Square root by hand
Create a program that will calculate n digits of the square root of a number.
The program should continue forever (or until the number of digits is specified) calculating and outputting each decimal digit in succession. The program should be a "spigot algorithm" generating the digits of the number sequentially from left to right providing increasing precision as the algorithm proceeds.
C#
<lang csharp>using System; using static System.Math; using static System.Console; using BI = System.Numerics.BigInteger;
class Program {
static void Main(string[] args) {
BI i, j, k, d; i = 2; int n = -1; int n0 = -1;
j = (BI)Floor(Sqrt((double)i)); k = j; d = j;
DateTime st = DateTime.Now;
if (args.Length > 0) int.TryParse(args[0], out n);
if (n > 0) n0 = n; else n = 1;
do {
Write(d); i = (i - k * d) * 100; k = 20 * j;
for (d = 1; d <= 10; d++)
if ((k + d) * d > i) { d -= 1; break; }
j = j * 10 + d; k += d; if (n0 > 0) n--;
} while (n > 0);
if (n0 > 0) WriteLine("\nTime taken for {0} digits: {1}", n0, DateTime.Now - st); }
}</lang>
- Output:
14142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927557999505011527820605714701095599716059702745345968620147285174186408891986095523292304843087143214508397626036279952514079896872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471603868999706990048150305440277903164542478230684929369186215805784631115966687130130156185689872372Time taken for 500 digits: 00:00:00.0092331
Go
<lang go>package main
import (
"fmt" "math/big"
)
var one = big.NewInt(1) var ten = big.NewInt(10) var twenty = big.NewInt(20) var hundred = big.NewInt(100)
func sqrt(n int64, limit int) {
i := big.NewInt(n)
j := new(big.Int).Sqrt(i)
k := new(big.Int).Set(j)
d := new(big.Int).Set(j)
t := new(big.Int)
digits := 0
for digits < limit {
fmt.Print(d)
t.Mul(k, d)
i.Sub(i, t)
i.Mul(i, hundred)
k.Mul(j, twenty)
d.Set(one)
for d.Cmp(ten) <= 0 {
t.Add(k, d)
t.Mul(t, d)
if t.Cmp(i) > 0 {
d.Sub(d, one)
break
}
d.Add(d, one)
}
j.Mul(j, ten)
j.Add(j, d)
k.Add(k, d)
digits = digits + 1
}
fmt.Println()
}
func main() {
sqrt(2, 480) // enough for demo purposes
}</lang>
- Output:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871
Phix
...whereas this is a spigot algorithm! <lang Phix>-- based on https://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Decimal_(base_10) function bcd_sub(string a,b)
-- returns "a"-"b", coping with different lengths
-- (assumes a>=b, which it always will be here,
--- protected as it is by the bcd_le(b,a) call.)
integer c = 0, d = length(a)-length(b)
if d<0 then a = repeat('0',-d)&a
elsif d>0 then b = repeat('0', d)&b end if
for i=length(a) to 1 by -1 do
d = a[i]-b[i]-c
c = d<0
a[i] = d+c*10+'0'
end for
a = trim_head(a,"0") -- (note: "" equ "0")
return a
end function
function bcd_xp20x(string p, integer x)
-- returns x*(p*20+x)
integer c = 0, d, m = 1
p &= x+'0'
for i=length(p) to 1 by -1 do
d = (p[i]-'0')*m*x+c
p[i] = remainder(d,10)+'0'
c = floor(d/10)
m = 2
end for
if c then
p = (remainder(c,10)+'0')&p
c = floor(c/10)
if c then ?9/0 end if -- loop rqd?
end if
return p
end function
function bcd_le(string a,b)
-- returns a<=b numerically, taking care of different lengths
integer d = length(a)-length(b)
if d<0 then
a = repeat('0',-d)&a
elsif d>0 then
b = repeat('0',d)&b
end if
return a<=b
end function
function spigot_sqrt(string s, integer maxlen=50)
-- returns the square root of a positive string number to any precision
if find('-',s) or s="" then ?9/0 end if
integer dot = find('.',s)
if dot=0 then dot = length(s)+1 end if
if remainder(dot,2)=0 then s = "0"&s end if
dot += 1
string res = "", p = "", c = ""
integer i = 1
while true do -- (until (i>length && carry=0) or > maxlen)
if (i<=length(s) and s[i]='.')
or (i >length(s) and dot) then
res &= "."
dot = 0
i += 1
end if
c &= iff(i<=length(s)?s[i]:'0') &
iff(i<length(s)?s[i+1]:'0')
for x=9 to 0 by -1 do
string y = bcd_xp20x(p,x)
if bcd_le(y,c) then
c = bcd_sub(c,y)
res &= x+'0'
p &= x+'0'
exit
end if
if x=0 then ?9/0 end if -- (sanity check)
end for
i += 2
if (c="" and i>length(s)) or length(res)>maxlen then exit end if
end while
return res
end function ?spigot_sqrt("152.2756") ?spigot_sqrt("15241.383936") string r = spigot_sqrt("2",500) puts(1,join_by(r,1,100,""))</lang>
- Output:
(the final "2" was re-joined up by hand)
"12.34" "123.456" 1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157 2735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571 4701095599716059702745345968620147285174186408891986095523292304843087143214508397626036279952514079 8968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884 71603868999706990048150305440277903164542478230684929369186215805784631115966687130130156185689872372
REXX
This REXX version handles non-negative numbers less than unity, and may suppress superfluous trailing zeros.
It also handles the placing of a decimal point (if needed). <lang rexx>/*REXX program computes the square root by the old "by pen─n'─paper" (hand) method.*/ signal on halt /*handle the case of user interrupt. */ parse arg xx digs . /*obtain optional arguments from the CL*/ if xx== | xx=="," then xx= 2 /*Not specified? Then use the default.*/ if digs== | digs=="," then digs= 400 /* " " " " " " */ numeric digits digs + digs % 2 /*ensure enough decimal digits for calc*/ call sqrtHand xx, digs /*invoke the function for sqrt by hand.*/ halt: say /*pgm comes here for exact sqrt or HALT*/ exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ iSqrt: procedure; parse arg z; q= 1; r= 0; do while q<=z; q= q*4; end
do while q>1; q= q%4; _= z-r-q; r= r%2; if _>=0 then do; z= _; r= r+q; end; end
return r /*R is the integer square root of Z. */
/*──────────────────────────────────────────────────────────────────────────────────────*/ sqrtHand: parse arg x,places; parse value iSqrt(x) with j 1 k 1 ? /*j, k, ? ≡ iSqrt(x) */
if ?==0 then ?= /*handle the case of sqrt < 1. */
if j*j=x then do; say j; return; end /*have we found the exact sqrt?*/
L= length(?) /*L: used to place dec. point.*/
if L==0 then call charout , . /*handle dec. point for i < 1. */
do spit=1 for places
call charout , ?
if L>0 then do; call charout , .; L= 0; end /*process dec. point*/
if ?== then ?= 0 /*ensure the ? is a valid dig.*/
x= (x - k*?) * 100; ?= 1
k= j * 20
do while ?<=10
if (k + ?)*? > x then do; ?= ? - 1; leave; end
else ?= ? + 1
end /*while ?≤10*/
j= ? + j*10
k= ? + k
end /*spit*/
return</lang>
- output when using the default inputs:
1.41421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157 2735013846230912297024924836055850737212644121497099935831413222665927505592755799950501152782060571 4701095599716059702745345968620147285174186408891986095523292304843087143214508397626036279952514079 8968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884 7160386899970699004815030544027790316454247823068492936918621580578463111596668713013015618568987237 2
- output when using the inputs of: .9 80
.9486832980505137995996680633298155601158665417975650480572514558377783315917714
- output when using the inputs of: 625
25
Visual Basic .NET
This is "spigot like", but not a true spigot, just an implementation of the "by hand" method of computing the square root, in this case, of two.<lang vbnet>Imports System.Math, System.Console, BI = System.Numerics.BigInteger
Module Module1
Sub Main(ByVal args As String())
Dim i, j, k, d As BI : i = 2
j = CType(Floor(Sqrt(CDbl(i))), BI) : k = j : d = j
Dim n As Integer = -1, n0 As Integer = -1,
st As DateTime = DateTime.Now
If args.Length > 0 Then Integer.TryParse(args(0), n)
If n > 0 Then n0 = n Else n = 1
Do
Write(d) : i = (i - k * d) * 100 : k = 20 * j
For d = 1 To 10
If (k + d) * d > i Then d -= 1 : Exit For
Next
j = j * 10 + d : k += d : If n0 > 0 Then n = n - 1
Loop While n > 0
If n0 > 0 Then WriteLine (VbLf & "Time taken for {0} digits: {1}", n0, DateTime.Now - st)
End Sub
End Module</lang>
- Output:
Execute without any command line parameters for it to run until it crashes (due to BigInteger variables eating up available memory). Output with command line parameter of 500:
14142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727350138462309122970249248360558507372126441214970999358314132226659275055927557999505011527820605714701095599716059702745345968620147285174186408891986095523292304843087143214508397626036279952514079896872533965463318088296406206152583523950547457502877599617298355752203375318570113543746034084988471603868999706990048150305440277903164542478230684929369186215805784631115966687130130156185689872372 Time taken for 500 digits: 00:00:00.0263710
Wren
<lang ecmascript>import "/big" for BigInt
var sqrt = Fn.new { |n, limit|
var i = BigInt.new(n)
var j = i.isqrt
var k = j
var d = j
var digits = 0
while (digits < limit) {
System.write(d)
i = (i - k*d) * 100
k = j * 20
d = BigInt.one
while (d <= 10) {
if ((k + d)*d > i) {
d = d.dec
break
}
d = d.inc
}
j = j*10 + d
k = k + d
digits = digits + 1
}
System.print()
}
sqrt.call(2, 480) // enough for demo purposes</lang>
- Output:
141421356237309504880168872420969807856967187537694807317667973799073247846210703885038753432764157273501384623091229702492483605585073721264412149709993583141322266592750559275579995050115278206057147010955997160597027453459686201472851741864088919860955232923048430871432145083976260362799525140798968725339654633180882964062061525835239505474575028775996172983557522033753185701135437460340849884716038689997069900481503054402779031645424782306849293691862158057846311159666871