Re: musl mathematical functions

Re: musl mathematical functions

[Szabolcs Nagy]

On 08/01/2020 13:29, paul zimmermann wrote:
>        Dear Szabolcs,
> 
> my colleague Jens Gustedt told me that you lead the development of mathematical
> functions in musl.
> 
> I just tried our mpcheck tool (https://gforge.inria.fr/projects/mpcheck) which
> checks the accuracy of mathematical functions, by comparing them to MPFR (which
> is supposed to give correct rounding).

thanks!

CCing the musl list as it should be discussed there.

> 
> For the GNU libc here is what I get for example for double precision
> (with 10000 random inputs per function):
> 
> zimmerma@tomate:~/svn/mpcheck$ ./mpcheck-double --seed=588493
> GCC: 9.2.1 20200104
> GNU libc version: 2.29
> GNU libc release: stable
> MPFR 3.1.6
> ...
> Max. errors : 3.59 (nearest), 5.80 (directed) [seed=588493]
> Incorrect roundings: 483084 (basic 0)
> Wrong side of directed rounding: 245029
> Wrong monotonicity: 31701
> Wrong errno: 992 (suppressed 992)
> Wrong inexact: 11 (suppressed 11)
> Wrong underflow: 42 (suppressed 42)
> 
> This means (among other things) that the maximal error found on those random
> inputs is 3.59 ulps for rounding to nearest, and 5.80 ulps for directed
> rounding.
> 
> With musl (revision 70d8060) I get:
> 
> zimmerma@tomate:~/svn/mpcheck$ ./mpcheck-double --seed=588493
> GCC: 9.2.1 20200104
> MPFR 3.1.6
> ...
> Max. errors : 5.30 (nearest), 1.44e19 (directed) [seed=588493]
> Incorrect roundings: 407422 (basic 0)
> Wrong side of directed rounding: 130496
> Wrong errno: 131411 (suppressed 10901)
> Wrong inexact: 125 (suppressed 125)
> Wrong overflow: 16 (suppressed 0)
> Wrong underflow: 178 (suppressed 108)
> 
> We get a slightly larger maximal error for rounding to nearest (5.30 instead
> of 3.59 for the GNU libc) but a huge maximal error for directed rounding.
> 
> The 1.44e19 error is obtained for the "sin" function, with input
> x=4.2725660088821189e2 and rounding upwards.

yes, this is a known issue (the math tests i use with
musl finds this, but it's suppressed for now
https://repo.or.cz/w/libc-test.git
https://github.com/ARM-software/optimized-routines
)

these issues come from fdlibm via freebsd which
does not support non-nearest rounding in the trig
arg reduction code (and possibly in other places).
http://git.musl-libc.org/cgit/musl/tree/src/math/__rem_pio2.c#n120
(note the comment: assume round-to-nearest)

i haven't fixed this because i don't have a good
solution: the key broken part is something like

  y = round(x/p)
  z -= y*p
  /* i.e. z = x mod p, and z in [-p/2,p/2] */
  return poly(z)

the problem is that the fast and portable way to
do round relies on the current rounding mode and
z can end up in the range [-p,p] with directed
rounding, but the poly approx only works on
[-p/2,p/2].

some targets have single instruction round that's
independent of the rounding mode, but most targets
don't.

changing fenv is slower than just calling round or
rint, and doing an external call is already too
expensive.

one can do tricks such that rounding mode has
less effect on arg reduction, e.g. add

  if (z > p/2 || z < -p/2) /* do something */

or if branches are too expensive, instead of

  Shift = 0x1.8p52
  y = x/p + Shift - Shift

implement round as e.g.

 Shift = 0x1800000000.8p0
 t = x/p + Shift
 tbits = representation_as_uint64(t)
 y = (double)(int32_t)(tbits >> 16)

then z is in [-p/2 - p/2^-16, p/2 + p/2^16]
in all rounding modes and the polynomial can
be made to work on that interval.

the downside is that these tricks make the
code slower and more importantly all such
tricks break symmetry: x and -x can have
different arg reduction result.

now i lean towards fixing it in a way that's
least expensive in the nearest-rounding case
(at least for |x| < 100, beyond that performance
does not matter much) and only care about
symmetry in nearest rounding mode, this should
be doable by adding a few ifs in the critical
path that never trigger with nearest rounding.

but other ideas are welcome.

thanks.

> 
> Indeed with the following program:
> 
> #include <stdio.h>
> #include <stdlib.h>
> #include <math.h>
> #include <fenv.h>
> 
> int
> main (int argc, char *argv[])
> {
>   double x = atof (argv[1]), y;
>   fesetround (FE_UPWARD);
>   y = sin (x);
>   printf ("sin(%.16e) = %.16e\n", x, y);
> }
> 
> I get with the GNU libc:
> 
> $ ./a.out 4.2725660088821189e2
> sin(4.2725660088821190e+02) = 1.1766512962000004e-14
> 
> and with musl:
> 
> $ ./a.out 4.2725660088821189e2
> sin(4.2725660088821190e+02) = -2.2563645396544984e-11
> 
> which is indeed very far from the correctly rounded result.
> 
> Best regards,
> Paul Zimmermann
> 
> 
> 

Re: Re: musl mathematical functions

[Rich Felker]

On Wed, Jan 08, 2020 at 03:28:54PM +0000, Szabolcs Nagy wrote:
> On 08/01/2020 13:29, paul zimmermann wrote:
> >        Dear Szabolcs,
> > 
> > my colleague Jens Gustedt told me that you lead the development of mathematical
> > functions in musl.
> > 
> > I just tried our mpcheck tool (https://gforge.inria.fr/projects/mpcheck) which
> > checks the accuracy of mathematical functions, by comparing them to MPFR (which
> > is supposed to give correct rounding).
> 
> thanks!
> 
> CCing the musl list as it should be discussed there.
> 
> > 
> > For the GNU libc here is what I get for example for double precision
> > (with 10000 random inputs per function):
> > 
> > zimmerma@tomate:~/svn/mpcheck$ ./mpcheck-double --seed=588493
> > GCC: 9.2.1 20200104
> > GNU libc version: 2.29
> > GNU libc release: stable
> > MPFR 3.1.6
> > ...
> > Max. errors : 3.59 (nearest), 5.80 (directed) [seed=588493]
> > Incorrect roundings: 483084 (basic 0)
> > Wrong side of directed rounding: 245029
> > Wrong monotonicity: 31701
> > Wrong errno: 992 (suppressed 992)
> > Wrong inexact: 11 (suppressed 11)
> > Wrong underflow: 42 (suppressed 42)
> > 
> > This means (among other things) that the maximal error found on those random
> > inputs is 3.59 ulps for rounding to nearest, and 5.80 ulps for directed
> > rounding.
> > 
> > With musl (revision 70d8060) I get:
> > 
> > zimmerma@tomate:~/svn/mpcheck$ ./mpcheck-double --seed=588493
> > GCC: 9.2.1 20200104
> > MPFR 3.1.6
> > ...
> > Max. errors : 5.30 (nearest), 1.44e19 (directed) [seed=588493]
> > Incorrect roundings: 407422 (basic 0)
> > Wrong side of directed rounding: 130496
> > Wrong errno: 131411 (suppressed 10901)
> > Wrong inexact: 125 (suppressed 125)
> > Wrong overflow: 16 (suppressed 0)
> > Wrong underflow: 178 (suppressed 108)
> > 
> > We get a slightly larger maximal error for rounding to nearest (5.30 instead
> > of 3.59 for the GNU libc) but a huge maximal error for directed rounding.
> > 
> > The 1.44e19 error is obtained for the "sin" function, with input
> > x=4.2725660088821189e2 and rounding upwards.
> 
> yes, this is a known issue (the math tests i use with
> musl finds this, but it's suppressed for now
> https://repo.or.cz/w/libc-test.git
> https://github.com/ARM-software/optimized-routines
> )
> 
> these issues come from fdlibm via freebsd which
> does not support non-nearest rounding in the trig
> arg reduction code (and possibly in other places).
> http://git.musl-libc.org/cgit/musl/tree/src/math/__rem_pio2.c#n120
> (note the comment: assume round-to-nearest)
> 
> i haven't fixed this because i don't have a good
> solution: the key broken part is something like
> 
>   y = round(x/p)
>   z -= y*p
>   /* i.e. z = x mod p, and z in [-p/2,p/2] */
>   return poly(z)
> 
> the problem is that the fast and portable way to
> do round relies on the current rounding mode and
> z can end up in the range [-p,p] with directed
> rounding, but the poly approx only works on
> [-p/2,p/2].
> 
> some targets have single instruction round that's
> independent of the rounding mode, but most targets
> don't.
> 
> changing fenv is slower than just calling round or
> rint, and doing an external call is already too
> expensive.
> 
> one can do tricks such that rounding mode has
> less effect on arg reduction, e.g. add
> 
>   if (z > p/2 || z < -p/2) /* do something */
> 
> or if branches are too expensive, instead of
> 
> [...]
> 
> now i lean towards fixing it in a way that's
> least expensive in the nearest-rounding case
> (at least for |x| < 100, beyond that performance
> does not matter much) and only care about
> symmetry in nearest rounding mode, this should
> be doable by adding a few ifs in the critical
> path that never trigger with nearest rounding.
> 
> but other ideas are welcome.

I'm in favor of a fix that's an entirely predictable branch on default
rounding mode; on most modern cpus it should have zero measurable
performance cost. If you're concerned it might matter, we can look at
some measurements.

Rich

Re: musl mathematical functions

[paul zimmermann]

       Dear Szabolcs,

thank you for your answer.

I understand the issues of slowing down the code and/or breaking symmetry,
but in my opinion the ordering should be:

   accuracy >> symmetry >> speed

where "x >> y" means that "x is more important than y".

Maybe you can find some tricks in the "Handbook of Floating-Point Arithmetic"?

Note that our mpcheck tool can also check for symmetry.

Anyway, if you do some changes, I'll be happy to run mpcheck again and send
you the new results.

Best regards,
Paul

> From: Szabolcs Nagy <Szabolcs.Nagy@arm.com>
> CC: nd <nd@arm.com>, "jens.gustedt@inria.fr" <jens.gustedt@inria.fr>,
> 	"Vincent.Lefevre@ens-lyon.fr" <Vincent.Lefevre@ens-lyon.fr>,
> 	"musl@lists.openwall.com" <musl@lists.openwall.com>
> Thread-Topic: musl mathematical functions
> Thread-Index: AQHVxieg5o3AZI5d3UWouuHlhctYy6fg5FoA
> Date: Wed, 8 Jan 2020 15:28:54 +0000
> user-agent: Mozilla/5.0 (X11; Linux aarch64; rv:60.0) Gecko/20100101
>  Thunderbird/60.9.0
> nodisclaimer: True
> Original-Authentication-Results: spf=none (sender IP is )
>  smtp.mailfrom=Szabolcs.Nagy@arm.com; 
> 
> On 08/01/2020 13:29, paul zimmermann wrote:
> >        Dear Szabolcs,
> > 
> > my colleague Jens Gustedt told me that you lead the development of mathematical
> > functions in musl.
> > 
> > I just tried our mpcheck tool (https://gforge.inria.fr/projects/mpcheck) which
> > checks the accuracy of mathematical functions, by comparing them to MPFR (which
> > is supposed to give correct rounding).
> 
> thanks!
> 
> CCing the musl list as it should be discussed there.
> 
> > 
> > For the GNU libc here is what I get for example for double precision
> > (with 10000 random inputs per function):
> > 
> > zimmerma@tomate:~/svn/mpcheck$ ./mpcheck-double --seed=588493
> > GCC: 9.2.1 20200104
> > GNU libc version: 2.29
> > GNU libc release: stable
> > MPFR 3.1.6
> > ...
> > Max. errors : 3.59 (nearest), 5.80 (directed) [seed=588493]
> > Incorrect roundings: 483084 (basic 0)
> > Wrong side of directed rounding: 245029
> > Wrong monotonicity: 31701
> > Wrong errno: 992 (suppressed 992)
> > Wrong inexact: 11 (suppressed 11)
> > Wrong underflow: 42 (suppressed 42)
> > 
> > This means (among other things) that the maximal error found on those random
> > inputs is 3.59 ulps for rounding to nearest, and 5.80 ulps for directed
> > rounding.
> > 
> > With musl (revision 70d8060) I get:
> > 
> > zimmerma@tomate:~/svn/mpcheck$ ./mpcheck-double --seed=588493
> > GCC: 9.2.1 20200104
> > MPFR 3.1.6
> > ...
> > Max. errors : 5.30 (nearest), 1.44e19 (directed) [seed=588493]
> > Incorrect roundings: 407422 (basic 0)
> > Wrong side of directed rounding: 130496
> > Wrong errno: 131411 (suppressed 10901)
> > Wrong inexact: 125 (suppressed 125)
> > Wrong overflow: 16 (suppressed 0)
> > Wrong underflow: 178 (suppressed 108)
> > 
> > We get a slightly larger maximal error for rounding to nearest (5.30 instead
> > of 3.59 for the GNU libc) but a huge maximal error for directed rounding.
> > 
> > The 1.44e19 error is obtained for the "sin" function, with input
> > x=4.2725660088821189e2 and rounding upwards.
> 
> yes, this is a known issue (the math tests i use with
> musl finds this, but it's suppressed for now
> https://repo.or.cz/w/libc-test.git
> https://github.com/ARM-software/optimized-routines
> )
> 
> these issues come from fdlibm via freebsd which
> does not support non-nearest rounding in the trig
> arg reduction code (and possibly in other places).
> http://git.musl-libc.org/cgit/musl/tree/src/math/__rem_pio2.c#n120
> (note the comment: assume round-to-nearest)
> 
> i haven't fixed this because i don't have a good
> solution: the key broken part is something like
> 
>   y = round(x/p)
>   z -= y*p
>   /* i.e. z = x mod p, and z in [-p/2,p/2] */
>   return poly(z)
> 
> the problem is that the fast and portable way to
> do round relies on the current rounding mode and
> z can end up in the range [-p,p] with directed
> rounding, but the poly approx only works on
> [-p/2,p/2].
> 
> some targets have single instruction round that's
> independent of the rounding mode, but most targets
> don't.
> 
> changing fenv is slower than just calling round or
> rint, and doing an external call is already too
> expensive.
> 
> one can do tricks such that rounding mode has
> less effect on arg reduction, e.g. add
> 
>   if (z > p/2 || z < -p/2) /* do something */
> 
> or if branches are too expensive, instead of
> 
>   Shift = 0x1.8p52
>   y = x/p + Shift - Shift
> 
> implement round as e.g.
> 
>  Shift = 0x1800000000.8p0
>  t = x/p + Shift
>  tbits = representation_as_uint64(t)
>  y = (double)(int32_t)(tbits >> 16)
> 
> then z is in [-p/2 - p/2^-16, p/2 + p/2^16]
> in all rounding modes and the polynomial can
> be made to work on that interval.
> 
> the downside is that these tricks make the
> code slower and more importantly all such
> tricks break symmetry: x and -x can have
> different arg reduction result.
> 
> now i lean towards fixing it in a way that's
> least expensive in the nearest-rounding case
> (at least for |x| < 100, beyond that performance
> does not matter much) and only care about
> symmetry in nearest rounding mode, this should
> be doable by adding a few ifs in the critical
> path that never trigger with nearest rounding.
> 
> but other ideas are welcome.
> 
> thanks.
> 
> > 
> > Indeed with the following program:
> > 
> > #include <stdio.h>
> > #include <stdlib.h>
> > #include <math.h>
> > #include <fenv.h>
> > 
> > int
> > main (int argc, char *argv[])
> > {
> >   double x = atof (argv[1]), y;
> >   fesetround (FE_UPWARD);
> >   y = sin (x);
> >   printf ("sin(%.16e) = %.16e\n", x, y);
> > }
> > 
> > I get with the GNU libc:
> > 
> > $ ./a.out 4.2725660088821189e2
> > sin(4.2725660088821190e+02) = 1.1766512962000004e-14
> > 
> > and with musl:
> > 
> > $ ./a.out 4.2725660088821189e2
> > sin(4.2725660088821190e+02) = -2.2563645396544984e-11
> > 
> > which is indeed very far from the correctly rounded result.
> > 
> > Best regards,
> > Paul Zimmermann
> > 
> > 
> > 
> 

Re: Re: musl mathematical functions

[Szabolcs Nagy]

* paul zimmermann <Paul.Zimmermann@inria.fr> [2020-01-10 17:01:43 +0100]:
>        Dear Szabolcs,
> 
> thank you for your answer.
> 
> I understand the issues of slowing down the code and/or breaking symmetry,
> but in my opinion the ordering should be:
> 
>    accuracy >> symmetry >> speed
> 
> where "x >> y" means that "x is more important than y".

what do you think about directed rounding mode behaviour?

i think libm functions are extremely rarely used with
non-nearest rounding mode so i think

  NR accuracy >> DR accuracy >> NR symmetry >> NR speed
  >> DR symmetry >> DR speed

where NR is nearest rounding and DR is directed rounding.

and by accuracy i just mean correct behavirour with respect
to exceptions and results (i.e. small ulp errors).

> 
> Maybe you can find some tricks in the "Handbook of Floating-Point Arithmetic"?
> 
> Note that our mpcheck tool can also check for symmetry.
> 
> Anyway, if you do some changes, I'll be happy to run mpcheck again and send
> you the new results.

thanks.

> 
> Best regards,
> Paul
> 
> > From: Szabolcs Nagy <Szabolcs.Nagy@arm.com>
> > CC: nd <nd@arm.com>, "jens.gustedt@inria.fr" <jens.gustedt@inria.fr>,
> > 	"Vincent.Lefevre@ens-lyon.fr" <Vincent.Lefevre@ens-lyon.fr>,
> > 	"musl@lists.openwall.com" <musl@lists.openwall.com>
> > Thread-Topic: musl mathematical functions
> > Thread-Index: AQHVxieg5o3AZI5d3UWouuHlhctYy6fg5FoA
> > Date: Wed, 8 Jan 2020 15:28:54 +0000
> > user-agent: Mozilla/5.0 (X11; Linux aarch64; rv:60.0) Gecko/20100101
> >  Thunderbird/60.9.0
> > nodisclaimer: True
> > Original-Authentication-Results: spf=none (sender IP is )
> >  smtp.mailfrom=Szabolcs.Nagy@arm.com; 
> > 
> > On 08/01/2020 13:29, paul zimmermann wrote:
> > >        Dear Szabolcs,
> > > 
> > > my colleague Jens Gustedt told me that you lead the development of mathematical
> > > functions in musl.
> > > 
> > > I just tried our mpcheck tool (https://gforge.inria.fr/projects/mpcheck) which
> > > checks the accuracy of mathematical functions, by comparing them to MPFR (which
> > > is supposed to give correct rounding).
> > 
> > thanks!
> > 
> > CCing the musl list as it should be discussed there.
> > 
> > > 
> > > For the GNU libc here is what I get for example for double precision
> > > (with 10000 random inputs per function):
> > > 
> > > zimmerma@tomate:~/svn/mpcheck$ ./mpcheck-double --seed=588493
> > > GCC: 9.2.1 20200104
> > > GNU libc version: 2.29
> > > GNU libc release: stable
> > > MPFR 3.1.6
> > > ...
> > > Max. errors : 3.59 (nearest), 5.80 (directed) [seed=588493]
> > > Incorrect roundings: 483084 (basic 0)
> > > Wrong side of directed rounding: 245029
> > > Wrong monotonicity: 31701
> > > Wrong errno: 992 (suppressed 992)
> > > Wrong inexact: 11 (suppressed 11)
> > > Wrong underflow: 42 (suppressed 42)
> > > 
> > > This means (among other things) that the maximal error found on those random
> > > inputs is 3.59 ulps for rounding to nearest, and 5.80 ulps for directed
> > > rounding.
> > > 
> > > With musl (revision 70d8060) I get:
> > > 
> > > zimmerma@tomate:~/svn/mpcheck$ ./mpcheck-double --seed=588493
> > > GCC: 9.2.1 20200104
> > > MPFR 3.1.6
> > > ...
> > > Max. errors : 5.30 (nearest), 1.44e19 (directed) [seed=588493]
> > > Incorrect roundings: 407422 (basic 0)
> > > Wrong side of directed rounding: 130496
> > > Wrong errno: 131411 (suppressed 10901)
> > > Wrong inexact: 125 (suppressed 125)
> > > Wrong overflow: 16 (suppressed 0)
> > > Wrong underflow: 178 (suppressed 108)
> > > 
> > > We get a slightly larger maximal error for rounding to nearest (5.30 instead
> > > of 3.59 for the GNU libc) but a huge maximal error for directed rounding.
> > > 
> > > The 1.44e19 error is obtained for the "sin" function, with input
> > > x=4.2725660088821189e2 and rounding upwards.
> > 
> > yes, this is a known issue (the math tests i use with
> > musl finds this, but it's suppressed for now
> > https://repo.or.cz/w/libc-test.git
> > https://github.com/ARM-software/optimized-routines
> > )
> > 
> > these issues come from fdlibm via freebsd which
> > does not support non-nearest rounding in the trig
> > arg reduction code (and possibly in other places).
> > http://git.musl-libc.org/cgit/musl/tree/src/math/__rem_pio2.c#n120
> > (note the comment: assume round-to-nearest)
> > 
> > i haven't fixed this because i don't have a good
> > solution: the key broken part is something like
> > 
> >   y = round(x/p)
> >   z -= y*p
> >   /* i.e. z = x mod p, and z in [-p/2,p/2] */
> >   return poly(z)
> > 
> > the problem is that the fast and portable way to
> > do round relies on the current rounding mode and
> > z can end up in the range [-p,p] with directed
> > rounding, but the poly approx only works on
> > [-p/2,p/2].
> > 
> > some targets have single instruction round that's
> > independent of the rounding mode, but most targets
> > don't.
> > 
> > changing fenv is slower than just calling round or
> > rint, and doing an external call is already too
> > expensive.
> > 
> > one can do tricks such that rounding mode has
> > less effect on arg reduction, e.g. add
> > 
> >   if (z > p/2 || z < -p/2) /* do something */
> > 
> > or if branches are too expensive, instead of
> > 
> >   Shift = 0x1.8p52
> >   y = x/p + Shift - Shift
> > 
> > implement round as e.g.
> > 
> >  Shift = 0x1800000000.8p0
> >  t = x/p + Shift
> >  tbits = representation_as_uint64(t)
> >  y = (double)(int32_t)(tbits >> 16)
> > 
> > then z is in [-p/2 - p/2^-16, p/2 + p/2^16]
> > in all rounding modes and the polynomial can
> > be made to work on that interval.
> > 
> > the downside is that these tricks make the
> > code slower and more importantly all such
> > tricks break symmetry: x and -x can have
> > different arg reduction result.
> > 
> > now i lean towards fixing it in a way that's
> > least expensive in the nearest-rounding case
> > (at least for |x| < 100, beyond that performance
> > does not matter much) and only care about
> > symmetry in nearest rounding mode, this should
> > be doable by adding a few ifs in the critical
> > path that never trigger with nearest rounding.
> > 
> > but other ideas are welcome.
> > 
> > thanks.
> > 
> > > 
> > > Indeed with the following program:
> > > 
> > > #include <stdio.h>
> > > #include <stdlib.h>
> > > #include <math.h>
> > > #include <fenv.h>
> > > 
> > > int
> > > main (int argc, char *argv[])
> > > {
> > >   double x = atof (argv[1]), y;
> > >   fesetround (FE_UPWARD);
> > >   y = sin (x);
> > >   printf ("sin(%.16e) = %.16e\n", x, y);
> > > }
> > > 
> > > I get with the GNU libc:
> > > 
> > > $ ./a.out 4.2725660088821189e2
> > > sin(4.2725660088821190e+02) = 1.1766512962000004e-14
> > > 
> > > and with musl:
> > > 
> > > $ ./a.out 4.2725660088821189e2
> > > sin(4.2725660088821190e+02) = -2.2563645396544984e-11
> > > 
> > > which is indeed very far from the correctly rounded result.
> > > 
> > > Best regards,
> > > Paul Zimmermann
> > > 
> > > 
> > > 
> > 

Re: Re: musl mathematical functions

[paul zimmermann]

       Dear Szabolcs,

> i think libm functions are extremely rarely used with
> non-nearest rounding mode so i think
> 
>   NR accuracy >> DR accuracy >> NR symmetry >> NR speed
>   >> DR symmetry >> DR speed
> 
> where NR is nearest rounding and DR is directed rounding.

yes this makes sense.

> and by accuracy i just mean correct behavirour with respect
> to exceptions and results (i.e. small ulp errors).

note that if directed rounding is used to implement interval
arithmetic, it is very important to have the return value on
the right side with respect to the exact value (at the cost
of a few ulps of accuracy).

Paul

Re: [musl] Re: musl mathematical functions

[Szabolcs Nagy]

* paul zimmermann <Paul.Zimmermann@inria.fr> [2020-01-10 19:35:08 +0100]:
> > i think libm functions are extremely rarely used with
> > non-nearest rounding mode so i think
> > 
> >   NR accuracy >> DR accuracy >> NR symmetry >> NR speed
> >   >> DR symmetry >> DR speed
> > 
> > where NR is nearest rounding and DR is directed rounding.
> 
> yes this makes sense.
> 
> > and by accuracy i just mean correct behavirour with respect
> > to exceptions and results (i.e. small ulp errors).
> 
> note that if directed rounding is used to implement interval
> arithmetic, it is very important to have the return value on
> the right side with respect to the exact value (at the cost
> of a few ulps of accuracy).

getting on the right side would regress the performance
for all users for something theoretical (existing math
libraries don't support it). at least i think it would
require accessing fenv (changing the rounding mode) or
other expensive operation in the hot code path.
(expensive rounding mode change is one of the reasons
interval arithmetics is not practical, lack of compiler
support for fenv access is another, the instruction set
and language semantics should be designed differently
for it to be practical.)

i'd recommend using an arbitrary precision library or a
correctly rounded math library (e.g. tr 18661-4 reserves
separate cr prefixed symbols for that), not libc functions
for interval arithmetic.

large ulp errors can usually be fixed without significant
penalty for nearest rounding. e.g. i'm considering a fix
for trig arg reduction with two additional branches (i
think this can be simplified with more code changes and
the cost can be eliminated on targets with rounding mode
independent rounding instructions)

diff --git a/src/math/__rem_pio2.c b/src/math/__rem_pio2.c
index d403f81c..80fd72c8 100644
--- a/src/math/__rem_pio2.c
+++ b/src/math/__rem_pio2.c
@@ -36,6 +36,7 @@
  */
 static const double
 toint   = 1.5/EPS,
+pio4    = 0x1.921fb54442d18p-1,
 invpio2 = 6.36619772367581382433e-01, /* 0x3FE45F30, 0x6DC9C883 */
 pio2_1  = 1.57079632673412561417e+00, /* 0x3FF921FB, 0x54400000 */
 pio2_1t = 6.07710050650619224932e-11, /* 0x3DD0B461, 0x1A626331 */
@@ -122,6 +123,17 @@ medium:
 		n = (int32_t)fn;
 		r = x - fn*pio2_1;
 		w = fn*pio2_1t;  /* 1st round, good to 85 bits */
+		if (predict_false(r - w < -pio4)) {
+			n--;
+			fn--;
+			r = x - fn*pio2_1;
+			w = fn*pio2_1t;
+		} else if (predict_false(r - w > pio4)) {
+			n++;
+			fn++;
+			r = x - fn*pio2_1;
+			w = fn*pio2_1t;
+		}
 		y[0] = r - w;
 		u.f = y[0];
 		ey = u.i>>52 & 0x7ff;