, because the value of the subexpression in the program depends on the expression it is embedded in.
A final comment on subexpressions: since converting decimal constants to binary is an operation, the evaluation rule also affects the interpretation of decimal constants. This is especially important for constants like 0. Another potential grey area occurs when a language includes exponentiation as one of its built-in operations. Unlike the basic arithmetic operations, the value of exponentiation is not always obvious [Kahan and Coonen ].
However, One definition might be to use the method shown in section Infinity. For example, to determine the value of a b , consider non-constant analytic functions f and g with the property that f x a and g x b as x 0.
If f x g x always approaches the same limit, then this should be the value of a b. In the case of 1. However, the IEEE standard says nothing about how these features are to be accessed from a programming language. Some of the IEEE capabilities can be accessed through a library of subroutine calls. For example the IEEE standard requires that square root be exactly rounded, and the square root function is often implemented directly in hardware.
This functionality is easily accessed via a library square root routine. However, other aspects of the standard are not so easily implemented as subroutines. For example, most computer languages specify at most two floating-point types, while the IEEE standard has four different precisions although the recommended configurations are single plus single-extended or single, double, and double-extended.
Infinity provides another example. But that might make them unusable in places that require constant expressions, such as the initializer of a constant variable. A more subtle situation is manipulating the state associated with a computation, where the state consists of the rounding modes, trap enable bits, trap handlers and exception flags. One approach is to provide subroutines for reading and writing the state. In addition, a single call that can atomically set a new value and return the old value is often useful.
As the examples in the section Flags show, a very common pattern of modifying IEEE state is to change it only within the scope of a block or subroutine. Thus the burden is on the programmer to find each exit from the block, and make sure the state is restored.
Language support for setting the state precisely in the scope of a block would be very useful here. Modula-3 is one language that implements this idea for trap handlers [Nelson ]. There are a number of minor points that need to be considered when implementing the IEEE standard in a language.
Although the IEEE standard defines the basic floating-point operations to return a NaN if any operand is a NaN, this might not always be the best definition for compound operations. For example when computing the appropriate scale factor to use in plotting a graph, the maximum of a set of values must be computed.
In this case it makes sense for the max operation to simply ignore NaNs. Finally, rounding can be a problem.
The IEEE standard defines rounding very precisely, and it depends on the current value of the rounding modes. This sometimes conflicts with the definition of implicit rounding in type conversions or the explicit round function in languages. This means that programs which wish to use IEEE rounding can't use the natural language primitives, and conversely the language primitives will be inefficient to implement on the ever increasing number of IEEE machines.
Compiler texts tend to ignore the subject of floating-point. For example Aho et al. However, these two expressions do not have the same semantics on a binary machine, because 0. Although it does qualify the statement that any algebraic identity can be used when optimizing code by noting that optimizers should not violate the language definition, it leaves the impression that floating-point semantics are not very important.
This is designed to give an estimate for machine epsilon. Avoiding this kind of "optimization" is so important that it is worth presenting one more very useful algorithm that is totally ruined by it.
Many problems, such as numerical integration and the numerical solution of differential equations involve computing sums with many terms. Because each addition can potentially introduce an error as large as. A simple way to correct for this is to store the partial summand in a double precision variable and to perform each addition using double precision. If the calculation is being done in single precision, performing the sum in double precision is easy on most computer systems.
However, if the calculation is already being done in double precision, doubling the precision is not so simple. One method that is sometimes advocated is to sort the numbers and add them from smallest to largest. However, there is a much more efficient method which dramatically improves the accuracy of sums, namely. Comparing this with the error in the Kahan summation formula shows a dramatic improvement.
Each summand is perturbed by only 2 e , instead of perturbations as large as ne in the simple formula. Details are in, Errors In Summation. These examples can be summarized by saying that optimizers should be extremely cautious when applying algebraic identities that hold for the mathematical real numbers to expressions involving floating-point variables.
Another way that optimizers can change the semantics of floating-point code involves constants. In the expression 1. Because this constant cannot be represented exactly in binary, the inexact exception should be raised. In addition, the underflow flag should to be set if the expression is evaluated in single precision.
Since the constant is inexact, its exact conversion to binary depends on the current value of the IEEE rounding modes. Thus an optimizer that converts 1. However, constants like Despite these examples, there are useful optimizations that can be done on floating-point code. First of all, there are algebraic identities that are valid for floating-point numbers.
However, even these simple identities can fail on a few machines such as CDC and Cray supercomputers. Instruction scheduling and in-line procedure substitution are two other potentially useful optimizations. Perhaps they have in mind that floating-point numbers model real numbers and should obey the same laws that real numbers do.
The problem with real number semantics is that they are extremely expensive to implement. Every time two n bit numbers are multiplied, the product will have 2 n bits.
An algorithm that involves thousands of operations such as solving a linear system will soon be operating on numbers with many significant bits, and be hopelessly slow. The implementation of library functions such as sin and cos is even more difficult, because the value of these transcendental functions aren't rational numbers. Exact integer arithmetic is often provided by lisp systems and is handy for some problems. However, exact floating-point arithmetic is rarely useful.
Since these bounds hold for almost all commercial hardware, it would be foolish for numerical programmers to ignore such algorithms, and it would be irresponsible for compiler writers to destroy these algorithms by pretending that floating-point variables have real number semantics. The topics discussed up to now have primarily concerned systems implications of accuracy and precision. Trap handlers also raise some interesting systems issues. The IEEE standard strongly recommends that users be able to specify a trap handler for each of the five classes of exceptions, and the section Trap Handlers , gave some applications of user defined trap handlers.
In the case of invalid operation and division by zero exceptions, the handler should be provided with the operands, otherwise, with the exactly rounded result. Depending on the programming language being used, the trap handler might be able to access other variables in the program as well.
For all exceptions, the trap handler must be able to identify what operation was being performed and the precision of its destination. The IEEE standard assumes that operations are conceptually serial and that when an interrupt occurs, it is possible to identify the operation and its operands. On machines which have pipelining or multiple arithmetic units, when an exception occurs, it may not be enough to simply have the trap handler examine the program counter.
Hardware support for identifying exactly which operation trapped may be necessary. Another problem is illustrated by the following program fragment. Suppose the second multiply raises an exception, and the trap handler wants to use the value of a. On hardware that can do an add and multiply in parallel, an optimizer would probably move the addition operation ahead of the second multiply, so that the add can proceed in parallel with the first multiply.
It would not be reasonable for a compiler to avoid this kind of optimization, because every floating-point operation can potentially trap, and thus virtually all instruction scheduling optimizations would be eliminated. This problem can be avoided by prohibiting trap handlers from accessing any variables of the program directly. Instead, the handler can be given the operands or result as an argument. But there are still problems.
If the multiply traps, its argument z could already have been overwritten by the addition, especially since addition is usually faster than multiply. Computer systems that support the IEEE standard must provide some way to save the value of z , either in hardware or by having the compiler avoid such a situation in the first place.
Kahan has proposed using presubstitution instead of trap handlers to avoid these problems. In this method, the user specifies an exception and the value he wants to be used as the result when the exception occurs. Using presubstitution, the user would specify that when an invalid operation occurs, the value 1 should be used. Connect with D. I allow to create an account.
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