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From: Terje Mathisen <terje.mathisen@tmsw.no>
Newsgroups: comp.lang.c,comp.arch
Subject: Re: Radians Or Degrees?
Date: Fri, 15 Mar 2024 11:23:45 +0100
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Michael, I for the main part agree with you here, i.e. calculating=20
sin(x) with x larger than 2^53 or so, is almost certainly stupid.

Actually using and depending upon the result is more stupid.

OTOH, it is and have always been, a core principle of ieee754 that basic =

operations (FADD/FSUB/FMUL/FDIV/FSQRT) shall assume that the inputs are=20
exact (no fractional ulp uncertainty), and that we from that starting=20
point must deliver a correctly rounded version of the infinitely precise =

exact result of the operation.

Given the latter, it is in fact very tempting to see if that basic=20
result rule could be applied to more of the non-core operations, but I=20
cannot foresee any situation where I would use it myself: If I find=20
myself in a situation where the final fractional ulp is important, then=20
I would far rather switch to doing the operation in fp128.

Terje

Michael S wrote:
> On Fri, 23 Feb 2024 11:10:00 +0100
> Terje Mathisen <terje.mathisen@tmsw.no> wrote:
>=20
>> MitchAlsup1 wrote:
>>> Steven G. Kargl wrote:
>>>> Agreed a programmer should use what is required by the problem
>>>> that they are solving.=C2=A0 I'll note that SW implementations have
>>>> their sets of tricks (e.g., use of double-double arithmetic to
>>>> achieve double precision).
>>>
>>> To get near IEEE desired precision, one HAS TO use more than 754
>>> precision.
>>
>> There are groups who have shown that exactly rounded trancendental
>> functions are in fact achievable with maybe 3X reduced performance.
>>
>=20
> At which cost in tables sizes?
>=20
>=20
>> There is a suggestion on the table to make that a (probably optional
>> imho) feature for an upcoming ieee754 revision.
>>
>> Terje
>>
>=20
> The critical point here is definition of what considered exact. If
> 'exact' is measured only on y side of y=3Dfoo(x), disregarding
> possible imprecision on the x side then you are very likely to end up
> with results that are slower to calculate, but not at all more useful
> from point of view of engineer or physicist. Exactly like Payne-Hanek
> or Mitch's equivalent of Payne-Hanek.
>=20
> The definition of 'exact' should be:
> For any finite-precision function foo(x) lets designate the same
> mathematical function calculated with infinite precision as Foo(x).
> Let's designate an operation of rounding of infinite-precision number t=
o
> desired finite precision as Rnd(). Rounding is done in to-nearest mode.=

> Unlike in the case of basic operations, ties are allowed to be broken i=
n
> any direction.
> The result of y=3Dfoo(x) for finite-precision number x considered
> exact if *at least one* two conditions is true:
> (1=3DY-clause) Rnd(Foo(x)) =3D=3D y
> (2=3DX-clause) There exist an infinite precision number X for which
> both Foo(X) =3D=3D y and Rnd(X) =3D=3D x.
>=20
> As follows from the (2), it is possible and not uncommon that more
> than one finite-precision number y is accepted exact result of foo(x).
>=20
> If Committee omits the 2nd clause then the whole proposition will be no=
t
> just not useful, but harmful.
>=20


--=20
- <Terje.Mathisen at tmsw.no>
"almost all programming can be viewed as an exercise in caching"