```Issue #13219 has been updated by Nathan Zook.

~~~
# Core Algorithm by Paul Zimmerman, article entitled
# Karatsuba Square Root
# https://hal.inria.fr/inria-00072854/document
# Presumably used in GMP.

# Personal computations derived from implementation details (page 5)
# n >= b**4 / 4
# b = 2**64**k
# n * 4 >= b ** 4
# n.length + 2 >= b.length * 4
# (n.length + 2) >> 2 >= b.length
# (n.length + 2) >> 2 >= 64 * k
# ((n.length + 2) >> 2) / 64 = k

def sqrtrem(n)
nlength = n.bit_length
if nlength >= (64 * 4 - 2)
bbits = (n.bit_length + 2 >> 2) / 64 * 64
elsif nlength >= (32 * 4 - 2)
bbits = (n.bit_length + 2 >> 2) / 32 * 32
elsif nlength >= (16 * 4 - 2)
bbits = (n.bit_length + 2 >> 2) / 16 * 16
else # single word now -- my computer has 64-bit mantissas!
s = Math.sqrt(n).to_i
r = n - s * s
return [s, r]
end

b = 1 << bbits
a3 =  n >> bbits * 3
a2 = (n >> bbits * 2) & bmask
a1 = (n >> bbits    ) & bmask

i, j = sqrtrem((a3 << bbits) + a2)
q, u = ((j << bbits) + a1).divmod(i << 1)
s = (i << bbits) + q
r = (u << bbits) + a0 - q ** 2

if r < 0
r += (s << 1) - 1
s -= 1
end

[s, r]
end

def sqrt_z(n)
s, r = sqrtrem(n)
s
end

Benchmark.ips do |x|
n = 10 ** 5000
x.report("iroot2") { n.iroot2 }
x.report("sqrt_z") { sqrt_z(n) }
end

Warming up --------------------------------------
iroot2     1.000  i/100ms
sqrt_z   987.000  i/100ms
Calculating -------------------------------------
iroot2      8.665  (¡Þ 0.0%) i/s -     44.000  in   5.078551s
sqrt_z     10.015k (¡Þ 1.6%) i/s -     50.337k in   5.027514s
=>
~~~

I've been at this for way too long today.

A few notes:
1) Assuming no architectural weirdness, it looks like--for this PARTICULAR case--Zimmerman's algorithm is doing about 4x better than what you are calling newtons_fast.  First, this is not a surprise, as no one uses Newton's method this way.  What is interesting is that Zimmerman's algorithm is designed to be good on GMP, and I am using stock ruby.
2) That hack at the top can almost certainly be dramatically improved (It's my contribution).  This is a demo of Zimmerman's work--which doesn't even really want to talk about numbers smaller than 2**254.  Please don't bother looking at performance under 1000 bits until that is cleaned up.
3) In a note that Zimmerman included in his article, he indicated that Newton-Raphson would probably be faster.  As I stated, I have been unable to find an implementation, and I am loathe to present anything without high confidence that it is correct.  I will do some more looking tomorrow.
4) In some sense it is silly to do timing tests like this.  We have no idea how much time is being spent in ruby and how much in C.  It is entirely likely that we could have 10x speed improvements (or much, much more) across certain ranges for some of these methods by dropping into C.  As I said, it is a question of how much work we want to do.  At a minimum, we should do a GMP-linked build and see what Zimmerman's work can really do.  Again, I don't know the legal considerations, and I would like guidance before attempting to reimplement an MP library from scratch.

----------------------------------------
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-63146

* Author: Jabari Zakiya
* Status: Open
* 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
return root if bitn_mask == 0
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
return root if bitn_mask == 0
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.

```
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 (¡Þ 5.5%) i/s -    271.218k in   5.017552s
intsqrt1(n)     55.872k (¡Þ 5.4%) i/s -    283.140k in   5.082953s
Math.sqrt(n).to_i      5.278M (¡Þ 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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