From ken@turtlevax.UUCP (Ken Turkowski) Sun Nov 18 15:41:29 1984
Date: 18 Nov 84 20:41:29 GMT

> >> 	It would also be nice if sin( x ) and cos( x ) could be
> >> 	computed simultaneously with reduced cost.  I doubt if this is
> >> 	possible but would like to know if it is.
>
> > 	sine = sin(x);
> > 	cosine = sqrt(1.0 - sine*sine);
>
> I wish people would check things out before posting them.  On our
> 4.2BSD system, the sqrt() approach is actually SLOWER than using
> cos().  (Under UNIX System V it is very slightly faster.)  Now,
> if I wanted cos(x)^2, I would certainly use 1 - sin(x)^2, which
> I already knew about (I too went to high school).
>
> What I had in mind was more along the lines of a CORDIC algorithm
> or some other obscure but useful approach.

In Tom Duff's SIGGRAPH '84 tutorial on Numerical Methods for Computer
Graphics, he writes (quote until end of article):

Calculating Euclidean distance probably accounts for 90% of the square
roots expended in computer graphics applications.  Aside from the
obvious geometric uses, most illumination models require that the unit
normal to each surface be computed at each pixel.

[Moler-Morrison, "Replacing Square Roots by Pythagorean Sums", IBM
Journal of Research and Development", vol. 27, no. 6, pp. 577-581, Nov.
1983] describes an algorithm for computing sqrt(a^2+b^2) which is fast,
robust and portable.  The naive approach to this problem (just writing
sqrt(a*a+b*b) ) has the problem that for many cases when the result is
a representable floating point number, the intermediate results may
overflow or underflow.  This seriously restricts the usable range of a
and b.  Moler and Morrison's method never causes harmful overflow or
underflow when the result is in range.  The algorithm has _____cubic
convergence, and depending on implementation details may be even faster
than sqrt(a*a+b*b).  Here is a C function that implements their
algorithm:

double hypot(p,q)
double p,q;
{
    double r, s;
    if (p < 0) p = -p;
    if (q < 0) q = -q;
    if (p < q) { r = p; p = q; q = r; }
    if (p == 0) return 0;
    for (;;) {
	r = q / p;
	r *= r;
	s = r + 4;
	if (s == 4)
	    return p;
	r /= s;
	p += 2 * r * p;
	q *= r;
    }
}

This routine produces a result good to 6.5 digits after 2 iterations,
to 20 digits after 3 iterations, and to 62 digits after 4 iterations.
In normal use you would probably unwind the loop as many times as you
need and omit the test altogether.  Moler and Morrison's paper
describes the generalization of this algorithm to n dimensions, and
exhibits a related square root algorithm which also has cubic
convergence.  [Dubrelle, "A Class of Numerical Methods for the
Computation of Pythagorean Sums", IBM Journal of Research and
Development, vol. 27, no. 6, pp. 582-589, Nov. 1983] gives a geometric
justification of the algorithm and presents a set of generalizations
that have arbitrarily large order of convergence.
--
Ken Turkowski @ CADLINC, Palo Alto (soon Menlo Park), CA
UUCP: {amd,decwrl,flairvax,nsc}!turtlevax!ken
ARPA: turtlevax!ken@DECWRL.ARPA
