Issue #13219 has been updated by Marcus Stollsteimer.


Jabari Zakiya wrote:
> I noticed in the other squareroot methods they raise an error for ``n < 0``
> whereas my version returns 'nil' when taking an even root of a negative integer.

I strongly agree with nobu's implementation. That's exactly the behavior I was about to suggest when I noticed it was already there.

> The reason I chose to return 'nil' is because in some applications where I'm
> processing lots of data I don't want the program to bomb when encountering
> negative integers, I want to continue and just choose how to process those conditions.

It doesn't have to bomb.

1. for typical use cases (number theory, etc.) you will have only positive values anyway
2. you can check your data before using the method
3. you can use a begin...rescue block and handle the exception appropriately

Returning `nil` for a mathematical operation doesn't make much sense. And a silent failure you are not aware of will bite you much more than an exception.

> First, mathematically it is perfectly allowable to take the even root of a negative number,
> it's just a complex number, so conceptually this shouldn't be an error, just unwanted for this method.

"Not an error, but unwanted"???

The point is that by premise the method doesn't make _any_ sense in the complex domain. It's supposed to return the largest integer (natural number) less or equal to the square root. This wouldn't work for complex numbers at all: how would you truncate to an integer? how would you do the comparison? (Complex numbers are not ordered.)

Therefore, isqrt only does make sense for natural numbers. (Different from the "normal" sqrt.)


----------------------------------------
Feature #13219: bug in Math.sqrt(n).to_i, to compute integer squareroot,  new word to accurately fix it
https://bugs.ruby-lang.org/issues/13219#change-63187

* Author: Jabari Zakiya
* Status: Closed
* Priority: Normal
* Assignee: 
* Target version: 
----------------------------------------
In doing a math application using **Math.sqrt(n).to_i** to compute integer squareroots 
of integers I started noticing errors for numbers > 10**28.


I coded an algorithm that accurately computes the integer squareroot for arbirary sized numbers
but its significantly slower than the math library written in C.

Thus, I request a new method **Math.intsqrt(n)** be created, that is coded in C and part of the
core library, that will compute the integer squareroots of integers accurately and fast.

Here is working highlevel code to accurately compute the integer squareroot.

```
def intsqrt(n)
  bits_shift = (n.to_s(2).size)/2 + 1
  bitn_mask = root = 1 << bits_shift
  while true
    root ^= bitn_mask if (root * root) > n
    bitn_mask >>= 1
    return root if bitn_mask == 0
    root |= bitn_mask
  end
end

def intsqrt1(n)
  return n if n | 1 == 1   # if n is 0 or 1
  bits_shift = (Math.log2(n).ceil)/2 + 1
  bitn_mask = root = 1 << bits_shift
  while true
    root ^= bitn_mask if (root * root) > n
    bitn_mask >>= 1
    return root if bitn_mask == 0
    root |= bitn_mask
  end
end

require "benchmark/ips"

Benchmark.ips do |x|
  n = 10**40
  puts "integer squareroot tests for n = #{n}"
  x.report("intsqrt(n)"       ) { intsqrt(n)  }
  x.report("intsqrt1(n)"      ) { intsqrt1(n) }
  x.report("Math.sqrt(n).to_i") { Math.sqrt(n).to_i }
  x.compare!
end
```
Here's why it needs to be done in C.

```
2.4.0 :178 > load 'intsqrttest.rb'
integer squareroot tests for n = 10000000000000000000000000000000000000000
Warming up --------------------------------------
          intsqrt(n)     5.318k i/100ms
         intsqrt1(n)     5.445k i/100ms
   Math.sqrt(n).to_i   268.281k i/100ms
Calculating -------------------------------------
          intsqrt(n)     54.219k (± 5.5%) i/s -    271.218k in   5.017552s
         intsqrt1(n)     55.872k (± 5.4%) i/s -    283.140k in   5.082953s
   Math.sqrt(n).to_i      5.278M (± 6.1%) i/s -     26.560M in   5.050707s

Comparison:
   Math.sqrt(n).to_i:  5278477.8 i/s
         intsqrt1(n):    55871.7 i/s - 94.47x  slower
          intsqrt(n):    54219.4 i/s - 97.35x  slower

 => true 
2.4.0 :179 > 

```
Here are examples of math errors using **Math.sqrt(n).to_i** run on Ruby-2.4.0.

```
2.4.0 :101 > n = 10**27; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c   
1000000000000000000000000000
31622776601683
999999999999949826038432489
31622776601683
999999999999949826038432489
31622776601683
999999999999949826038432489
 => nil 
2.4.0 :102 > n = 10**28; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c 
10000000000000000000000000000
100000000000000
10000000000000000000000000000
100000000000000
10000000000000000000000000000
100000000000000
10000000000000000000000000000
 => nil 
2.4.0 :103 > n = 10**29; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c 
100000000000000000000000000000
316227766016837
99999999999999409792567484569
316227766016837
99999999999999409792567484569
316227766016837
99999999999999409792567484569
 => nil 
2.4.0 :104 > n = 10**30; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c  
1000000000000000000000000000000
1000000000000000
1000000000000000000000000000000
1000000000000000
1000000000000000000000000000000
1000000000000000
1000000000000000000000000000000
 => nil 
2.4.0 :105 > n = 10**31; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c 
10000000000000000000000000000000
3162277660168379
9999999999999997900254631487641
3162277660168379
9999999999999997900254631487641
3162277660168379
9999999999999997900254631487641
 => nil 
2.4.0 :106 > n = 10**32; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c 
100000000000000000000000000000000
10000000000000000
100000000000000000000000000000000
10000000000000000
100000000000000000000000000000000
10000000000000000
100000000000000000000000000000000
 => nil 
2.4.0 :107 > n = 10**33; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c
1000000000000000000000000000000000
31622776601683793
999999999999999979762122758866849
31622776601683793
999999999999999979762122758866849
31622776601683792
999999999999999916516569555499264
 => nil 
2.4.0 :108 > n = 10**34; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c
10000000000000000000000000000000000
100000000000000000
10000000000000000000000000000000000
100000000000000000
10000000000000000000000000000000000
100000000000000000
10000000000000000000000000000000000
 => nil 
2.4.0 :109 > n = 10**35; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c
100000000000000000000000000000000000
316227766016837933
99999999999999999873578871987712489
316227766016837933
99999999999999999873578871987712489
316227766016837952
100000000000000011890233980627554304
 => nil 
2.4.0 :110 > n = 10**36; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c
1000000000000000000000000000000000000
1000000000000000000
1000000000000000000000000000000000000
1000000000000000000
1000000000000000000000000000000000000
1000000000000000000
1000000000000000000000000000000000000
 => nil 
2.4.0 :111 > n = 10**37; puts n, (a = intsqrt(n)), a*a, (b = intsqrt1(n)), b*b, (c = Math.sqrt(n).to_i), c*c
10000000000000000000000000000000000000
3162277660168379331
9999999999999999993682442519108007561
3162277660168379331
9999999999999999993682442519108007561
3162277660168379392
10000000000000000379480317059650289664
 => nil 
2.4.0 :112 > 
```



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