From: "Dann Corbit" <dcorbit@solutionsiq.com>
Subject: Re: Quick square rooting algorithm
Date: Tue, 21 Dec 1999 22:44:35 -0800
Newsgroups: sci.math.num-analysis

"Oleg Soloviev" <gps@chollian.net> wrote in message
news:Ti_74.2728$dq2.128141@news2.bora.net...
> Need advice for implementing quick square rooting algorithm - wish to use
> for graphic purposes - Are any methods quicker and better than square least
> method/polinom approximation?

If you only need integer precision, I find this one is hard to beat.

unsigned long   isqrt26(unsigned long input)
{
    unsigned long   s = 0;
    unsigned long   s21 = 1073741824;

    if (s21 <= input)
        s += 32768, input -= s21;
    s21 = (s << 15) + 268435456;
    if (s21 <= input)
        s += 16384, input -= s21;
    s21 = (s << 14) + 67108864;
    if (s21 <= input)
        s += 8192, input -= s21;
    s21 = (s << 13) + 16777216;
    if (s21 <= input)
        s += 4096, input -= s21;
    s21 = (s << 12) + 4194304;
    if (s21 <= input)
        s += 2048, input -= s21;
    s21 = (s << 11) + 1048576;
    if (s21 <= input)
        s += 1024, input -= s21;
    s21 = (s << 10) + 262144;
    if (s21 <= input)
        s += 512, input -= s21;
    s21 = (s << 9) + 65536;
    if (s21 <= input)
        s += 256, input -= s21;
    s21 = (s << 8) + 16384;
    if (s21 <= input)
        s += 128, input -= s21;
    s21 = (s << 7) + 4096;
    if (s21 <= input)
        s += 64, input -= s21;
    s21 = (s << 6) + 1024;
    if (s21 <= input)
        s += 32, input -= s21;
    s21 = (s << 5) + 256;
    if (s21 <= input)
        s += 16, input -= s21;
    s21 = (s << 4) + 64;
    if (s21 <= input)
        s += 8, input -= s21;
    s21 = (s << 3) + 16;
    if (s21 <= input)
        s += 4, input -= s21;
    s21 = (s << 2) + 4;
    if (s21 <= input)
        s += 2, input -= s21;
    s21 = (s << 1) + 1;
    if (s21 <= input)
        s++;
    return s;
}
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From: "Dann Corbit" <dcorbit@solutionsiq.com>
Subject: Re: Calculating length of a line without square root
Date: Thu, 30 Dec 1999 12:16:48 -0800
Newsgroups: sci.math.num-analysis

That one is slow.  Try isqrt28 or isqrt29 instead:

unsigned long   isqrt28(unsigned long input)
{
    unsigned long   s = 0;
    unsigned long   s21 = 1073741824;

    if (s21 <= input)
        s |= 32768, input -= s21;
    s21 = (s << 15) + 268435456;
    if (s21 <= input)
        s |= 16384, input -= s21;
    s21 = (s << 14) + 67108864;
    if (s21 <= input)
        s |= 8192, input -= s21;
    s21 = (s << 13) + 16777216;
    if (s21 <= input)
        s |= 4096, input -= s21;
    s21 = (s << 12) + 4194304;
    if (s21 <= input)
        s |= 2048, input -= s21;
    s21 = (s << 11) + 1048576;
    if (s21 <= input)
        s |= 1024, input -= s21;
    s21 = (s << 10) + 262144;
    if (s21 <= input)
        s |= 512, input -= s21;
    s21 = (s << 9) + 65536;
    if (s21 <= input)
        s |= 256, input -= s21;
    s21 = (s << 8) + 16384;
    if (s21 <= input)
        s |= 128, input -= s21;
    s21 = (s << 7) + 4096;
    if (s21 <= input)
        s |= 64, input -= s21;
    s21 = (s << 6) + 1024;
    if (s21 <= input)
        s |= 32, input -= s21;
    s21 = (s << 5) + 256;
    if (s21 <= input)
        s |= 16, input -= s21;
    s21 = (s << 4) + 64;
    if (s21 <= input)
        s |= 8, input -= s21;
    s21 = (s << 3) + 16;
    if (s21 <= input)
        s |= 4, input -= s21;
    s21 = (s << 2) + 4;
    if (s21 <= input)
        s |= 2, input -= s21;
    s21 = (s << 1) + 1;
    if (s21 <= input)
        s++;
    return s;
}

/* Log base 2 estimator */
int             qlog2(unsigned long n)
{
    register int    i = (n & 0xffff0000) ? 16 : 0;
    if ((n >>= i) & 0xff00)
        i |= 8, n >>= 8;
    if (n & 0xf0)
        i |= 4, n >>= 4;
    if (n & 0xc)
        i |= 2, n >>= 2;
    return (i | (n >> 1));
}
unsigned long   isqrt29(unsigned long x)
{
    unsigned long   xn,
                    xs;
    if (x <= 1)
        return (x);
    xn = x >> (qlog2(x) / 2);   // First guess is x shifted by half its
    // highest bit position
    do {
        xs = xn;                // Save present Xn
        xn = (xn + x / xn) / 2; // Compiler outputs right shifts for /2^n
    } while (xn - xs > 1);
    return (xn);
}

If you just need an approximiation, use this:

// Integer Square Root function
// Uses factoring to find square root
// A 256 entry table used to work out the square root of the 7 or 8 most
// significant bits.  A power of 2 used to approximate the rest.
// Based on an 80386 Assembly implementation by Arne Steinarson
unsigned const char sqq_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
};


// Really fast, but not real accurate.
unsigned long   isqrt08(unsigned long n)
{
    if (n >= 0x10000)
        if (n >= 0x1000000)
            if (n >= 0x10000000)
                if (n >= 0x40000000)
                    return (sqq_table[n >> 24] << 8);
                else
                    return (sqq_table[n >> 22] << 7);
            else if (n >= 0x4000000)
                return (sqq_table[n >> 20] << 6);
            else
                return (sqq_table[n >> 18] << 5);
        else if (n >= 0x100000)
            if (n >= 0x400000)
                return (sqq_table[n >> 16] << 4);
            else
                return (sqq_table[n >> 14] << 3);
        else if (n >= 0x40000)
            return (sqq_table[n >> 12] << 2);
        else
            return (sqq_table[n >> 10] << 1);
    else if (n >= 0x100)
        if (n >= 0x1000)
            if (n >= 0x4000)
                return (sqq_table[n >> 8]);
            else
                return (sqq_table[n >> 6] >> 1);
        else if (n >= 0x400)
            return (sqq_table[n >> 4] >> 2);
        else
            return (sqq_table[n >> 2] >> 3);
    else if (n >= 0x10)
        if (n >= 0x40)
            return (sqq_table[n] >> 4);
        else
            return (sqq_table[n << 2] << 5);
    else if (n >= 0x4)
        return (sqq_table[n >> 4] << 6);
    else
        return (sqq_table[n >> 6] << 7);
}

HTH.
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