> For moderate arguments final adjustment to
> satisfy requirements of ISQRT may add nontrivial cost.
That's not entirely true: with following patch
and testcase
[approxSqrt(x) for x in (1..2*10^5)];
it's 0.86s v.s. 0.80s.
Well, for small arguments we can even hard code answers.
So I propose to
1. change the documentation to \spad{-1 < x - sqrt(n) <= 0}
2. apply following check
3. use ISQRT for INT/NNI/PI
--- a/src/algebra/intfact.spad
+++ b/src/algebra/intfact.spad
@@ -300,11 +300,16 @@
n := n quo (4::I)
s := approxSqrt shift(a, -2 * n)
s := shift(s, n)
- return ((1 + s + a quo s) quo two)
+ new := ((1 + s + a quo s) quo two)
+ err0 := new*new - a
+ if err0 > 0 then new:=new-1
+ return new
-- initial approximation for the root is within a factor of 2
(new, old) := (shift(1, n quo two), 1)
while new ~= old repeat
(new, old) := ((1 + new + a quo new) quo two, new)
+ err0 := new*new - a
+ err0 > 0 => new-1
new
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