Address loss of precision issues in complex arcsin, arcsinh, arctan, arctanh

Address loss of precision issues in complex arcsin, arcsinh, arctan, arctanh

The complex-valued arc* functions (that -> 0 for z -> 0) have loss of
precision issues for small arguments. This patch addresses this
by switching to a series expansion in this regime.

Applied in: http://scipy.org/scipy/numpy/changeset/6520

[cournape, 2009-02-28 14:48:52]

A few nitpicks + question concerning the precision of the test

http://codereview.appspot.com/22054/diff/1/2
File numpy/core/src/umath_funcs.inc.src (right):

http://codereview.appspot.com/22054/diff/1/2#newcode102
Line 102: #ifdef SERIES_HORNER_TERM
Nitpick: I prefer the macro to include the templated type in its name

http://codereview.appspot.com/22054/diff/1/2#newcode358
Line 358: SERIES_HORNER_TERM(r, &x2, 1./6);
maybe those constants (here and below) could be defined in a predefined table ?

http://codereview.appspot.com/22054/diff/1/2#newcode454
Line 454: return;
The macro for the HORNER scheme should be undefined at this point to avoid
namespace pollution

http://codereview.appspot.com/22054/diff/1/3
File numpy/core/tests/test_umath.py (right):

http://codereview.appspot.com/22054/diff/1/3#newcode430
Line 430: x = np.logspace(-300, -1e-9, 2000)
Are the tests only for double imp ? Would be good to test for single and long
extended precision

[pav, 2009-02-28 16:22:17]

http://codereview.appspot.com/22054/diff/1/2
File numpy/core/src/umath_funcs.inc.src (right):

http://codereview.appspot.com/22054/diff/1/2#newcode102
Line 102: #ifdef SERIES_HORNER_TERM
On 2009/02/28 14:48:52, cournape wrote:
> Nitpick: I prefer the macro to include the templated type in its name

Done.

http://codereview.appspot.com/22054/diff/1/2#newcode358
Line 358: SERIES_HORNER_TERM(r, &x2, 1./6);
On 2009/02/28 14:48:52, cournape wrote:
> maybe those constants (here and below) could be defined in a predefined table
?

I'm not sure if this gains anything in clarity. And it's probably best if the
constants are kept near the point where they are used.

http://codereview.appspot.com/22054/diff/1/2#newcode454
Line 454: return;
On 2009/02/28 14:48:52, cournape wrote:
> The macro for the HORNER scheme should be undefined at this point to avoid
> namespace pollution

Done.

http://codereview.appspot.com/22054/diff/1/3
File numpy/core/tests/test_umath.py (right):

http://codereview.appspot.com/22054/diff/1/3#newcode430
Line 430: x = np.logspace(-300, -1e-9, 2000)
On 2009/02/28 14:48:52, cournape wrote:
> Are the tests only for double imp ? Would be good to test for single and long
> extended precision

Added.

[charlesr.harris, 2009-02-28 16:52:39]

Thanks for fixing these.

http://codereview.appspot.com/22054/diff/1/2
File numpy/core/src/umath_funcs.inc.src (left):

http://codereview.appspot.com/22054/diff/1/2#oldcode319
Line 319: * return nc_neg(nc_prodi(nc_log(nc_sum(nc_prod(nc_i,x),
I've started replacing these expressions with more readable versions here and
there. Maybe

returns -i*log(i*x + sqrt(1 - x**2))

http://codereview.appspot.com/22054/diff/1/2
File numpy/core/src/umath_funcs.inc.src (right):

http://codereview.appspot.com/22054/diff/1/2#newcode346
Line 346: } else {
Nitpick: Else like so

}
else {
}

I've been cleaning up the code style throughout the c files but haven't got to
the ufunc_func file yet, so this would make one less line to edit.

http://codereview.appspot.com/22054/diff/1/2#newcode347
Line 347: /* Small arguments: series expansion, to avoid loss of precision
Another nitpick, multiline comments like

/*
 * blah
 * blah
 */

http://codereview.appspot.com/22054/diff/1/2#newcode358
Line 358: SERIES_HORNER_TERM(r, &x2, 1./6);
Maybe 25./42.? You are telling the compiler make a cast, but it might be a bit
clearer to make it explicit all the way through.