Issue #13219 has been updated by Jabari Zakiya.


So the clear winner, and new champeen is...**newtons_fast**, at least for squareroots.

Just goes to show, don't f**k with Newton!

```
class Integer
  def irootn(n)
    return nil if self < 0 && n.even?
    raise "root n is < 2 or not an Integer" unless n.is_a?(Integer) && n > 1
    num  = self.abs
    bits_shift = (num.bit_length)/n + 2
    root, bitn_mask = 0, (1 << bits_shift)
    until (bitn_mask >>= 1) == 0
      root |= bitn_mask
      root ^= bitn_mask if root**n > num
    end
    root *= (self < 0 ? -1 : 1)
  end

  def iroot2; irootn(2) end    
end

def isqrt1(n)
    r = 0
    x = 1 << ((n.bit_length >> 1 ) << 1)
    until x == 0
        t = r | x
        r >>= 1
        (n -= t; r |= x) if n >= t
        x >>= 2
    end
    r
end

def newtons_fast(n)  # Newton's method with n < initial_x <= 2*n
  raise if n<0
  return n if n<=1
  x = 1<<((n.bit_length+1)/2)
  y = (x + n/x)/2
  while y < x
    x = y
    y = (x + n/x)/2
  end
  x
end

Benchmark.ips do |x|
  exp = 400; n = 10**exp
  puts "integer squareroot tests for n = 10**#{exp}"
  x.report("iroot2"     ) { n.iroot2    }
  x.report("isqrt1(n)"  ) { isqrt1(n)   }
  x.report("newtons_fast(n)") { newtons_fast(n)
  x.compare!
end
```

```
2.4.0 :138 > load 'irootstest.rb'                                      
integer squareroot tests for n = 10**5
Warming up --------------------------------------
              iroot2    85.000k i/100ms
           isqrt1(n)    87.646k i/100ms
     newtons_fast(n)   191.796k i/100ms
Calculating -------------------------------------
              iroot2      1.014M ( 2.9%) i/s -      5.100M in   5.031392s
           isqrt1(n)      1.049M ( 3.0%) i/s -      5.259M in   5.018645s
     newtons_fast(n)      2.846M ( 4.6%) i/s -     14.385M in   5.066760s

Comparison:
     newtons_fast(n):  2845709.1 i/s
           isqrt1(n):  1048817.1 i/s - 2.71x  slower
              iroot2:  1014498.5 i/s - 2.81x  slower

 => true 
2.4.0 :139 > load 'irootstest.rb'
integer squareroot tests for n = 10**50
Warming up --------------------------------------
              iroot2     4.077k i/100ms
           isqrt1(n)     3.992k i/100ms
     newtons_fast(n)    24.377k i/100ms
Calculating -------------------------------------
              iroot2     41.189k ( 6.1%) i/s -    207.927k in   5.071463s
           isqrt1(n)     40.293k ( 3.4%) i/s -    203.592k in   5.058807s
     newtons_fast(n)    260.960k ( 3.4%) i/s -      1.316M in   5.050191s

Comparison:
     newtons_fast(n):   260959.7 i/s
              iroot2:    41188.8 i/s - 6.34x  slower
           isqrt1(n):    40292.7 i/s - 6.48x  slower

 => true 
2.4.0 :140 > load 'irootstest.rb'
integer squareroot tests for n = 10**500
Warming up --------------------------------------
              iroot2    74.000  i/100ms
           isqrt1(n)   127.000  i/100ms
     newtons_fast(n)     3.775k i/100ms
Calculating -------------------------------------
              iroot2    741.359  ( 4.6%) i/s -      3.700k in   5.001895s
           isqrt1(n)      1.258k ( 7.0%) i/s -      6.350k in   5.077988s
     newtons_fast(n)     37.762k ( 4.5%) i/s -    188.750k in   5.008756s

Comparison:
     newtons_fast(n):    37761.5 i/s
           isqrt1(n):     1257.6 i/s - 30.03x  slower
              iroot2:      741.4 i/s - 50.94x  slower

 => true 
2.4.0 :141 > load 'irootstest.rb'
integer squareroot tests for n = 10**5000
Warming up --------------------------------------
              iroot2     1.000  i/100ms
           isqrt1(n)     5.000  i/100ms
     newtons_fast(n)   340.000  i/100ms
Calculating -------------------------------------
              iroot2     11.196  ( 8.9%) i/s -     56.000  in   5.013699s
           isqrt1(n)     53.056  ( 3.8%) i/s -    265.000  in   5.002572s
     newtons_fast(n)      3.402k ( 3.1%) i/s -     17.000k in   5.002400s

Comparison:
     newtons_fast(n):     3401.9 i/s
           isqrt1(n):       53.1 i/s - 64.12x  slower
              iroot2:       11.2 i/s - 303.85x  slower

```

----------------------------------------
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-63139

* 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
    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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