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math: compute Erfcinv accurately for small arguments - #82104

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shogo82148:issue-82102-erfcinv
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math: compute Erfcinv accurately for small arguments#82104
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shogo82148:issue-82102-erfcinv

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@shogo82148 shogo82148 commented Oct 9, 2026 •
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Erfcinv(x) was implemented as Erfinv(1-x).
For small x, computing 1-x discards most of the information in x,
and the large derivative of Erfinv near 1 amplifies the rounding error.
For x <= 2**-54, 1-x rounds to 1 and Erfcinv returned +Inf.
Tiny negative arguments such as -1e-40, which are outside the domain,
also returned +Inf instead of NaN.

Split the rational approximations of Erfinv into helpers
and give Erfcinv its own implementation. It checks the domain directly,
uses the identity Erfcinv(x) = -Erfcinv(2-x) for x > 1 (2-x is exact there),
and evaluates the tail approximation from x itself instead of from 1-x.

Fixes #82102

@shogo82148
math: compute Erfcinv accurately for small arguments
8c25c45 
Erfcinv(x) was implemented as Erfinv(1-x). For small x, computing
1-x discards most of the information in x, and the large derivative
of Erfinv near 1 amplifies the rounding error. For x <= 2**-54,
1-x rounds to 1 and Erfcinv returned +Inf. Tiny negative arguments
such as -1e-40, which are outside the domain, also returned +Inf
instead of NaN.

Split the rational approximations of Erfinv into helpers and give
Erfcinv its own implementation. It checks the domain directly, uses
the identity Erfcinv(x) = -Erfcinv(2-x) for x > 1 (2-x is exact
there), and evaluates the tail approximation from x itself instead
of from 1-x.

Fixes golang#82102
@shogo82148
shogo82148 force-pushed the issue-82102-erfcinv branch 2 times, most recently from 7d177bf to 8c25c45 Compare October 9, 2026 17:27
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gopherbot commented Oct 9, 2026

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This PR (HEAD: 8c25c45) has been imported to Gerrit for code review.

Please visit Gerrit at https://go-review.googlesource.com/c/go/+/848186.

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math: Erfcinv loses precision and returns +Inf for small arguments

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