On Mon, Oct 15, 2001 at 03:52:06PM +0200, Kent Karlsson wrote:
> > Simon Peyton-Jones:
> > > Russell O'Connor suggests:
> > > | but sinh and cosh can easily be defined in terms of exp
> > > | sinh x = (exp(x) - exp(-x))/2
> > > | cosh x = (exp(x) + exp(-x))/2
...
> > > This looks pretty reasonable to me. We should have default methods
> > > for anything we can.
> Mathematically, yes. Numerically, no. Even if 'exp' is implemented
> with high accuracy, the suggested defaults may return a very
> inaccurate (in ulps) result. Take sinh near zero. sinh(x) with x
> very close to 0 should return x. With the above 'default' sinh(x)
> will return exactly 0 for a relatively wide interval around 0, which
> is the wrong result except for 0 itself.
Hmm, on these grounds the current default definition for tanh x is
even worse behaved:
tanh x = sinh x / cosh x
For moderately large floating point x, this will overflow.
Frankly, I don't think the whole discussion matters very much; nobody
who cares will use the default definitions. But remember to think
about branch cuts.
And why not go further?
cos x = (exp (i*x) + exp (-i*x))/2 where i = sqrt (-1)
etc.
> In general, this is why LIA-2 (Language Independent Arithmetic, part
> 2, "Elementary numerical functions", ISO/IEC 10967-2:2001) rarely
> attempts to define one numerical operation in terms of other
> numerical operations. That is done only when the relationship is
> exact (even if the operations themselves are inexact). That is not
> the case for the abovementioned operations. But it is the case for
> the relationship between the complex sin operation and the complex
> sinh operation, for instance. (Complex will be covered by LIA-3.)
This sounds like a very interesting standard. I am constantly annoyed
by ISO's attempts to hide their standards; one might wonder what the
purpose is of having unavailable "standards". Is the content
available somewhere?
Best,
Dylan Thurston
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