SquareRoot.java/*
* Integer Square Root function
* Contributors include Arne Steinarson for the basic approximation idea, Dann
* Corbit and Mathew Hendry for the first cut at the algorithm, Lawrence Kirby
* for the rearrangement, improvments and range optimization, Paul Hsieh
* for the round-then-adjust idea, Tim Tyler, for the Java port
* and Jeff Lawson for a bug-fix and some code to improve accuracy.
*
*
* v0.02 - 2003/09/07
*/
/**
* Faster replacements for (int)(java.lang.Math.sqrt(integer))
*/
public class SquareRoot {
final static int[] table = {
0, 16, 22, 27, 32, 35, 39, 42, 45, 48, 50, 53, 55, 57,
59, 61, 64, 65, 67, 69, 71, 73, 75, 76, 78, 80, 81, 83,
84, 86, 87, 89, 90, 91, 93, 94, 96, 97, 98, 99, 101, 102,
103, 104, 106, 107, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118,
119, 120, 121, 122, 123, 124, 125, 126, 128, 128, 129, 130, 131, 132,
133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 144, 145,
146, 147, 148, 149, 150, 150, 151, 152, 153, 154, 155, 155, 156, 157,
158, 159, 160, 160, 161, 162, 163, 163, 164, 165, 166, 167, 167, 168,
169, 170, 170, 171, 172, 173, 173, 174, 175, 176, 176, 177, 178, 178,
179, 180, 181, 181, 182, 183, 183, 184, 185, 185, 186, 187, 187, 188,
189, 189, 190, 191, 192, 192, 193, 193, 194, 195, 195, 196, 197, 197,
198, 199, 199, 200, 201, 201, 202, 203, 203, 204, 204, 205, 206, 206,
207, 208, 208, 209, 209, 210, 211, 211, 212, 212, 213, 214, 214, 215,
215, 216, 217, 217, 218, 218, 219, 219, 220, 221, 221, 222, 222, 223,
224, 224, 225, 225, 226, 226, 227, 227, 228, 229, 229, 230, 230, 231,
231, 232, 232, 233, 234, 234, 235, 235, 236, 236, 237, 237, 238, 238,
239, 240, 240, 241, 241, 242, 242, 243, 243, 244, 244, 245, 245, 246,
246, 247, 247, 248, 248, 249, 249, 250, 250, 251, 251, 252, 252, 253,
253, 254, 254, 255
};
/**
* A faster replacement for (int)(java.lang.Math.sqrt(x)).
Completely accurate for x < 2147483648 (i.e. 2^31)...
*/
static int sqrt(int x) {
int xn;
if (x >= 0x10000) {
if (x >= 0x1000000) {
if (x >= 0x10000000) {
if (x >= 0x40000000) {
xn = table[x >> 24] << 8;
}
else {
xn = table[x >> 22] << 7;
}
} else {
if (x >= 0x4000000) {
xn = table[x >> 20] << 6;
} else {
xn = table[x >> 18] << 5;
}
}
xn = (xn + 1 + (x / xn)) >> 1;
xn = (xn + 1 + (x / xn)) >> 1;
return ((xn * xn) > x) ? --xn : xn;
} else {
if (x >= 0x100000) {
if (x >= 0x400000) {
xn = table[x >> 16] << 4;
} else {
xn = table[x >> 14] << 3;
}
} else {
if (x >= 0x40000) {
xn = table[x >> 12] << 2;
} else {
xn = table[x >> 10] << 1;
}
}
xn = (xn + 1 + (x / xn)) >> 1;
return ((xn * xn) > x) ? --xn : xn;
}
} else {
if (x >= 0x100) {
if (x >= 0x1000) {
if (x >= 0x4000) {
xn = (table[x >> 8]) + 1;
} else {
xn = (table[x >> 6] >> 1) + 1;
}
} else {
if (x >= 0x400) {
xn = (table[x >> 4] >> 2) + 1;
} else {
xn = (table[x >> 2] >> 3) + 1;
}
}
return ((xn * xn) > x) ? --xn : xn;
} else {
if (x >= 0) {
return table[x] >> 4;
}
}
}
illegalArgument();
return -1;
}
/**
* A faster replacement for (int)(java.lang.Math.sqrt(x)).
* Completely accurate for x < 2147483648 (i.e. 2^31)...
* Adjusted to more closely approximate
* "(int)(java.lang.Math.sqrt(x) + 0.5)"
* by Jeff Lawson.
*/
static int accurateSqrt(int x) {
int xn;
if (x >= 0x10000) {
if (x >= 0x1000000) {
if (x >= 0x10000000) {
if (x >= 0x40000000) {
xn = table[x >> 24] << 8;
} else {
xn = table[x >> 22] << 7;
}
} else {
if (x >= 0x4000000) {
xn = table[x >> 20] << 6;
} else {
xn = table[x >> 18] << 5;
}
}
xn = (xn + 1 + (x / xn)) >> 1;
xn = (xn + 1 + (x / xn)) >> 1;
return adjustment(x, xn);
} else {
if (x >= 0x100000) {
if (x >= 0x400000) {
xn = table[x >> 16] << 4;
} else {
xn = table[x >> 14] << 3;
}
} else {
if (x >= 0x40000) {
xn = table[x >> 12] << 2;
} else {
xn = table[x >> 10] << 1;
}
}
xn = (xn + 1 + (x / xn)) >> 1;
return adjustment(x, xn);
}
} else {
if (x >= 0x100) {
if (x >= 0x1000) {
if (x >= 0x4000) {
xn = (table[x >> 8]) + 1;
} else {
xn = (table[x >> 6] >> 1) + 1;
}
} else {
if (x >= 0x400) {
xn = (table[x >> 4] >> 2) + 1;
}
else {
xn = (table[x >> 2] >> 3) + 1;
}
}
return adjustment(x, xn);
} else {
if (x >= 0) {
return adjustment(x, table[x] >> 4);
}
}
}
illegalArgument();
return -1;
}
private static int adjustment(int x, int xn) {
// Added by Jeff Lawson:
// need to test:
// if |xn * xn - x| > |x - (xn-1) * (xn-1)| then xn-1 is more accurate
// if |xn * xn - x| > |(xn+1) * (xn+1) - x| then xn+1 is more accurate
// or, for all cases except x == 0:
// if |xn * xn - x| > x - xn * xn + 2 * xn - 1 then xn-1 is more accurate
// if |xn * xn - x| > xn * xn + 2 * xn + 1 - x then xn+1 is more accurate
int xn2 = xn * xn;
// |xn * xn - x|
int comparitor0 = xn2 - x;
if (comparitor0 < 0) {
comparitor0 = -comparitor0;
}
int twice_xn = xn << 1;
// |x - (xn-1) * (xn-1)|
int comparitor1 = x - xn2 + twice_xn - 1;
if (comparitor1 < 0) {
// need to correct for x == 0 case?
comparitor1 = -comparitor1;
// only gets here when x == 0
}
// |(xn+1) * (xn+1) - x|
int comparitor2 = xn2 + twice_xn + 1 - x;
if (comparitor0 > comparitor1) {
return (comparitor1 > comparitor2) ? ++xn : --xn;
}
return (comparitor0 > comparitor2) ? ++xn : xn;
}
/**
* A *much* faster replacement for (int)(java.lang.Math.sqrt(x)).
* Completely accurate for x < 289...
*/
static int fastSqrt(int x) {
if (x >= 0x10000) {
if (x >= 0x1000000) {
if (x >= 0x10000000) {
if (x >= 0x40000000) {
return (table[x >> 24] << 8);
} else {
return (table[x >> 22] << 7);
}
} else if (x >= 0x4000000) {
return (table[x >> 20] << 6);
} else {
return (table[x >> 18] << 5);
}
} else if (x >= 0x100000) {
if (x >= 0x400000) {
return (table[x >> 16] << 4);
} else {
return (table[x >> 14] << 3);
}
} else if (x >= 0x40000) {
return (table[x >> 12] << 2);
} else {
return (table[x >> 10] << 1);
}
} else if (x >= 0x100) {
if (x >= 0x1000) {
if (x >= 0x4000) {
return (table[x >> 8]);
} else {
return (table[x >> 6] >> 1);
}
} else if (x >= 0x400) {
return (table[x >> 4] >> 2);
} else {
return (table[x >> 2] >> 3);
}
} else if (x >= 0) {
return table[x] >> 4;
}
illegalArgument();
return -1;
}
private static void illegalArgument() {
throw new IllegalArgumentException("Attemt to take the square root of negative number");
}
/** From http://research.microsoft.com/~hollasch/cgindex/math/introot.html
* where it is presented by Ben Discoe (rodent@netcom.COM)
* Not terribly speedy...
*/
/*
static int unrolled_sqrt(int x) {
int v;
int t = 1<<30;
int r = 0;
int s;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
t >>= 2;
s = t + r;
r>>= 1;
if (s <= x) {
x -= s;
r |= t;
}
return r;
}
*/
/**
* Mark Borgerding's algorithm...
* Not terribly speedy...
*/
/*
static int mborg_sqrt(int val) {
int guess=0;
int bit = 1 << 15;
do {
guess ^= bit;
// check to see if we can set this bit without going over sqrt(val)...
if (guess * guess > val )
guess ^= bit;
// it was too much, unset the bit...
} while ((bit >>= 1) != 0);
return guess;
}
*/
/**
* Taken from http://www.jjj.de/isqrt.cc
* Code not tested well...
* Attributed to: http://www.tu-chemnitz.de/~arndt/joerg.html / email: arndt@physik.tu-chemnitz.de
* Slow.
*/
/*
final static int BITS = 32;
final static int NN = 0; // range: 0...BITSPERLONG/2
final static int test_sqrt(int x) {
int i;
int a = 0;
// accumulator...
int e = 0;
// trial product...
int r;
r=0;
// remainder...
for (i=0; i < (BITS/2) + NN; i+
{
r <<= 2;
r += (x >> (BITS - 2));
x <<= 2;
a <<= 1;
e = (a << 1)+1;
if(r >= e)
{
r -= e;
a++;
}
}
return a;
}
*/
/*
// Totally hopeless performance...
static int test_sqrt(int n) {
float r = 2.0F;
float s = 0.0F;
for(; r < (float)n / r; r *= 2.0F);
for(s = (r + (float)n / r) / 2.0F; r - s > 1.0F; s = (r + (float)n / r) / 2.0F) {
r = s;
}
return (int)s;
}
*/
}
Maintained by John Loomis, updated Sat Mar 16 12:52:37 2019