Date: Sun, 16 Sep 2012 16:00:05 -0500 From: Stephen Montgomery-Smith <stephen@missouri.edu> To: Bruce Evans <brde@optusnet.com.au> Cc: freebsd-numerics@freebsd.org Subject: Re: Complex arg-trig functions Message-ID: <50563DD5.4060303@missouri.edu> In-Reply-To: <20120917041848.F3504@besplex.bde.org> References: <5017111E.6060003@missouri.edu> <20120814201105.T934@besplex.bde.org> <502A780B.2010106@missouri.edu> <20120815223631.N1751@besplex.bde.org> <502C0CF8.8040003@missouri.edu> <20120906221028.O1542@besplex.bde.org> <5048D00B.8010401@missouri.edu> <504D3CCD.2050006@missouri.edu> <504FF726.9060001@missouri.edu> <20120912191556.F1078@besplex.bde.org> <20120912225847.J1771@besplex.bde.org> <50511B40.3070009@missouri.edu> <20120913204808.T1964@besplex.bde.org> <5051F59C.6000603@missouri.edu> <20120914014208.I2862@besplex.bde.org> <50526050.2070303@missouri.edu> <20120914212403.H1983@besplex.bde.org> <50538E28.6050400@missouri.edu> <20120915231032.C2669@besplex.bde.org> <50548E15.3010405@missouri.edu> <5054C027.2040008@missouri.edu> <5054C200.7090307@missouri.edu> <20120916041132.D6344@besplex.bde.org> <50553424.2080902@missouri.edu> <20120916134730.Y957@besplex.bde.org> <5055ECA8.2080008@missouri.edu> <20120917022614.R2943@besplex.bde.org> <20120917041848.F3504@besplex.b! de.org>
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On 09/16/2012 02:53 PM, Bruce Evans wrote:
> On Mon, 17 Sep 2012, Bruce Evans wrote:
>
>> On Sun, 16 Sep 2012, Stephen Montgomery-Smith wrote:
>>
>>> On 09/16/2012 12:14 AM, Bruce Evans wrote:
>>>> On Sat, 15 Sep 2012, Stephen Montgomery-Smith wrote:
>>>>
>>>>> One more thing I would like an opinion on.
>>>>>
>>>>> In my code I check for |z| being small, and then use the
>>>>> approximations:
>>>>> casinh(z) = z
>>>>> cacos(z) = Pi - z
>>>>
>>>> Actually Pi/2 - z.
>>>>
>>>>> catanh(z) = z
>>> So all things being said and done, I am going to remove the use of
>>> these approximations.
>
> It gives the expected pessimizations, and unexpected accuracy improvements
> and unimprovements. On amd64:
I got unexpected accuracy improvements as well! I thought it might just
be a coincidence, so I ignored it.
> @ @@ -313,5 +323,6 @@
> @ return (cpackf(copysignf(0, x), y+y));
> @ if (isinf(y))
> @ - return (cpackf(copysignf(0, x), copysignf(m_pi_2, y)));
> @ + return (cpackf(copysignf(0, x),
> @ + copysignf(m_pi_2 + tiny, y)));
> @ if (x == 0)
> @ return (cpackf(x, y+y));
> @ @@ -320,9 +331,16 @@
> @ @ if (isinf(x) || isinf(y))
> @ - return (cpackf(copysignf(0, x), copysignf(m_pi_2, y)));
> @ + return (cpackf(copysignf(0, x), copysignf(m_pi_2 + tiny, y)));
> @ @ if (ax > RECIP_EPSILON || ay > RECIP_EPSILON)
> @ if ((int)(1+tiny)==1)
> @ - return (cpackf(copysignf(real_part_reciprocal(ax, ay),
> x), copysignf(m_pi_2, y)));
> @ + return (cpackf(
> @ + copysignf(real_part_reciprocal(ax, ay), x),
> @ + copysignf(m_pi_2 + tiny, y)));
> @ +
> @ + /* XXX the numbers are related to sqrt(6 * FLT_EPSILON). */
> @ + if (ax < 2048 * FLT_EPSILON && ay < 2048 * FLT_EPSILON)
> @ + if ((int)ax==0 && (int)ay==0)
> @ + return (z);
> @ @ if (ax == 1 && ay < FLT_EPSILON) {
I implemented all the m_pi_2 + tiny changes.
Let me still ponder the |z| being small issue. Or you can put that code
back in when it is committed.
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