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The Re-layout Shapes command works best with connected drawings, such as flowcharts, network diagrams, organization charts, and tree diagrams. As well as when shapes are connected in the correct order. For example, in a top-to-bottom drawing, the connector's begin point should be connected to the top shape, and the connector's end point should be connected to the bottom shape.
If you're editing an existing document and you want to insert a multi-column layout somewhere in the middle of text you've already got typed, the "this point forward" method may result in a temporary and fixable but still infuriating mess.
There are two different IEEE standards for floating-point computation. IEEE 754 is a binary standard that requires = 2, p = 24 for single precision and p = 53 for double precision [IEEE 1987]. It also specifies the precise layout of bits in a single and double precision. IEEE 854 allows either = 2 or = 10 and unlike 754, does not specify how floating-point numbers are encoded into bits [Cody et al. 1984]. It does not require a particular value for p, but instead it specifies constraints on the allowable values of p for single and double precision. The term IEEE Standard will be used when discussing properties common to both standards.
Some compiler writers view restrictions which prohibit converting (x + y) + z to x + (y + z) as irrelevant, of interest only to programmers who use unportable tricks. 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 2n bits. Every time two n bit numbers with widely spaced exponents are added, the number of bits in the sum is n + the space between the exponents. The sum could have up to (emax - emin) + n bits, or roughly 2·emax + 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.
The increasing acceptance of the IEEE floating-point standard means that codes that utilize features of the standard are becoming ever more portable. The section The IEEE Standard, gave numerous examples illustrating how the features of the IEEE standard can be used in writing practical floating-point codes.
The preceding paper has shown that floating-point arithmetic must be implemented carefully, since programmers may depend on its properties for the correctness and accuracy of their programs. In particular, the IEEE standard requires a careful implementation, and it is possible to write useful programs that work correctly and deliver accurate results only on systems that conform to the standard. The reader might be tempted to conclude that such programs should be portable to all IEEE systems. Indeed, portable software would be easier to write if the remark "When a program is moved between two machines and both support IEEE arithmetic, then if any intermediate result differs, it must be because of software bugs, not from differences in arithmetic," were true.
Several of the examples in the preceding paper depend on some knowledge of the way floating-point arithmetic is rounded. In order to rely on examples such as these, a programmer must be able to predict how a program will be interpreted, and in particular, on an IEEE system, what the precision of the destination of each arithmetic operation may be. Alas, the loophole in the IEEE standard's definition of destination undermines the programmer's ability to know how a program will be interpreted. Consequently, several of the examples given above, when implemented as apparently portable programs in a high-level language, may not work correctly on IEEE systems that normally deliver results to destinations with a different precision than the programmer expects. Other examples may work, but proving that they work may lie beyond the average programmer's ability.
In this section, we classify existing implementations of IEEE 754 arithmetic based on the precisions of the destination formats they normally use. We then review some examples from the paper to show that delivering results in a wider precision than a program expects can cause it to compute wrong results even though it is provably correct when the expected precision is used. We also revisit one of the proofs in the paper to illustrate the intellectual effort required to cope with unexpected precision even when it doesn't invalidate our programs. These examples show that despite all that the IEEE standard prescribes, the differences it allows among different implementations can prevent us from writing portable, efficient numerical software whose behavior we can accurately predict. To develop such software, then, we must first create programming languages and environments that limit the variability the IEEE standard permits and allow programmers to express the floating-point semantics upon which their programs depend.
Some algorithms that depend on correct rounding can fail with double-rounding. In fact, even some algorithms that don't require correct rounding and work correctly on a variety of machines that don't conform to IEEE 754 can fail with double-rounding. The most useful of these are the portable algorithms for performing simulated multiple precision arithmetic mentioned in the section Exactly Rounded Operations. For example, the procedure described in Theorem 6 for splitting a floating-point number into high and low parts doesn't work correctly in double-rounding arithmetic: try to split the double precision number 252 + 3 × 226 - 1 into two parts each with at most 26 bits. When each operation is rounded correctly to double precision, the high order part is 252 + 227 and the low order part is 226 - 1, but when each operation is rounded first to extended double precision and then to double precision, the procedure produces a high order part of 252 + 228 and a low order part of -226 - 1. The latter number occupies 27 bits, so its square can't be computed exactly in double precision. Of course, it would still be possible to compute the square of this number in extended double precision, but the resulting algorithm would no longer be portable to single/double systems. Also, later steps in the multiple precision multiplication algorithm assume that all partial products have been computed in double precision. Handling a mixture of double and extended double variables correctly would make the implementation significantly more expensive.
Likewise, portable algorithms for adding multiple precision numbers represented as arrays of double precision numbers can fail in double-rounding arithmetic. These algorithms typically rely on a technique similar to Kahan's summation formula. As the informal explanation of the summation formula given on Errors In Summation suggests, if s and y are floating-point variables with |s| |y| and we compute: t = s + y; e = (s - t) + y;
Round results correctly to both the precision and range of the double format. This strict enforcement of double precision would be most useful for programs that test either numerical software or the arithmetic itself near the limits of both the range and precision of the double format. Such careful test programs tend to be difficult to write in a portable way; they become even more difficult (and error prone) when they must employ dummy subroutines and other tricks to force results to be rounded to a particular format. Thus, a programmer using an extended-based system to develop robust software that must be portable to all IEEE 754 implementations would quickly come to appreciate being able to emulate the arithmetic of single/double systems without extraordinary effort.
Of course, to find this solution, the programmer must know that double expressions may be evaluated in extended precision, that the ensuing double-rounding problem can cause the algorithm to malfunction, and that extended precision may be used instead according to Theorem 14. A more obvious solution is simply to specify that each expression be rounded correctly to double precision. On extended-based systems, this merely requires changing the rounding precision mode, but unfortunately, the C99 standard does not provide a portable way to do this. (Early drafts of the Floating-Point C Edits, the working document that specified the changes to be made to the C90 standard to support floating-point, recommended that implementations on systems with rounding precision modes provide fegetprec and fesetprec functions to get and set the rounding precision, analogous to the fegetround and fesetround functions that get and set the rounding direction. This recommendation was removed before the changes were made to the C99 standard.)
The idea that IEEE 754 prescribes precisely the result a given program must deliver is nonetheless appealing. Many programmers like to believe that they can understand the behavior of a program and prove that it will work correctly without reference to the compiler that compiles it or the computer that runs it. In many ways, supporting this belief is a worthwhile goal for the designers of computer systems and programming languages. Unfortunately, when it comes to floating-point arithmetic, the goal is virtually impossible to achieve. The authors of the IEEE standards knew that, and they didn't attempt to achieve it. As a result, despite nearly universal conformance to (most of) the IEEE 754 standard throughout the computer industry, programmers of portable software must continue to cope with unpredictable floating-point arithmetic. 2b1af7f3a8