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math/big: fix comment in divRecursiveStep
There appears to be a typo in the description of the recursive division algorithm. Two things seem suspicious with the original comment: 1. It is talking about choosing s, but s doesn't appear anywhere in the equation. 2. The math in the equation is incorrect. Where B = len(v)/2 s = B - 1 Proof that it is incorrect: len(v) - B >= B + 1 len(v) - len(v)/2 >= len(v)/2 + 1 This doesn't hold if len(v) is even, e.g. 10: 10 - 10/2 >= 10/2 + 1 10 - 5 >= 5 + 1 5 >= 6 // this is false The new equation will be the following, which will be mathematically correct: len(v) - s >= B + 1 len(v) - (len(v)/2 - 1) >= len(v)/2 + 1 len(v) - len(v)/2 + 1 >= len(v)/2 + 1 len(v) - len(v)/2 >= len(v)/2 This holds if len(v) is even or odd. e.g. 10 10 - 10/2 >= 10/2 10 - 5 >= 5 5 >= 5 e.g. 11 11 - 11/2 >= 11/2 11 - 5 >= 5 6 >= 5 Change-Id: If77ce09286cf7038637b5dfd0fb7d4f828023f56 Reviewed-on: https://go-review.googlesource.com/c/go/+/287372 Run-TryBot: Katie Hockman <katie@golang.org> Reviewed-by: Filippo Valsorda <filippo@golang.org> Trust: Katie Hockman <katie@golang.org>
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@ -881,7 +881,7 @@ func (z nat) divRecursiveStep(u, v nat, depth int, tmp *nat, temps []*nat) {
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// then floor(u1/v1) >= floor(u/v)
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// then floor(u1/v1) >= floor(u/v)
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//
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//
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// Moreover, the difference is at most 2 if len(v1) >= len(u/v)
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// Moreover, the difference is at most 2 if len(v1) >= len(u/v)
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// We choose s = B-1 since len(v)-B >= B+1 >= len(u/v)
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// We choose s = B-1 since len(v)-s >= B+1 >= len(u/v)
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s := (B - 1)
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s := (B - 1)
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// Except for the first step, the top bits are always
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// Except for the first step, the top bits are always
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// a division remainder, so the quotient length is <= n.
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// a division remainder, so the quotient length is <= n.
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