[9] RFR of 8032027: Add BigInteger square root methods

[joe darcy]

Here is a trick to consider for future performance tuning: all large 
floating-point numbers are integers. Once the size of the positive 
exponent exceeds the number of bits of precision, the value must be an 
integer. For double, that means values greater than 2^54 are integers 
and the full double exponent range goes out to 2^1023.

Consequently, if a BigInteger is less than 2^1023 and the spread of 1 
bits in the BigInteger is contained within a double , the result could 
be directly calculated as

     BigInteger(Math.floor(Math.sqrt(bi.doubleValue())))

without needing any iterations on the BigInteger side. The bit spread 
could be able to be calculated from something like

     bi.bitLenth() - bi.getLowestSetBit()

HTH,

-Joe

On 10/2/2015 2:29 PM, Brian Burkhalter wrote:
> On Oct 2, 2015, at 2:16 PM, Louis Wasserman <wasserman.louis at gmail.com> wrote:
>
>> I'm pretty sure we could potentially public-domain our implementation, if that were an issue.
> Personally, if it’s much better than mine (or what mine could be revised to be) I’d be happy to have the better outcome.
>
>>> The implementation I have here is effectively the one from Hacker’s Delight (2nd ed.). The initial guess is intended to be larger than the true result in order to simplify the termination condition of that algorithm as the monotonic convergence cannot have any oscillation between potential terminal values.
>> This isn't a problem.  The arithmetic mean of *any* two nonnegative numbers is always greater than their geometric mean, so for *any* nonnegative a, (a + x/a)/2 >= sqrt(a * x/a) = sqrt(x).
> On Oct 2, 2015, at 2:18 PM, Louis Wasserman <lowasser at google.com> wrote:
>
>> (https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means proves that the arithmetic mean is >= the geometric mean.)
> Got it.
>
>> So once you do *one* Newton iteration, the convergence is guaranteed to be monotonically decreasing after that point.
>>
>> Newton's method doubles the number of correct digits with every iteration.  So you're paying one extra Newton iteration, but in exchange you're getting -handwave- 50 correct bits to start out with.  That *more* than pays for itself.
>>
>>> There is certainly some room here for improvement in the range of input values less than Double.MAX_VALUE but this is a first stab at the implementation so hopefully such improvement may be brought in later if it is not in the first pass.
>> It doesn't matter whether the input is bigger than Double.MAX_VALUE.  You can just find some even number, s, such that x.shiftRight(s) < 2^52.  Then use
>>
>> doubleToBigInteger(Math.sqrt(x.shiftRight(s))).shiftLeft(s / 2)
>>
>> as your starting estimate.  You're still getting ~50 correct bits to start your Newton iteration.
> Excellent suggestion. I’ll look into revising it accordingly.
>
> Initially I had the thing broken into three ranges: 4 <= x <= Long.MAX_VALUE, Long.MAX_VALUE < x <= Double.MAX_VALUE, and Double.MAX_VALUE < x but found that this was lame and pointless.
>
> Thanks,
>
> Brian