[PATCH 2/6] Remove slow paths from sin/cos

[Wilco Dijkstra]

Joseph Myers wrote:
> On Mon, 12 Mar 2018, Wilco Dijkstra wrote:
>
>> Though these patches keep the ULP accuracy across the full range as is, 
>> we could agree on higher ULP errors for large/huge range reduction cases 
>> in the future. The main complexity is for certain rare inputs which happen to
>> be extremely close to an integer multiple of PI/2, and those few cases mean
>> you need significant extra work to guarantee 0.5 ULP error bound on range
>> reduction.
>
> I don't think the work for having an error bound not much more than 0.5ulp 
> on the final result of sin/cos for large arguments is significantly 
> different from the work for having an error bound of say 3ulp (the 
> testsuite has a global maximum of 9ulp (16ulp for IBM long double) beyond 
> which it will not accept errors even if those large errors are listed in 
> libm-test-ulps files - that bound is simply based on the errors 
> empirically observed at present, for functions other than Bessel functions 
> and cpow which are known to have cases with much larger errors).

Having to do another reduction step can increase latency significantly. To give
an example, I had to increase accuracy of the first-level range reduction because
a handful huge multiples of PI (6-7 digits) had 2-3ULP reduction error. The additional
accuracy means range reduction became 50% slower while being way too accurate
for almost all inputs.

Obviously it would be possible redesign it from scratch using a much smaller input
range and FMA when available etc.

Wilco