ES6 accuracy of special functions
Carl Shapiro
carl.shapiro at gmail.com
Mon Jul 28 10:49:15 PDT 2014
Dear es-discuss,
In ECMA-262, section 15.8.2
<http://www.ecma-international.org/publications/files/ECMA-ST/Ecma-262.pdf>,
the note allows implementations to choose appropriate algorithms for the
evaluation of the special functions and it is recommended but not required
to use the algorithms from fdlibm <http://netlib.org/fdlibm>.
Since this is a recommendation and not a requirement implementations
compute incorrect results for some values. This produces things where
Math.cos(Math.pow(2,120)) doesn’t even have the correct sign or basic
identities like sin(-x) = -sin(x) don’t hold for all finite values of x.
This spreadsheet
<https://docs.google.com/spreadsheets/d/1t2jrptAvaQetDIYPD8GKc90Dni2dT3FuHgKKFF-eJHw/edit?usp=sharing>
gives some results from various browsers on some selected functions.
This lack of precision makes it very difficult to port numerical
applications from C or Java to Javascript. It also forces every serious
numerical Javascript application to test against every browser and platform
for correct behaviour. This seems a major disservice to the web platform
and Javascript in particular.
Since the specification recommends using the algorithms from fdlibm, which,
I believe produces results that are accurate to < 1 ulp, why not make this
a requirement? As the spreadsheet shows, many browsers already achieve
correct results. Porting fdlibm to Javascript is not particularly difficult
provided a couple of key routines are available. (My colleague has done
this for the trig functions, except for the hairy case of the Payne-Hanek
pi reduction routine.)
Note also that Java requires that many special function be accurate to < 1
ulp. (See http://docs.oracle.com/javase/8/docs/api/java/lang/Math.html)
Specifying a similar requirement for Javascript should not be too onerous
on existing implementations. Java is an existence proof that these
requirements can work.
While having an accuracy requirement is good in itself, it’s also important
that the functions are semi-monotonic to match the mathematical functions.
This is also a requirement in Java. It is known that applications using
divided differences behave incorrectly when functions are not monotonic
when they should be.
Kindly,
Carl
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