From: John H Meyers on
On Sun, 27 Aug 2006 19:27:21 -0500:

> All TI calculators have a 14 digits accuracy...

They may retain a 14-digit mantissa,
but that doesn't mean as many digits are accurate,
because for sines of angles near 180 degrees
(pi radians), it's usually more like saying
"many calcs are accurate to plus/minus 1E-14,"
but since the true answer is near zero,
the *relative* inaccuracy becomes much greater.

Joe Horn and others have noted that HP calcs need to know pi
to about 25 places to get the right answers
(all *significant* digits correct) to similar problems:

24 digits of PI [1997/07/29]

pi to 24 significant digits;
TI sin(x) bug HP doesn't have [2003/10/27]

Also of interest:

Question on Rads [1997/03/08]

PDQ Unleashed: No More Limitations [2004/10/07]
(contains pi to 64 decimals, for some reason :)

[r->] [OFF]
From: Veli-Pekka Nousiainen on
Fv wrote:
> Explanation:
> Since the slope of sin is 1 where it crosses the x axis,
> sin (Pi + tiny number) = tiny number
> In this case the "tiny number" is simply = 3.141592654 - Pi =
> 4.1020676153735661672049711580283060062489417902505540769218359371379100137....*10^-10
> So, of course the TI answer is correct too, since it corresponds to
> 14 sig. dig.. To justify the accuracy given by HP, both Pi and
> 3.141592654 would have to be given to about 22 digits (that would be
> something like 3.141592654000000000000 and 3.141592653589793238463). Of
> course HP
> does not in general compute to this accuracy. HP algorithm gives an
> accurate answer that is not justified by the number of digits it can
> handle in general, and this is OK in a calculator. It would be
> perfect if HP could give arbitrary accuracy numbers or machine
> accuracy results depending on a setting.

LongFloat library does this. It uses DIGITS
and it can also useinterval arithmetic, letting you to see the accuracy
The 49g+/50G are pretty fast with it - enough for my usage
(good bye laptop...)

From: GWB on
Veli-Pekka Nousiainen wrote:

> LongFloat library does this. It uses DIGITS
> and it can also useinterval arithmetic, letting you to see the accuracy
> The 49g+/50G are pretty fast with it - enough for my usage
> (good bye laptop...)

I should have used my 49G: I have LongFloat installed on it and it
really does the job nicely. I've just computed both expressions to 50



Some digits were lost when calculating the trigonometric identity but
this is still good enough for verifying the first 40 or so digits.


From: GWB on

Zeno wrote:
> In article <1156729581.785305.19530(a)>, GWB
> <gerson.w.barbosa(a)> wrote:
> > GWB wrote:
> > > Zeno wrote:
> > > > The HP cals are much more accurate when compuitng Trig functions.
> > > >
> > > > An example, is to computer the SIN of exactly 3.141592654 (NOT Pi, but
> > > > the just given rounding of it) radians....the HPs get the correct
> > > > answer to 12 significant digits, while all other brand do not even come
> > > > close in accuracy.
> > > >
> > > > The correct answer is
> > > > -4.10206761537 E-10
> > > > which only HPs give.
> > >
> > > Surely they are, but I fear the correct answer is
> > >
> > > -4,10206857034707E-10 (according to my own RPN calc written in Delphi)
> > >
> > > or
> > >
> > > -4,10206857035E-10 to 12 significant places
> > >
> > > Anyway, the HP-48/49/50 is not bad. It appears TI gives -4.102E-10
> > >
> >
> > Oops, I was wrong! Even Delphi extended type slips on this.
> >
> > Recalculating sin(3.141592654) as 3 sin(3.141592654/3) -
> > 4(sin(3.141592654/3))^3 I obtained -4.10206535406132E-10 (in Delphi).
> > This should be a more accurate result since the sine function would
> > have no problem in the pi/3 boundary. But I am not sure about this
> > result either because of rounding errors and other issues I am not
> > aware of. Could someone compute both sin(3.141592654) and the
> > trigonometric identity above in Maple to 30 places so we can see how
> > many digits match?
> >
> > Thanks,
> >
> > Gerson.
> >
> Sorry, but both your Delphi answer are wrong
> What i stated before is correct

It appears I have a buggy Pentium III 500 MHz. On it my Dephi program
returns sin(3.141592654) as -4.10206857034707E-10. On other two
computers I have at home (AMD Athlon(TM)XP 1800+ and Intel DX-4 100)
the answers are -4.10206761624916E-10, not so accurate as the HP-48GX
because of the reasons explained elsewhere in this thread but much
better than the previous result.

By the way, does anyone know a standard test for the Pentium III bug?
(according to a test I performed years ago, my processor was not
buggy). Thanks!



From: Zeno on
In article <op.te0il8f9nn735j(a)>, John H Meyers
<jhmeyers(a)nomail.invalid> wrote:


In the above mentioned thread, there is a a part that claims the HPs
get a wrong answer and the Casio gets it right. But in fact, the Hps
get it right, and the Casios do not. The text from the thread is

"Balazs Fischer wrote:

> ---===> Quoting Nicholas Bodley to A Suehiro <===---

> NB> This matter of other makes giving "nice" ("pretty?"), but
> NB> theoretically-incorrect results has me wondering: Is it possible
> NB> to "trick" these other makes so that the results they give are
> NB> more-obviously wrong?

> Try sin(1146408/364913) on a calc that beautifies its results (in rad mode)
> and you should get 0. My TI-35 does give the wrong answer while the HP has no
> problems.

Unfortunately, the Casio CFX-9800G (the color-display calculator)
gets an answer of -1.6E-12 for this, and the HP48 gets -2.0676E-13.
The real answer is -1.610740019899030939776779...E-12, so I'm afraid
Casio wins this contest hands down.

Sorry. Pick a better example next time.

> Now you can guess how I came up with this number :-).

You obtained the fraction by 'pi' ->NUM ->Q in STD mode.

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