Message-ID: <364CAD3F.79419487@cyberoptics.com>
From: Eric Rudd <rudd AT cyberoptics DOT com>
Organization: CyberOptics
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Subject: Re: high precision nonlinear math functions ?
References: <4 DOT 1 DOT 19981112171650 DOT 00a26d70 AT hal DOT nt DOT tuwien DOT ac DOT at>
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Date: Fri, 13 Nov 1998 16:05:51 -0600
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Anton Helm wrote:

> I've some problems with numerical precision of some math functions.
>
> log(tan(b/2.0 + PI/4.0))

There is a lot of cancellation in this formula for small b.  I would
suggest that you rewrite your algorithm to avoid such problems.  For
instance, one could use the formula for the tangent of the sum of two
angles to rewrite the above expression as

   log1p(2.*tan(0.5*b)/(1. - tan(0.5*b))

to avoid the cancellation.

> asin(e * sin(b))

Similar problems occur if e*sin(b) is near 1.  In general, one should try
pretty hard to come up with a good computational form, before resorting to
extended precision.  It is rarely necessary.

-Eric Rudd
rudd AT cyberoptics DOT com