Fixes for bugs in sinpi/cospi/tanpi

patch to fix half-cycle trigonometric functions

Paul Zimmermann, a MPFR developer, contacted me about large errors in
the half-cycle trigonometric functions. I'

Fixes for bugs in sinpi/cospi/tanpi

patch to fix half-cycle trigonometric functions

Paul Zimmermann, a MPFR developer, contacted me about large errors in
the half-cycle trigonometric functions. I've have investigated these
issues and developed the attached patch. The float, double, and ld80
(long double) changes have been tested.

Caveat emptor: The ld128 changes have not been compiled. The ld128
changes have not been tested. I do not have access to a system that
uses ld128 floating point.

Here is an itemized list of changes:

* lib/msun/src/math_private.h:
. Add fast floor macros to compute the integer part of |x| for
0 <= |x| 01xp(N-1), where N is the precision of the type of x.
These macros are used in the half-cycle trigonometric functions
(e.g., sinpi(x)).
. The FFLOOR80 macros is used with the Intel 80-bit extended double
functions. This macors corrects an off-by-one error, which led to
enormous error for |x| > 0x1p32.

* lib/msun/src/s_cospif.c:
* lib/msun/src/s_cospi.c:
* lib/msun/ld80/s_cospil.c:
. Update Copyright years.
. Use FFLOOR*() macro to get integer part of |x|.
. Correct handle the range 0x1p(N-1) <= |x| < 0x1pN. Here, one needs
to determine if the integral value of |x| is even or odd to choose
+1 or -1. If |x| >= 0x1pN, always return +1.

* lib/msun/src/s_sinpif.c:
* lib/msun/src/s_sinpi.c:
* lib/msun/ld80/s_sinpil.c:
. Update Copyright years.
. Use FFLOOR*() macro to get integer part of |x|.

* lib/msun/src/s_tanpif.c:
* lib/msun/src/s_tanpi.c:
* lib/msun/ld80/s_tanpil.c:
. Update Copyright years.
. For +-0.5, return +-inf. Previously, tanpi[fl]() returned an NaN.
. Use FFLOOR*() to get integer part of |x|. Need to determine if the
integer part is even or odd. This is used to set +-0 for |x|
integral
and +-inf for (n+1/2).
. For 0x1p(N-1) <= |x| < 0x1pN need to determine if x is an even or
odd
integer to select +0 or -0. For |x| >= 0x1pN, it is always an even
integer, select 0.
. Note, tanpi[fl](x) is an odd function, so one needs to consider
tanpi[fl](-|x|) = - tanpi[fl](|x|).

* lib/msun/ld128/s_cospil.c:
* lib/msun/ld128/s_sinpil.c:
* lib/msun/ld128/s_tanpil.c:
. Update Copyright years.
. These routines use an FFLOOR128 macros, which likely should be
replaced by a bit twiddling algorithm.
. The same considerations above are applied to 0x1p112 <= |x| <
0x1p113,
and |x| >= 0x1p113 cases.
. Note, even and odd determination used fmodl(x,2.), which is likely
slow.

PR: 272742
MFC after: 1 week

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[LIBM] implementations of sinpi[fl], cospi[fl], and tanpi[fl]

Both IEEE-754 2008 and ISO/IEC TS 18661-4 define the half-cycle
trignometric functions cospi, sinpi, and tanpi. The attached
patch impl

[LIBM] implementations of sinpi[fl], cospi[fl], and tanpi[fl]

Both IEEE-754 2008 and ISO/IEC TS 18661-4 define the half-cycle
trignometric functions cospi, sinpi, and tanpi. The attached
patch implements cospi[fl], sinpi[fl], and tanpi[fl]. Limited
testing on the cospi and sinpi reveal a max ULP less than 0.89;
while tanpi is more problematic with a max ULP less than 2.01
in the interval [0,0.5]. The algorithms used in these functions
are documented in {ks}_cospi.c, {ks}_sinpi.c, and s_tanpi.c.

Note. I no longer have access to a system with ld128 and
adequate support to compile and test the ld128 implementations
of these functions. Given the almost complete lack of input from
others on improvements to libm, I doubt that anyone cares. If
someone does care, the ld128 files contain a number of FIXME comments,
and in particular, while the polynomial coefficients are given
I did not update the polynomial algorithms to properly use the
coefficients.

PR: 218514
MFC after: 2 weeks

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