What's the algorithm behind Math.pow() in Java

Question

I just started with Java and as a first project I am writing a program that finds roots(cube root in this case) of a given number. Presently I am trying out Newton-Ralphson

Answer 1

This is a fun question. If you look into the the source code for Java's Math class, you will find that it calls StrictMath.pow(double1, double2), and StrictMath's signature is public static native double pow(double a, double b);

So, in the end, it is a truly native call that might differ depending on the platform. However, there is an implementation somewhere, and it isn't very easy to look at. Here is the description of the function and the code for the function itself:

Note

Looking through the math, trying to understand it might inevitably lead to even more questions. But, by searching through this Github on Java Math Function Source Code and glancing out the mathematical summaries, you can definitely understand the native functions better. Happy Exploring :)

Method Description

Method:  Let x =  2   * (1+f)
      1. Compute and return log2(x) in two pieces:
              log2(x) = w1 + w2,
         where w1 has 53-24 = 29 bit trailing zeros.
      2. Perform y*log2(x) = n+y' by simulating muti-precision
         arithmetic, where |y'|<=0.5.
      3. Return x**y = 2**n*exp(y'*log2)

Special Cases

      1.  (anything) ** 0  is 1
      2.  (anything) ** 1  is itself
      3.  (anything) ** NAN is NAN
      4.  NAN ** (anything except 0) is NAN
      5.  +-(|x| > 1) **  +INF is +INF
      6.  +-(|x| > 1) **  -INF is +0
      7.  +-(|x| < 1) **  +INF is +0
      8.  +-(|x| < 1) **  -INF is +INF
      9.  +-1         ** +-INF is NAN
      10. +0 ** (+anything except 0, NAN)               is +0
      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
      12. +0 ** (-anything except 0, NAN)               is +INF
      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
      15. +INF ** (+anything except 0,NAN) is +INF
      16. +INF ** (-anything except 0,NAN) is +0
      17. -INF ** (anything)  = -0 ** (-anything)
      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
      19. (-anything except 0 and inf) ** (non-integer) is NAN

Accuracy

       pow(x,y) returns x**y nearly rounded. In particular
                      pow(integer,integer)
       always returns the correct integer provided it is
       representable.

Source Code

#ifdef __STDC__
        double __ieee754_pow(double x, double y)
#else
        double __ieee754_pow(x,y)
        double x, y;
#endif
{
        double z,ax,z_h,z_l,p_h,p_l;
        double y1,t1,t2,r,s,t,u,v,w;
        int i0,i1,i,j,k,yisint,n;
        int hx,hy,ix,iy;
        unsigned lx,ly;

        i0 = ((*(int*)&one)>>29)^1; i1=1-i0;
        hx = __HI(x); lx = __LO(x);
        hy = __HI(y); ly = __LO(y);
        ix = hx&0x7fffffff;  iy = hy&0x7fffffff;

    /* y==zero: x**0 = 1 */
        if((iy|ly)==0) return one;

    /* +-NaN return x+y */
        if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
           iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
                return x+y;

    /* determine if y is an odd int when x < 0
     * yisint = 0       ... y is not an integer
     * yisint = 1       ... y is an odd int
     * yisint = 2       ... y is an even int
     */
        yisint  = 0;
        if(hx<0) {
            if(iy>=0x43400000) yisint = 2; /* even integer y */
            else if(iy>=0x3ff00000) {
                k = (iy>>20)-0x3ff;        /* exponent */
                if(k>20) {
                    j = ly>>(52-k);
                    if((j<<(52-k))==ly) yisint = 2-(j&1);
                } else if(ly==0) {
                    j = iy>>(20-k);
                    if((j<<(20-k))==iy) yisint = 2-(j&1);
                }
            }
        }

    /* special value of y */
        if(ly==0) {
            if (iy==0x7ff00000) {       /* y is +-inf */
                if(((ix-0x3ff00000)|lx)==0)
                    return  y - y;      /* inf**+-1 is NaN */
                else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
                    return (hy>=0)? y: zero;
                else                    /* (|x|<1)**-,+inf = inf,0 */
                    return (hy<0)?-y: zero;
            }
            if(iy==0x3ff00000) {        /* y is  +-1 */
                if(hy<0) return one/x; else return x;
            }
            if(hy==0x40000000) return x*x; /* y is  2 */
            if(hy==0x3fe00000) {        /* y is  0.5 */
                if(hx>=0)       /* x >= +0 */
                return sqrt(x);
            }
        }

        ax   = fabs(x);
    /* special value of x */
        if(lx==0) {
            if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
                z = ax;                 /*x is +-0,+-inf,+-1*/
                if(hy<0) z = one/z;     /* z = (1/|x|) */
                if(hx<0) {
                    if(((ix-0x3ff00000)|yisint)==0) {
                        z = (z-z)/(z-z); /* (-1)**non-int is NaN */
                    } else if(yisint==1)
                        z = -1.0*z;             /* (x<0)**odd = -(|x|**odd) */
                }
                return z;
            }
        }

        n = (hx>>31)+1;

    /* (x<0)**(non-int) is NaN */
        if((n|yisint)==0) return (x-x)/(x-x);

        s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
        if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */

    /* |y| is huge */
        if(iy>0x41e00000) { /* if |y| > 2**31 */
            if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
                if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
                if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
            }
        /* over/underflow if x is not close to one */
            if(ix<0x3fefffff) return (hy<0)? s*huge*huge:s*tiny*tiny;
            if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*tiny*tiny;
        /* now |1-x| is tiny <= 2**-20, suffice to compute
           log(x) by x-x^2/2+x^3/3-x^4/4 */
            t = ax-one;         /* t has 20 trailing zeros */
            w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
            u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
            v = t*ivln2_l-w*ivln2;
            t1 = u+v;
            __LO(t1) = 0;
            t2 = v-(t1-u);
        } else {
            double ss,s2,s_h,s_l,t_h,t_l;
            n = 0;
        /* take care subnormal number */
            if(ix<0x00100000)
                {ax *= two53; n -= 53; ix = __HI(ax); }
            n  += ((ix)>>20)-0x3ff;
            j  = ix&0x000fffff;
        /* determine interval */
            ix = j|0x3ff00000;          /* normalize ix */
            if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
            else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
            else {k=0;n+=1;ix -= 0x00100000;}
            __HI(ax) = ix;

        /* compute ss = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
            u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
            v = one/(ax+bp[k]);
            ss = u*v;
            s_h = ss;
            __LO(s_h) = 0;
        /* t_h=ax+bp[k] High */
            t_h = zero;
            __HI(t_h)=((ix>>1)|0x20000000)+0x00080000+(k<<18);
            t_l = ax - (t_h-bp[k]);
            s_l = v*((u-s_h*t_h)-s_h*t_l);
        /* compute log(ax) */
            s2 = ss*ss;
            r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
            r += s_l*(s_h+ss);
            s2  = s_h*s_h;
            t_h = 3.0+s2+r;
            __LO(t_h) = 0;
            t_l = r-((t_h-3.0)-s2);
        /* u+v = ss*(1+...) */
            u = s_h*t_h;
            v = s_l*t_h+t_l*ss;
        /* 2/(3log2)*(ss+...) */
            p_h = u+v;
            __LO(p_h) = 0;
            p_l = v-(p_h-u);
            z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
            z_l = cp_l*p_h+p_l*cp+dp_l[k];
        /* log2(ax) = (ss+..)*2/(3*log2) = n + dp_h + z_h + z_l */
            t = (double)n;
            t1 = (((z_h+z_l)+dp_h[k])+t);
            __LO(t1) = 0;
            t2 = z_l-(((t1-t)-dp_h[k])-z_h);
        }

    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
        y1  = y;
        __LO(y1) = 0;
        p_l = (y-y1)*t1+y*t2;
        p_h = y1*t1;
        z = p_l+p_h;
        j = __HI(z);
        i = __LO(z);
        if (j>=0x40900000) {                            /* z >= 1024 */
            if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
                return s*huge*huge;                     /* overflow */
            else {
                if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
            }
        } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
            if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
                return s*tiny*tiny;             /* underflow */
            else {
                if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
            }
        }
    /*
     * compute 2**(p_h+p_l)
     */
        i = j&0x7fffffff;
        k = (i>>20)-0x3ff;
        n = 0;
        if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
            n = j+(0x00100000>>(k+1));
            k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
            t = zero;
            __HI(t) = (n&~(0x000fffff>>k));
            n = ((n&0x000fffff)|0x00100000)>>(20-k);
            if(j<0) n = -n;
            p_h -= t;
        }
        t = p_l+p_h;
        __LO(t) = 0;
        u = t*lg2_h;
        v = (p_l-(t-p_h))*lg2+t*lg2_l;
        z = u+v;
        w = v-(z-u);
        t  = z*z;
        t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
        r  = (z*t1)/(t1-two)-(w+z*w);
        z  = one-(r-z);
        j  = __HI(z);
        j += (n<<20);
        if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
        else __HI(z) += (n<<20);
        return s*z;
}