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From: Szabolcs Nagy <nsz@port70.net>
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Subject: Re: x87 asin and acos
Date: Sat, 23 Feb 2019 21:30:31 +0100
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* Shane Seelig <stseelig@mail.com> [2019-02-23 09:21:08 -0500]:
> Currently 'asin' uses the algorithm:
> 	arcsin(x) == arctan(x/(sqrt((1-x)(1+x))))
> If the following algorithm were to be used instead, an 'fadd' could be
> removed.
> 	arcsin(x) == arctan(x/(sqrt(1-x**2)))

that change seems valid as far as the result is concerned.
(the worst case rounding error of the sqrt argument should
be around LDBL_EPS in both cases)

but the fenv behaviour is not valid:
for tiny x, x*x raises spurious underflow exception
for large x, x*x raises spurious overflow exception

(1-x)(1+x) avoids these issues.

note that there is not much performance difference between
the two expressions: 1-x and 1+x are independent computations
so the latency is 1 add + 1 mul in both cases, and the entire
function is likely dominated by fpsqrt followed by fpatan
both of which have huge latency compared to add or mul.