e_powf.c

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/* e_powf.c -- float version of e_pow.c.
 * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
 */

/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice 
 * is preserved.
 * ====================================================
 */

#include "math.h"
#include "math_private.h"

static const volatile float huge = 1.0e+30, tiny = 1.0e-30;

static const float
bp[] = {1.0, 1.5,},
dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
zero    =  0.0,
one =  1.0,
two =  2.0,
two24   =  16777216.0,  /* 0x4b800000 */
    /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
L1  =  6.0000002384e-01, /* 0x3f19999a */
L2  =  4.2857143283e-01, /* 0x3edb6db7 */
L3  =  3.3333334327e-01, /* 0x3eaaaaab */
L4  =  2.7272811532e-01, /* 0x3e8ba305 */
L5  =  2.3066075146e-01, /* 0x3e6c3255 */
L6  =  2.0697501302e-01, /* 0x3e53f142 */
P1   =  1.6666667163e-01, /* 0x3e2aaaab */
P2   = -2.7777778450e-03, /* 0xbb360b61 */
P3   =  6.6137559770e-05, /* 0x388ab355 */
P4   = -1.6533901999e-06, /* 0xb5ddea0e */
P5   =  4.1381369442e-08, /* 0x3331bb4c */
lg2  =  6.9314718246e-01, /* 0x3f317218 */
lg2_h  =  6.93145752e-01, /* 0x3f317200 */
lg2_l  =  1.42860654e-06, /* 0x35bfbe8c */
ovt =  4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
cp    =  9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
cp_h  =  9.6179199219e-01, /* 0x3f763800 =head of cp */
cp_l  =  4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
ivln2    =  1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
ivln2_h  =  1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/

float
powf(float x, float y)
{
    float z,ax,z_h,z_l,p_h,p_l;
    float yy1,t1,t2,r,s,t,u,v,w;
    int32_t i,j,k,yisint,n;
    int32_t hx,hy,ix,iy,is;

    GET_FLOAT_WORD(hx,x);
    GET_FLOAT_WORD(hy,y);
    ix = hx&0x7fffffff;  iy = hy&0x7fffffff;

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

    /* +-NaN return x+y */
    if(ix > 0x7f800000 ||
       iy > 0x7f800000)
        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>=0x4b800000) yisint = 2; /* even integer y */
        else if(iy>=0x3f800000) {
        k = (iy>>23)-0x7f;     /* exponent */
        j = iy>>(23-k);
        if((j<<(23-k))==iy) yisint = 2-(j&1);
        }       
    } 

    /* special value of y */
    if (iy==0x7f800000) {   /* y is +-inf */
        if (ix==0x3f800000)
            return  y - y;  /* inf**+-1 is NaN */
        else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
            return (hy>=0)? y: zero;
        else            /* (|x|<1)**-,+inf = inf,0 */
            return (hy<0)?-y: zero;
    } 
    if(iy==0x3f800000) {    /* y is  +-1 */
        if(hy<0) return one/x; else return x;
    }
    if(hy==0x40000000) return x*x; /* y is  2 */
    if(hy==0x3f000000) {    /* y is  0.5 */
        if(hx>=0)   /* x >= +0 */
        return sqrtf(x);    
    }

    ax   = fabsf(x);
    /* special value of x */
    if(ix==0x7f800000||ix==0||ix==0x3f800000){
        z = ax;         /*x is +-0,+-inf,+-1*/
        if(hy<0) z = one/z; /* z = (1/|x|) */
        if(hx<0) {
        if(((ix-0x3f800000)|yisint)==0) {
            z = (z-z)/(z-z); /* (-1)**non-int is NaN */
        } else if(yisint==1) 
            z = -z;     /* (x<0)**odd = -(|x|**odd) */
        }
        return z;
    }
    
    /* (x<0)**(non-int) is NaN */
    if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);

    /* |y| is huge */
    if(iy>0x4d000000) { /* if |y| > 2**27 */
    /* over/underflow if x is not close to one */
        if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny;
        if(ix>0x3f800007) return (hy>0)? huge*huge: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)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
        u = ivln2_h*t;  /* ivln2_h has 16 sig. bits */
        v = t*ivln2_l-w*ivln2;
        t1 = u+v;
        GET_FLOAT_WORD(is,t1);
        SET_FLOAT_WORD(t1,is&0xfffff000);
        t2 = v-(t1-u);
    } else {
        float s2,s_h,s_l,t_h,t_l;
        n = 0;
    /* take care subnormal number */
        if(ix<0x00800000)
        {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
        n  += ((ix)>>23)-0x7f;
        j  = ix&0x007fffff;
    /* determine interval */
        ix = j|0x3f800000;      /* normalize ix */
        if(j<=0x1cc471) k=0;    /* |x|<sqrt(3/2) */
        else if(j<0x5db3d7) k=1;    /* |x|<sqrt(3)   */
        else {k=0;n+=1;ix -= 0x00800000;}
        SET_FLOAT_WORD(ax,ix);

    /* compute s = 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]);
        s = u*v;
        s_h = s;
        GET_FLOAT_WORD(is,s_h);
        SET_FLOAT_WORD(s_h,is&0xfffff000);
    /* t_h=ax+bp[k] High */
        SET_FLOAT_WORD(t_h,((ix>>1)|0x20000000)+0x0040000+(k<<21));
        t_l = ax - (t_h-bp[k]);
        s_l = v*((u-s_h*t_h)-s_h*t_l);
    /* compute log(ax) */
        s2 = s*s;
        r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
        r += s_l*(s_h+s);
        s2  = s_h*s_h;
        t_h = (float)3.0+s2+r;
        GET_FLOAT_WORD(is,t_h);
        SET_FLOAT_WORD(t_h,is&0xfffff000);
        t_l = r-((t_h-(float)3.0)-s2);
    /* u+v = s*(1+...) */
        u = s_h*t_h;
        v = s_l*t_h+t_l*s;
    /* 2/(3log2)*(s+...) */
        p_h = u+v;
        GET_FLOAT_WORD(is,p_h);
        SET_FLOAT_WORD(p_h,is&0xfffff000);
        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) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
        t = (float)n;
        t1 = (((z_h+z_l)+dp_h[k])+t);
        GET_FLOAT_WORD(is,t1);
        SET_FLOAT_WORD(t1,is&0xfffff000);
        t2 = z_l-(((t1-t)-dp_h[k])-z_h);
    }

    s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
    if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
        s = -one;   /* (-ve)**(odd int) */

    /* split up y into yy1+y2 and compute (yy1+y2)*(t1+t2) */
    GET_FLOAT_WORD(is,y);
    SET_FLOAT_WORD(yy1,is&0xfffff000);
    p_l = (y-yy1)*t1+y*t2;
    p_h = yy1*t1;
    z = p_l+p_h;
    GET_FLOAT_WORD(j,z);
    if (j>0x43000000)               /* if z > 128 */
        return s*huge*huge;             /* overflow */
    else if (j==0x43000000) {           /* if z == 128 */
        if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
    }
    else if ((j&0x7fffffff)>0x43160000)     /* z <= -150 */
        return s*tiny*tiny;             /* underflow */
    else if (j==0xc3160000){            /* z == -150 */
        if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
    }
    /*
     * compute 2**(p_h+p_l)
     */
    i = j&0x7fffffff;
    k = (i>>23)-0x7f;
    n = 0;
    if(i>0x3f000000) {      /* if |z| > 0.5, set n = [z+0.5] */
        n = j+(0x00800000>>(k+1));
        k = ((n&0x7fffffff)>>23)-0x7f;  /* new k for n */
        SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
        n = ((n&0x007fffff)|0x00800000)>>(23-k);
        if(j<0) n = -n;
        p_h -= t;
    } 
    t = p_l+p_h;
    GET_FLOAT_WORD(is,t);
    SET_FLOAT_WORD(t,is&0xfffff000);
    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);
    GET_FLOAT_WORD(j,z);
    j += (n<<23);
    if((j>>23)<=0) z = scalbnf(z,n);    /* subnormal output */
    else SET_FLOAT_WORD(z,j);
    return s*z;
}