[Numpy-discussion] sinpi/cospi trigonometric functions

[Tom Programming]

Hi all,

(I am very new to this mail list so please cut me some slack)

trigonometric functions like sin(x) are usually implemented as:

1. Some very complicated function that does bit twiddling and basically computes the reminder of x by pi/2. An example in http://www.netlib.org/fdlibm/e_rem_pio2.c (that calls http://www.netlib.org/fdlibm/k_rem_pio2.c ). i.e. ~500 lines of branching C code. The complexity arises in part because for big values of x the subtraction becomes more and more ill defined, due to x being represented in binary base to which an irrational number has to subtracted and consecutive floating point values being more and more apart for higher absolute values.
2. A Taylor series for the small values of x, 
3. Plus some manipulation to get the correct branch, deal with subnormal numbers, deal with -0, etc...

If we used a function like sinpi(x) = sin(pi*x) part (1) can be greatly simplified, since it becomes trivial to separate the reminder of the division by pi/2. There are gains both in the accuracy and the performance.

In large parts of the code anyways there is a pi inside the argument of sin since it is common to compute something like sin(2*pi*f*t) etc. So I wonder if it is feasible to implement those functions in numpy.

To strengthen my argument I'll note that the IEEE standard, too, defines ( https://irem.univ-reunion.fr/IMG/pdf/ieee-754-2008.pdf ) the functions sinPi, cosPi, tanPi, atanPi, atan2Pi. And there are existing implementations, for example, in Julia ( https://github.com/JuliaLang/julia/blob/6aaedecc447e3d8226d5027fb13d0c3cbfbfea2a/base/special/trig.jl#L741-L745 ) and the Boost C++ Math library ( https://www.boost.org/doc/libs/1_54_0/boost/math/special_functions/sin_pi.hpp )

And that issue caused by apparently inexact calculations have been raised in the past in various forums ( https://stackoverflow.com/questions/20903384/numpy-sinpi-returns-negative-value https://stackoverflow.com/questions/51425732/how-to-make-sinpi-and-cospi-2-zero https://www.reddit.com/r/Python/comments/2g99wa/why_does_python_not_make_sinpi_0_just_really/ ... )

PS: to be nitpicky I see that most implementation implement sinpi as sin(pi*x) for small values of x, i.e. they multiply x by pi and then use the same coefficients for the Taylor series as the canonical sin. A multiply instruction could be spared, in my opinion, by storing different Taylor expansion number coefficients tailored for the sinpi function. It is not clear to me if it is not done because the performance gain is small, because I am wrong about something, or because those 6 constants of the Taylor expansion have a "sacred aura" about them and nobody wants to enter deeply into this.

PPS: I am aware that it could be seen as rude to request a feature from an open source project but I am asking if there is a point in providing these functions in the first place. I could try to provide implementations for them in some time if it is indeed a worthwhile effort


Yours,

Tom.