math/big: better initial guess for nat.sqrt

The proposed change introduces a better initial guess which is closer to the final value and therefore converges in fewer steps. Consider for example sqrt(8): previously the guess was 8, whereas now it is 4 (and the result is 2). All this change does is it computes the division by two more accurately while it keeps the guess ≥ √x.

Change-Id: I917248d734a7b0488d14a647a063f674e56c4e30
GitHub-Last-Rev: c06d9d4876c8e7d6739f0e4b687e370fe1e9aad7
GitHub-Pull-Request: golang/go#28981
Reviewed-on: https://go-review.googlesource.com/c/163866
Run-TryBot: Brad Fitzpatrick <bradfitz@golang.org>
TryBot-Result: Gobot Gobot <gobot@golang.org>
Reviewed-by: Robert Griesemer <gri@golang.org>

diff --git a/src/math/big/nat.go b/src/math/big/nat.go
index 1e4a3b09cf77166ad69260d585e4c492c99a490a..336633a2faa58086fae76b316a91d67c51725e5a 100644 (file)
--- a/src/math/big/nat.go
+++ b/src/math/big/nat.go
@@ -1345,7 +1345,7 @@ func (z nat) sqrt(x nat) nat {
        var z1, z2 nat
        z1 = z
        z1 = z1.setUint64(1)
-       z1 = z1.shl(z1, uint(x.bitLen()/2+1)) // must be ≥ √x
+       z1 = z1.shl(z1, uint(x.bitLen()+1)/2) // must be ≥ √x
        for n := 0; ; n++ {
                z2, _ = z2.div(nil, x, z1)
                z2 = z2.add(z2, z1)