Tan has poles along the real axis. In order to accurately calculate
the value near these poles, a range reduction by Pi is performed and
the result calculated via a Taylor series. The prior implementation
of range reduction used Cody-Waite range reduction in three parts.
This fails when x is too large to accurately calculate the partial
products in the summation accurately. Above this threshold, Payne-Hanek
range reduction using a multiple precision value of 1/Pi is required.
Additionally, the threshold used in math/trig_reduce.go for Payne-Hanek
range reduction was not set conservatively enough. The prior threshold
ensured that catastrophic failure did not occur where the argument x
would not actually be reduced below Pi/4. However, errors in reduction
begin to occur at values much lower when z = ((x - y*PI4A) - y*PI4B) - y*PI4C
is not exact because y*PI4A cannot be exactly represented as a float64.
reduceThreshold is lowered to the proper value.
Fixes #31566
Change-Id: I0f39a4171a5be44f64305f18dc57f6c29f19dba7
Reviewed-on: https://go-review.googlesource.com/c/go/+/172838 Reviewed-by: Rob Pike <r@golang.org>