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0 pointscperciva10y ago0 comments

Here are some actual examples:

Something to add to your list of examples: During the first Gulf war, 28 US soldiers died due to accumulated rounding errors in the Patriot Missile battery computers: https://www.ima.umn.edu/~arnold/disasters/patriot.html

(This was in fact a known issue, and operators had been instructed to reboot the computers every 8 hours. Unfortunately this instruction ignored the fact that, in the field, nobody wanted to be responsible for turning off their defensive systems for a minute.)

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notjpl10y ago

Yeah, that doesn't surprise me in the least, many high-tech military systems have MTBF/MTTF of a few hours at best. Also, that's what you get when you try to do radar time computations using 24-bit fixed point in Ada.

Back to astronomy, in many astronomy libraries (such as astropy library) computations regarding time are done using 2 doubles (about 106 bit precision). 1 double is not enough.

_brandmeyer_ also mentioned something important that I totally forgot - any trigonometric computation requires computing modulo-pi to an accuracy of 1 ulp, which requires storing PI to ~1144 bits for double precision (for numbers near pi) (see Kahan argument reduction paper).

Since Intel processsors don't reach the required precision for IEEE standard above pi/2, this modulo reduction is done in software to this day. gcc maintains a 1144 bit PI constant and does a 1144 bit modulo every time you compute a sine/cosine above pi.

TLDR - 344 decimal digits of PI are used. High-precision PI computation is surprisingly more common than we expect...

[1] http://docs.astropy.org/en/stable/time/index.html

[2] https://software.intel.com/en-us/blogs/2014/10/09/fsin-docum...

[3] https://gcc.gnu.org/gcc-4.3/changes.html

[4] http://www.csee.umbc.edu/~phatak/645/supl/Ng-ArgReduction.pd...

cpercivaOP10y ago

any trigonometric computation requires computing modulo-pi to an accuracy of 1 ulp

For the trigonometric function itself, sure. For any reasonable algorithm which uses the trigonometric function, no. If find yourself computing sin(10^6), you're not really trying to compute sin(10^6); you're trying to compute sin(x) for some value of x which you know lies between 10^6(1 - epsilon) and 10^6(1 + epsilon). So the extent to which trigonometric calculations can lose precision by not doing extra-precision argument reduction, that precision was already lost in computing the unreduced argument.

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