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glibc  2.9
e_powf.c
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00001 /* e_powf.c -- float version of e_pow.c.
00002  * Conversion to float by Ian Lance Taylor, Cygnus Support, ian@cygnus.com.
00003  */
00004 
00005 /*
00006  * ====================================================
00007  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
00008  *
00009  * Developed at SunPro, a Sun Microsystems, Inc. business.
00010  * Permission to use, copy, modify, and distribute this
00011  * software is freely granted, provided that this notice
00012  * is preserved.
00013  * ====================================================
00014  */
00015 
00016 #if defined(LIBM_SCCS) && !defined(lint)
00017 static char rcsid[] = "$NetBSD: e_powf.c,v 1.7 1996/04/08 15:43:44 phil Exp $";
00018 #endif
00019 
00020 #include "math.h"
00021 #include "math_private.h"
00022 
00023 static const float huge = 1.0e+30, tiny = 1.0e-30;
00024 
00025 #ifdef __STDC__
00026 static const float
00027 #else
00028 static float
00029 #endif
00030 bp[] = {1.0, 1.5,},
00031 dp_h[] = { 0.0, 5.84960938e-01,}, /* 0x3f15c000 */
00032 dp_l[] = { 0.0, 1.56322085e-06,}, /* 0x35d1cfdc */
00033 zero    =  0.0,
00034 one    =  1.0,
00035 two    =  2.0,
00036 two24  =  16777216.0,       /* 0x4b800000 */
00037        /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
00038 L1  =  6.0000002384e-01, /* 0x3f19999a */
00039 L2  =  4.2857143283e-01, /* 0x3edb6db7 */
00040 L3  =  3.3333334327e-01, /* 0x3eaaaaab */
00041 L4  =  2.7272811532e-01, /* 0x3e8ba305 */
00042 L5  =  2.3066075146e-01, /* 0x3e6c3255 */
00043 L6  =  2.0697501302e-01, /* 0x3e53f142 */
00044 P1   =  1.6666667163e-01, /* 0x3e2aaaab */
00045 P2   = -2.7777778450e-03, /* 0xbb360b61 */
00046 P3   =  6.6137559770e-05, /* 0x388ab355 */
00047 P4   = -1.6533901999e-06, /* 0xb5ddea0e */
00048 P5   =  4.1381369442e-08, /* 0x3331bb4c */
00049 lg2  =  6.9314718246e-01, /* 0x3f317218 */
00050 lg2_h  =  6.93145752e-01, /* 0x3f317200 */
00051 lg2_l  =  1.42860654e-06, /* 0x35bfbe8c */
00052 ovt =  4.2995665694e-08, /* -(128-log2(ovfl+.5ulp)) */
00053 cp    =  9.6179670095e-01, /* 0x3f76384f =2/(3ln2) */
00054 cp_h  =  9.6179199219e-01, /* 0x3f763800 =head of cp */
00055 cp_l  =  4.7017383622e-06, /* 0x369dc3a0 =tail of cp_h */
00056 ivln2    =  1.4426950216e+00, /* 0x3fb8aa3b =1/ln2 */
00057 ivln2_h  =  1.4426879883e+00, /* 0x3fb8aa00 =16b 1/ln2*/
00058 ivln2_l  =  7.0526075433e-06; /* 0x36eca570 =1/ln2 tail*/
00059 
00060 #ifdef __STDC__
00061        float __ieee754_powf(float x, float y)
00062 #else
00063        float __ieee754_powf(x,y)
00064        float x, y;
00065 #endif
00066 {
00067        float z,ax,z_h,z_l,p_h,p_l;
00068        float y1,t1,t2,r,s,t,u,v,w;
00069        int32_t i,j,k,yisint,n;
00070        int32_t hx,hy,ix,iy,is;
00071 
00072        GET_FLOAT_WORD(hx,x);
00073        GET_FLOAT_WORD(hy,y);
00074        ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
00075 
00076     /* y==zero: x**0 = 1 */
00077        if(iy==0) return one;
00078 
00079     /* x==+-1 */
00080        if(x == 1.0) return one;
00081        if(x == -1.0 && isinf(y)) return one;
00082 
00083     /* +-NaN return x+y */
00084        if(ix > 0x7f800000 ||
00085           iy > 0x7f800000)
00086               return x+y;
00087 
00088     /* determine if y is an odd int when x < 0
00089      * yisint = 0    ... y is not an integer
00090      * yisint = 1    ... y is an odd int
00091      * yisint = 2    ... y is an even int
00092      */
00093        yisint  = 0;
00094        if(hx<0) {
00095            if(iy>=0x4b800000) yisint = 2; /* even integer y */
00096            else if(iy>=0x3f800000) {
00097               k = (iy>>23)-0x7f;      /* exponent */
00098               j = iy>>(23-k);
00099               if((j<<(23-k))==iy) yisint = 2-(j&1);
00100            }
00101        }
00102 
00103     /* special value of y */
00104        if (iy==0x7f800000) {       /* y is +-inf */
00105            if (ix==0x3f800000)
00106                return  y - y;      /* inf**+-1 is NaN */
00107            else if (ix > 0x3f800000)/* (|x|>1)**+-inf = inf,0 */
00108                return (hy>=0)? y: zero;
00109            else                    /* (|x|<1)**-,+inf = inf,0 */
00110                return (hy<0)?-y: zero;
00111        }
00112        if(iy==0x3f800000) { /* y is  +-1 */
00113            if(hy<0) return one/x; else return x;
00114        }
00115        if(hy==0x40000000) return x*x; /* y is  2 */
00116        if(hy==0x3f000000) { /* y is  0.5 */
00117            if(hx>=0) /* x >= +0 */
00118            return __ieee754_sqrtf(x);
00119        }
00120 
00121        ax   = fabsf(x);
00122     /* special value of x */
00123        if(ix==0x7f800000||ix==0||ix==0x3f800000){
00124            z = ax;                 /*x is +-0,+-inf,+-1*/
00125            if(hy<0) z = one/z;     /* z = (1/|x|) */
00126            if(hx<0) {
00127               if(((ix-0x3f800000)|yisint)==0) {
00128                   z = (z-z)/(z-z); /* (-1)**non-int is NaN */
00129               } else if(yisint==1)
00130                   z = -z;          /* (x<0)**odd = -(|x|**odd) */
00131            }
00132            return z;
00133        }
00134 
00135     /* (x<0)**(non-int) is NaN */
00136        if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
00137 
00138     /* |y| is huge */
00139        if(iy>0x4d000000) { /* if |y| > 2**27 */
00140        /* over/underflow if x is not close to one */
00141            if(ix<0x3f7ffff8) return (hy<0)? huge*huge:tiny*tiny;
00142            if(ix>0x3f800007) return (hy>0)? huge*huge:tiny*tiny;
00143        /* now |1-x| is tiny <= 2**-20, suffice to compute
00144           log(x) by x-x^2/2+x^3/3-x^4/4 */
00145            t = x-1;         /* t has 20 trailing zeros */
00146            w = (t*t)*((float)0.5-t*((float)0.333333333333-t*(float)0.25));
00147            u = ivln2_h*t;   /* ivln2_h has 16 sig. bits */
00148            v = t*ivln2_l-w*ivln2;
00149            t1 = u+v;
00150            GET_FLOAT_WORD(is,t1);
00151            SET_FLOAT_WORD(t1,is&0xfffff000);
00152            t2 = v-(t1-u);
00153        } else {
00154            float s2,s_h,s_l,t_h,t_l;
00155            n = 0;
00156        /* take care subnormal number */
00157            if(ix<0x00800000)
00158               {ax *= two24; n -= 24; GET_FLOAT_WORD(ix,ax); }
00159            n  += ((ix)>>23)-0x7f;
00160            j  = ix&0x007fffff;
00161        /* determine interval */
00162            ix = j|0x3f800000;             /* normalize ix */
00163            if(j<=0x1cc471) k=0;    /* |x|<sqrt(3/2) */
00164            else if(j<0x5db3d7) k=1;       /* |x|<sqrt(3)   */
00165            else {k=0;n+=1;ix -= 0x00800000;}
00166            SET_FLOAT_WORD(ax,ix);
00167 
00168        /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
00169            u = ax-bp[k];           /* bp[0]=1.0, bp[1]=1.5 */
00170            v = one/(ax+bp[k]);
00171            s = u*v;
00172            s_h = s;
00173            GET_FLOAT_WORD(is,s_h);
00174            SET_FLOAT_WORD(s_h,is&0xfffff000);
00175        /* t_h=ax+bp[k] High */
00176            SET_FLOAT_WORD(t_h,((ix>>1)|0x20000000)+0x0040000+(k<<21));
00177            t_l = ax - (t_h-bp[k]);
00178            s_l = v*((u-s_h*t_h)-s_h*t_l);
00179        /* compute log(ax) */
00180            s2 = s*s;
00181            r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
00182            r += s_l*(s_h+s);
00183            s2  = s_h*s_h;
00184            t_h = (float)3.0+s2+r;
00185            GET_FLOAT_WORD(is,t_h);
00186            SET_FLOAT_WORD(t_h,is&0xfffff000);
00187            t_l = r-((t_h-(float)3.0)-s2);
00188        /* u+v = s*(1+...) */
00189            u = s_h*t_h;
00190            v = s_l*t_h+t_l*s;
00191        /* 2/(3log2)*(s+...) */
00192            p_h = u+v;
00193            GET_FLOAT_WORD(is,p_h);
00194            SET_FLOAT_WORD(p_h,is&0xfffff000);
00195            p_l = v-(p_h-u);
00196            z_h = cp_h*p_h;         /* cp_h+cp_l = 2/(3*log2) */
00197            z_l = cp_l*p_h+p_l*cp+dp_l[k];
00198        /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
00199            t = (float)n;
00200            t1 = (((z_h+z_l)+dp_h[k])+t);
00201            GET_FLOAT_WORD(is,t1);
00202            SET_FLOAT_WORD(t1,is&0xfffff000);
00203            t2 = z_l-(((t1-t)-dp_h[k])-z_h);
00204        }
00205 
00206        s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
00207        if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
00208            s = -one; /* (-ve)**(odd int) */
00209 
00210     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
00211        GET_FLOAT_WORD(is,y);
00212        SET_FLOAT_WORD(y1,is&0xfffff000);
00213        p_l = (y-y1)*t1+y*t2;
00214        p_h = y1*t1;
00215        z = p_l+p_h;
00216        GET_FLOAT_WORD(j,z);
00217        if (j>0x43000000)                         /* if z > 128 */
00218            return s*huge*huge;                          /* overflow */
00219        else if (j==0x43000000) {                 /* if z == 128 */
00220            if(p_l+ovt>z-p_h) return s*huge*huge; /* overflow */
00221        }
00222        else if ((j&0x7fffffff)>0x43160000)              /* z <= -150 */
00223            return s*tiny*tiny;                          /* underflow */
00224        else if ((u_int32_t) j==0xc3160000){             /* z == -150 */
00225            if(p_l<=z-p_h) return s*tiny*tiny;           /* underflow */
00226        }
00227     /*
00228      * compute 2**(p_h+p_l)
00229      */
00230        i = j&0x7fffffff;
00231        k = (i>>23)-0x7f;
00232        n = 0;
00233        if(i>0x3f000000) {          /* if |z| > 0.5, set n = [z+0.5] */
00234            n = j+(0x00800000>>(k+1));
00235            k = ((n&0x7fffffff)>>23)-0x7f; /* new k for n */
00236            SET_FLOAT_WORD(t,n&~(0x007fffff>>k));
00237            n = ((n&0x007fffff)|0x00800000)>>(23-k);
00238            if(j<0) n = -n;
00239            p_h -= t;
00240        }
00241        t = p_l+p_h;
00242        GET_FLOAT_WORD(is,t);
00243        SET_FLOAT_WORD(t,is&0xfffff000);
00244        u = t*lg2_h;
00245        v = (p_l-(t-p_h))*lg2+t*lg2_l;
00246        z = u+v;
00247        w = v-(z-u);
00248        t  = z*z;
00249        t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
00250        r  = (z*t1)/(t1-two)-(w+z*w);
00251        z  = one-(r-z);
00252        GET_FLOAT_WORD(j,z);
00253        j += (n<<23);
00254        if((j>>23)<=0) z = __scalbnf(z,n); /* subnormal output */
00255        else SET_FLOAT_WORD(z,j);
00256        return s*z;
00257 }