Re: [PATCH, Fortran] Extension: COTAN and degree-valued trig intrinsics with -fdec-math

[Tobias Burnus <tobias dot burnus at physik dot fu-berlin dot de>]

Fritz Reese wrote:
> Attached is a patch extending the GNU Fortran front-end to support
> some additional math intrinsics, enabled with a new compile flag
> -fdec-math. The flag adds the COTAN intrinsic (cotangent), as well as
> degree versions of all trigonometric intrinsics (SIND, TAND, ACOSD,
> etc...).
>
> +	/* There is no builtin mpc_cot, so compute x = 1 / tan (x).  */
> +	val = &result->value.complex;
> +	mpc_tan (*val, *val, GFC_MPC_RND_MODE);
> +	mpc_div (*val, one, *val, GFC_MPC_RND_MODE);

The internet remarks: TAN(x) for x -> pi/2 goes to +inf or -inf - while
COTAN(pi/2) == 0. For values near pi/2, using cos(x)/sin(x) should be
numerically better than 1/tan(x). [Cf. mpfr_sin_cos  and BUILT_IN_SINCOS.]

I am not sure how big the prevision difference really is. A quick test
showed results which didn't look to bad:

implicit none
real(8), volatile  :: x
x = 3.14159265358d0/2.0d0
print '(g0)', cotan(x), cos(x)/sin(x), cotan(x)-cos(x)/sin(x)
end

yields:

0.48965888601467483E-011
0.48965888601467475E-011
0.80779356694631609E-027

Usint N[1/Tan[314159265358/10^11/2],200], Mathematica shows
  4.8966192313216916397514812338... * 10^12.
I am not quite sure whether I should expect that already the 5th digit
differs from the results above.


> mpfr_set_d (tmp, 180.0l, GFC_RND_MODE);

With some fonts the L after 180.0 looks like a one; I was initially
puzzled why one uses not 180 but 180.01. Hence, I'd prefer an "L" instead
of an "l".

However, the second argument of mpfr_set_d is a "double" and 180.0 is
already a double. Hence, one can even completely get rid of the "l".



Regarding the decimal trigonometric functions, the internet suggests to
use
   sin (fmod ((x), 360) * M_PI / 180)
instead of
   sin (x * M_PI / 180)
to yield better results for angles which are larger than +/-360 degrees.


Cheers,

Tobias