Is super-accuracy matters?

[Albert Chan]

(10-07-2023 03:22 AM)Albert Chan Wrote:  HP50g, approx(pi - 355/113)     = -.00000026676

We want accurate sin(x), even for "unrealistic" x, because it may be used in "other" ways.
Example, if we have accurate sin(pi ≈ \(\pi\)), we can evaluate above accurately.

sin(pi) = sin(\(\pi\) - pi) ≈ \(\pi\) - pi      → \(\pi\) ≈ pi + sin(pi)

355 = 113*pi + ε      // ε = smod(355,pi). Since ε>0, replace smod with mod

(\(\pi\) - 355/113) ≈ ((pi + sin(pi))*113 - (113*pi + ε)) / 113

HP71B Wrote:>radians
>(sin(pi)*113 - mod(355,pi)) / 113 ! error = -1 ULP
-2.66764189063E-7