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*To*: agj@xxxxxxxxxxxx, will@xxxxxxxxxxx*Subject*: Re: numerical conditioning MAGNITUDE and /*From*: William D Clinger <will@xxxxxxxxxxx>*Date*: Thu, 22 Jun 2006 21:22:58 -0400*Cc*: srfi-77@xxxxxxxxxxxxxxxxx*Delivered-to*: srfi-77@xxxxxxxxxxxxxxxxx

Aubrey Jaffer wrote: > I got ahead of myself. (where is that strawman!) I never worry about losing a strawman, because I can always invent another one. Aubrey went on to provide an excellent example of why it is better to leave some things in the fabled "quality of implementation" category: > * that MAGNITUDE and / should work over their full ranges is too > obvious to state in a specification. > > How would such a constraint be expressed? The tangent function proved > a counterexample to general statements about the output range of > functions. But specifics seem workable: > > The procedure MAGNITUDE returns a finite real nonnegative number for > every argument whose (mathematical) magnitude is less than the > most-positive-finite-flonum in the implementation. > > The procedure / applied to z1 and z2 returns a finite number when > (/ (magnitude z1) (magnitude z2)) is less than the > most-positive-finite-flonum in the implementation. > > Both of these constraints are compatible with both polar and > rectangular representations of complex numbers. Unfortunately, both constraints are unrealistic and largely useless. They are unrealistic because, when the mathematically correct result is within 1 ulp of the largest flonum, they imply a cumulative roundoff error in the positive direction of less than 1 ulp. They are useless because the easiest ways to satisfy that absurdly small bound on the positive error involve techniques that tend to decrease rather than to increase the accuracy of these operations: change the rounding mode to round toward zero, use single precision to represent all inexact reals so double precision can be reserved for the use of library procedures that need extra precision to satisfy the constraints, and so on. Furthermore the stated constraints are concerned only with edge cases, and place no useful constraint on the accuracy of common cases. Legalistic attempts to require some precise quality of implementation often inspire an equally legalistic search for loopholes, with consequences opposite to those intended. Will

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