## Feature #13250

### Initial estimate for Integer#sqrt should be improved

**Description**

r57705, by Nobu, in response to issue #13219, added Integer#sqrt. The initial estimator used is `1 << (b-1)/2 | n >> (b+1)/2`

. For this style of estimator, `1 << (b-1)/2 | n >> (b+3)/2`

would be best. (Basically, there is a fencepost error in the current commit.)

However, much better initial estimates can be made. In particular, using the hardware floating point square root will almost certainly eliminate 3-4 rounds of approximation. Even for smaller numbers, this represents a substantial gain.

If for some reason, it is problematic to use fsqrt, a small table lookup can eliminate a couple of rounds.

#### Updated by Student (Nathan Zook) almost 3 years ago

Think about this some more, the fastest solution would be to actually to a N-R round in the floating point unit as well. I will have something in a few days.

#### Updated by nobu (Nobuyoshi Nakada) almost 3 years ago

Nathan Zook wrote:

The initial estimator used is

`1 << (b-1)/2 | n >> (b+1)/2`

. For this style of estimator,`1 << (b-1)/2 | n >> (b+3)/2`

would be best. (Basically, there is a fencepost error in the current commit.)

It is `(1 << (b-1)/2) | (n >> (b/2+1))`

(+1 after /2), which equals the latter formula.

#### Updated by jzakiya (Jabari Zakiya) almost 3 years ago

It would be nice if you backported this code into point releases of older versions too.

#### Updated by nobu (Nobuyoshi Nakada) almost 3 years ago

**Status**changed from*Open*to*Closed*

Applied in changeset r57713.

bignum.c: improve estimate

- bignum.c (estimate_initial_sqrt, rb_big_isqrt): improve initial estimate by sqrt(). [ruby-core:79754] [Feature #13250]