What Approximation Techniques Exist for Computing the Square Root?
if you want a cheap and dirty optimized power-series expansion (the coefficients for Taylor series converge slowly) for sqrt() and a bunch of other trancendentals, i have some code from long ago.  i used to sell this code, but no one has paid me for it for nearly a decade.  so i think i'll release it for public consumption.  this particular file was for an application where the processor had floating point (IEEE-754 single precision) and they had a C compiler and dev system, but they did not have (or they didn't want to link in) the stdlib that would have had the standard math functions.  they did not need perfect precision, but they wanted things to be fast.  you can pretty easily reverse engineer the code to see what the power series coefficients are and write your own code.  this code assumes IEEE-754 and masked off the bits for mantissa and exponent.

it appears that the "code markup" that SE has is unfriendly with angle characters (you know ">" or "&lt;"), so you'll probably have to hit "edit" to see all of it.

//
//    FILE: __functions.h
//
//    fast and approximate transcendental functions
//
//    copyright (c) 2004  Robert Bristow-Johnson
//
//    rbj@audioimagination.com
//


#ifndef __FUNCTIONS_H
#define __FUNCTIONS_H

#define TINY 1.0e-8
#define HUGE 1.0e8

#define PI              (3.1415926535897932384626433832795028841972)        /* pi */
#define ONE_OVER_PI     (0.3183098861837906661338147750939)
#define TWOPI           (6.2831853071795864769252867665590057683943)        /* 2*pi */
#define ONE_OVER_TWOPI  (0.15915494309189535682609381638)
#define PI_2            (1.5707963267948966192313216916397514420986)        /* pi/2 */
#define TWO_OVER_PI     (0.636619772367581332267629550188)
#define LN2             (0.6931471805599453094172321214581765680755)        /* ln(2) */
#define ONE_OVER_LN2    (1.44269504088896333066907387547)
#define LN10            (2.3025850929940456840179914546843642076011)        /* ln(10) */
#define ONE_OVER_LN10   (0.43429448190325177635683940025)
#define ROOT2           (1.4142135623730950488016887242096980785697)        /* sqrt(2) */
#define ONE_OVER_ROOT2  (0.707106781186547438494264988549)

#define DB_LOG2_ENERGY          (3.01029995663981154631945610163)           /* dB = DB_LOG2_ENERGY*__log2(energy) */
#define DB_LOG2_AMPL            (6.02059991327962309263891220326)           /* dB = DB_LOG2_AMPL*__log2(amplitude) */
#define ONE_OVER_DB_LOG2_AMPL   (0.16609640474436811218256075335)           /* amplitude = __exp2(ONE_OVER_DB_LOG2_AMPL*dB) */

#define LONG_OFFSET     4096L
#define FLOAT_OFFSET    4096.0



float   __sqrt(float x);

float   __log2(float x);
float   __exp2(float x);

float   __log(float x);
float   __exp(float x);

float   __pow(float x, float y);

float   __sin_pi(float x);
float   __cos_pi(float x);

float   __sin(float x);
float   __cos(float x);
float   __tan(float x);

float   __atan(float x);
float   __asin(float x);
float   __acos(float x);

float   __arg(float Imag, float Real);

float   __poly(float *a, int order, float x);
float   __map(float *f, float scaler, float x);
float   __discreteMap(float *f, float scaler, float x);

unsigned long __random();

#endif




//
//    FILE: __functions.c
//
//    fast and approximate transcendental functions
//
//    copyright (c) 2004  Robert Bristow-Johnson
//
//    rbj@audioimagination.com
//

#define STD_MATH_LIB 0

#include "__functions.h"

#if STD_MATH_LIB
#include "math.h"   // angle brackets don't work with SE markup
#endif




float   __sqrt(register float x)
    {
#if STD_MATH_LIB
    return (float) sqrt((double)x);
#else
    if (x &gt; 5.877471754e-39)
        {
        register float accumulator, xPower;
        register long intPart;
        register union {float f; long i;} xBits;

        xBits.f = x;

        intPart = ((xBits.i)&gt;&gt;23);                  /* get biased exponent */
        intPart -= 127;                             /* unbias it */

        x = (float)(xBits.i &amp; 0x007FFFFF);          /* mask off exponent leaving 0x800000*(mantissa - 1) */
        x *= 1.192092895507812e-07;                 /* divide by 0x800000 */

        accumulator =  1.0 + 0.49959804148061*x;
        xPower = x*x;
        accumulator += -0.12047308243453*xPower;
        xPower *= x;
        accumulator += 0.04585425015501*xPower;
        xPower *= x;
        accumulator += -0.01076564682800*xPower;

        if (intPart &amp; 0x00000001)
            {
            accumulator *= ROOT2;                   /* an odd input exponent means an extra sqrt(2) in the output */
            }

        xBits.i = intPart &gt;&gt; 1;                     /* divide exponent by 2, lose LSB */
        xBits.i += 127;                             /* rebias exponent */
        xBits.i &lt;&lt;= 23;                             /* move biased exponent into exponent bits */

        return accumulator * xBits.f;
        }
     else
        {
        return 0.0;
        }
#endif
    }




float   __log2(register float x)
    {
#if STD_MATH_LIB
    return (float) (ONE_OVER_LN2*log((double)x));
#else
    if (x &gt; 5.877471754e-39)
        {
        register float accumulator, xPower;
        register long intPart;

        register union {float f; long i;} xBits;

        xBits.f = x;

        intPart = ((xBits.i)&gt;&gt;23);                  /* get biased exponent */
        intPart -= 127;                             /* unbias it */

        x = (float)(xBits.i &amp; 0x007FFFFF);          /* mask off exponent leaving 0x800000*(mantissa - 1) */
        x *= 1.192092895507812e-07;                 /* divide by 0x800000 */

        accumulator = 1.44254494359510*x;
        xPower = x*x;
        accumulator += -0.71814525675041*xPower;
        xPower *= x;
        accumulator += 0.45754919692582*xPower;
        xPower *= x;
        accumulator += -0.27790534462866*xPower;
        xPower *= x;
        accumulator += 0.12179791068782*xPower;
        xPower *= x;
        accumulator += -0.02584144982967*xPower;

        return accumulator + (float)intPart;
        }
     else
        {
        return -HUGE;
        }
#endif
    }


float   __exp2(register float x)
    {
#if STD_MATH_LIB
    return (float) exp(LN2*(double)x);
#else
    if (x &gt;= -127.0)
        {
        register float accumulator, xPower;
        register union {float f; long i;} xBits;

        xBits.i = (long)(x + FLOAT_OFFSET) - LONG_OFFSET;       /* integer part */
        x -= (float)(xBits.i);                                  /* fractional part */

        accumulator = 1.0 + 0.69303212081966*x;
        xPower = x*x;
        accumulator += 0.24137976293709*xPower;
        xPower *= x;
        accumulator += 0.05203236900844*xPower;
        xPower *= x;
        accumulator += 0.01355574723481*xPower;

        xBits.i += 127;                                         /* bias integer part */
        xBits.i &lt;&lt;= 23;                                         /* move biased int part into exponent bits */

        return accumulator * xBits.f;
        }
     else
        {
        return 0.0;
        }
#endif
    }


float   __log(register float x)
    {
#if STD_MATH_LIB
    return (float) log((double)x);
#else
    return LN2*__log2(x);
#endif
    }

float   __exp(register float x)
    {
#if STD_MATH_LIB
    return (float) exp((double)x);
#else
    return __exp2(ONE_OVER_LN2*x);
#endif
    }

float   __pow(float x, float y)
    {
#if STD_MATH_LIB
    return (float) pow((double)x, (double)y);
#else
    return __exp2(y*__log2(x));
#endif
    }




float   __sin_pi(register float x)
    {
#if STD_MATH_LIB
    return (float) sin(PI*(double)x);
#else
    register float accumulator, xPower, xSquared;

    register long evenIntPart = ((long)(0.5*x + 1024.5) - 1024)&lt;&lt;1;
    x -= (float)evenIntPart;

    xSquared = x*x;
    accumulator = 3.14159265358979*x;
    xPower = xSquared*x;
    accumulator += -5.16731953364340*xPower;
    xPower *= xSquared;
    accumulator += 2.54620566822659*xPower;
    xPower *= xSquared;
    accumulator += -0.586027023087261*xPower;
    xPower *= xSquared;
    accumulator += 0.06554823491427*xPower;

    return accumulator;
#endif
    }


float   __cos_pi(register float x)
    {
#if STD_MATH_LIB
    return (float) cos(PI*(double)x);
#else
    register float accumulator, xPower, xSquared;

    register long evenIntPart = ((long)(0.5*x + 1024.5) - 1024)&lt;&lt;1;
    x -= (float)evenIntPart;

    xSquared = x*x;
    accumulator = 1.57079632679490*x;                       /* series for sin(PI/2*x) */
    xPower = xSquared*x;
    accumulator += -0.64596406188166*xPower;
    xPower *= xSquared;
    accumulator += 0.07969158490912*xPower;
    xPower *= xSquared;
    accumulator += -0.00467687997706*xPower;
    xPower *= xSquared;
    accumulator += 0.00015303015470*xPower;

    return 1.0 - 2.0*accumulator*accumulator;               /* cos(w) = 1 - 2*(sin(w/2))^2 */
#endif
    }


float   __sin(register float x)
    {
#if STD_MATH_LIB
    return (float) sin((double)x);
#else
    x *= ONE_OVER_PI;
    return __sin_pi(x);
#endif
    }

float   __cos(register float x)
    {
#if STD_MATH_LIB
    return (float) cos((double)x);
#else
    x *= ONE_OVER_PI;
    return __cos_pi(x);
#endif
    }

float   __tan(register float x)
    {
#if STD_MATH_LIB
    return (float) tan((double)x);
#else
    x *= ONE_OVER_PI;
    return __sin_pi(x)/__cos_pi(x);
#endif
    }




float   __atan(register float x)
    {
#if STD_MATH_LIB
    return (float) atan((double)x);
#else
    register float accumulator, xPower, xSquared, offset;

    offset = 0.0;

    if (x &lt; -1.0)
        {
        offset = -PI_2;
        x = -1.0/x;
        }
     else if (x &gt; 1.0)
        {
        offset = PI_2;
        x = -1.0/x;
        }
    xSquared = x*x;
    accumulator = 1.0;
    xPower = xSquared;
    accumulator += 0.33288950512027*xPower;
    xPower *= xSquared;
    accumulator += -0.08467922817644*xPower;
    xPower *= xSquared;
    accumulator += 0.03252232640125*xPower;
    xPower *= xSquared;
    accumulator += -0.00749305860992*xPower;

    return offset + x/accumulator;
#endif
    }


float   __asin(register float x)
    {
#if STD_MATH_LIB
    return (float) asin((double)x);
#else
    return __atan(x/__sqrt(1.0 - x*x));
#endif
    }

float   __acos(register float x)
    {
#if STD_MATH_LIB
    return (float) acos((double)x);
#else
    return __atan(__sqrt(1.0 - x*x)/x);
#endif
    }


float   __arg(float Imag, float Real)
    {
#if STD_MATH_LIB
    return (float) atan2((double)Imag, (double)Real);
#else
    register float accumulator, xPower, xSquared, offset, x;

    if (Imag &gt; 0.0)
        {
        if (Imag &lt;= -Real)
            {
            offset = PI;
            x = Imag/Real;
            }
         else if (Imag &gt; Real)
            {
            offset = PI_2;
            x = -Real/Imag;
            }
         else
            {
            offset = 0.0;
            x = Imag/Real;
            }
        }
     else
        {
        if (Imag &gt;= Real)
            {
            offset = -PI;
            x = Imag/Real;
            }
         else if (Imag &lt; -Real)
            {
            offset = -PI_2;
            x = -Real/Imag;
            }
         else
            {
            offset = 0.0;
            x = Imag/Real;
            }
        }

    xSquared = x*x;
    accumulator = 1.0;
    xPower = xSquared;
    accumulator += 0.33288950512027*xPower;
    xPower *= xSquared;
    accumulator += -0.08467922817644*xPower;
    xPower *= xSquared;
    accumulator += 0.03252232640125*xPower;
    xPower *= xSquared;
    accumulator += -0.00749305860992*xPower;

    return offset + x/accumulator;
#endif
    }




float   __poly(float *a, int order, float x)
    {
    register float accumulator = 0.0, xPower;
    register int n;

    accumulator = a[0];
    xPower = x;
    for (n=1; n&lt;=order; n++)
        {
        accumulator += a[n]*xPower;
        xPower *= x;
        }

    return accumulator;
    }


float   __map(float *f, float scaler, float x)
    {
    register long i;

    x *= scaler;

    i = (long)(x + FLOAT_OFFSET) - LONG_OFFSET;         /* round down without floor() */

    return f[i] + (f[i+1] - f[i])*(x - (float)i);       /* linear interpolate between points */
    }


float   __discreteMap(float *f, float scaler, float x)
    {
    register long i;

    x *= scaler;

    i = (long)(x + (FLOAT_OFFSET+0.5)) - LONG_OFFSET;   /* round to nearest */

    return f[i];
    }


unsigned long __random()
    {
    static unsigned long seed0 = 0x5B7A2775, seed1 = 0x80C7169F;

    seed0 += seed1;
    seed1 += seed0;

    return seed1;
    }