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[/] [or1k/] [trunk/] [linux/] [uClibc/] [libm/] [e_pow.c] - Blame information for rev 1765

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1 1325 phoenix
/* @(#)e_pow.c 5.1 93/09/24 */
2
/*
3
 * ====================================================
4
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
5
 *
6
 * Developed at SunPro, a Sun Microsystems, Inc. business.
7
 * Permission to use, copy, modify, and distribute this
8
 * software is freely granted, provided that this notice
9
 * is preserved.
10
 * ====================================================
11
 */
12
 
13
#if defined(LIBM_SCCS) && !defined(lint)
14
static char rcsid[] = "$NetBSD: e_pow.c,v 1.9 1995/05/12 04:57:32 jtc Exp $";
15
#endif
16
 
17
/* __ieee754_pow(x,y) return x**y
18
 *
19
 *                    n
20
 * Method:  Let x =  2   * (1+f)
21
 *      1. Compute and return log2(x) in two pieces:
22
 *              log2(x) = w1 + w2,
23
 *         where w1 has 53-24 = 29 bit trailing zeros.
24
 *      2. Perform y*log2(x) = n+y' by simulating muti-precision
25
 *         arithmetic, where |y'|<=0.5.
26
 *      3. Return x**y = 2**n*exp(y'*log2)
27
 *
28
 * Special cases:
29
 *      1.  (anything) ** 0  is 1
30
 *      2.  (anything) ** 1  is itself
31
 *      3.  (anything) ** NAN is NAN
32
 *      4.  NAN ** (anything except 0) is NAN
33
 *      5.  +-(|x| > 1) **  +INF is +INF
34
 *      6.  +-(|x| > 1) **  -INF is +0
35
 *      7.  +-(|x| < 1) **  +INF is +0
36
 *      8.  +-(|x| < 1) **  -INF is +INF
37
 *      9.  +-1         ** +-INF is NAN
38
 *      10. +0 ** (+anything except 0, NAN)               is +0
39
 *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
40
 *      12. +0 ** (-anything except 0, NAN)               is +INF
41
 *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
42
 *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
43
 *      15. +INF ** (+anything except 0,NAN) is +INF
44
 *      16. +INF ** (-anything except 0,NAN) is +0
45
 *      17. -INF ** (anything)  = -0 ** (-anything)
46
 *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
47
 *      19. (-anything except 0 and inf) ** (non-integer) is NAN
48
 *
49
 * Accuracy:
50
 *      pow(x,y) returns x**y nearly rounded. In particular
51
 *                      pow(integer,integer)
52
 *      always returns the correct integer provided it is
53
 *      representable.
54
 *
55
 * Constants :
56
 * The hexadecimal values are the intended ones for the following
57
 * constants. The decimal values may be used, provided that the
58
 * compiler will convert from decimal to binary accurately enough
59
 * to produce the hexadecimal values shown.
60
 */
61
 
62
#include "math.h"
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#include "math_private.h"
64
 
65
#ifdef __STDC__
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static const double
67
#else
68
static double
69
#endif
70
bp[] = {1.0, 1.5,},
71
dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
72
dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
73
zero    =  0.0,
74
one     =  1.0,
75
two     =  2.0,
76
two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
77
huge    =  1.0e300,
78
tiny    =  1.0e-300,
79
        /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
80
L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
81
L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
82
L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
83
L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
84
L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
85
L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
86
P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
87
P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
88
P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
89
P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
90
P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
91
lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
92
lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
93
lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
94
ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
95
cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
96
cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
97
cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
98
ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
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ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
100
ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
101
 
102
#ifdef __STDC__
103
        double __ieee754_pow(double x, double y)
104
#else
105
        double __ieee754_pow(x,y)
106
        double x, y;
107
#endif
108
{
109
        double z,ax,z_h,z_l,p_h,p_l;
110
        double y1,t1,t2,r,s,t,u,v,w;
111
        int32_t i,j,k,yisint,n;
112
        int32_t hx,hy,ix,iy;
113
        u_int32_t lx,ly;
114
 
115
        EXTRACT_WORDS(hx,lx,x);
116
        EXTRACT_WORDS(hy,ly,y);
117
        ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
118
 
119
    /* y==zero: x**0 = 1 */
120
        if((iy|ly)==0) return one;
121
 
122
    /* +-NaN return x+y */
123
        if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
124
           iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
125
                return x+y;
126
 
127
    /* determine if y is an odd int when x < 0
128
     * yisint = 0       ... y is not an integer
129
     * yisint = 1       ... y is an odd int
130
     * yisint = 2       ... y is an even int
131
     */
132
        yisint  = 0;
133
        if(hx<0) {
134
            if(iy>=0x43400000) yisint = 2; /* even integer y */
135
            else if(iy>=0x3ff00000) {
136
                k = (iy>>20)-0x3ff;        /* exponent */
137
                if(k>20) {
138
                    j = ly>>(52-k);
139
                    if((j<<(52-k))==ly) yisint = 2-(j&1);
140
                } else if(ly==0) {
141
                    j = iy>>(20-k);
142
                    if((j<<(20-k))==iy) yisint = 2-(j&1);
143
                }
144
            }
145
        }
146
 
147
    /* special value of y */
148
        if(ly==0) {
149
            if (iy==0x7ff00000) {       /* y is +-inf */
150
                if(((ix-0x3ff00000)|lx)==0)
151
                    return  y - y;      /* inf**+-1 is NaN */
152
                else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
153
                    return (hy>=0)? y: zero;
154
                else                    /* (|x|<1)**-,+inf = inf,0 */
155
                    return (hy<0)?-y: zero;
156
            }
157
            if(iy==0x3ff00000) {        /* y is  +-1 */
158
                if(hy<0) return one/x; else return x;
159
            }
160
            if(hy==0x40000000) return x*x; /* y is  2 */
161
            if(hy==0x3fe00000) {        /* y is  0.5 */
162
                if(hx>=0)        /* x >= +0 */
163
                return __ieee754_sqrt(x);
164
            }
165
        }
166
 
167
        ax   = fabs(x);
168
    /* special value of x */
169
        if(lx==0) {
170
            if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
171
                z = ax;                 /*x is +-0,+-inf,+-1*/
172
                if(hy<0) z = one/z;      /* z = (1/|x|) */
173
                if(hx<0) {
174
                    if(((ix-0x3ff00000)|yisint)==0) {
175
                        z = (z-z)/(z-z); /* (-1)**non-int is NaN */
176
                    } else if(yisint==1)
177
                        z = -z;         /* (x<0)**odd = -(|x|**odd) */
178
                }
179
                return z;
180
            }
181
        }
182
 
183
    /* (x<0)**(non-int) is NaN */
184
        if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
185
 
186
    /* |y| is huge */
187
        if(iy>0x41e00000) { /* if |y| > 2**31 */
188
            if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
189
                if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
190
                if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
191
            }
192
        /* over/underflow if x is not close to one */
193
            if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
194
            if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
195
        /* now |1-x| is tiny <= 2**-20, suffice to compute
196
           log(x) by x-x^2/2+x^3/3-x^4/4 */
197
            t = x-1;            /* t has 20 trailing zeros */
198
            w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
199
            u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
200
            v = t*ivln2_l-w*ivln2;
201
            t1 = u+v;
202
            SET_LOW_WORD(t1,0);
203
            t2 = v-(t1-u);
204
        } else {
205
            double s2,s_h,s_l,t_h,t_l;
206
            n = 0;
207
        /* take care subnormal number */
208
            if(ix<0x00100000)
209
                {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
210
            n  += ((ix)>>20)-0x3ff;
211
            j  = ix&0x000fffff;
212
        /* determine interval */
213
            ix = j|0x3ff00000;          /* normalize ix */
214
            if(j<=0x3988E) k=0;          /* |x|<sqrt(3/2) */
215
            else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
216
            else {k=0;n+=1;ix -= 0x00100000;}
217
            SET_HIGH_WORD(ax,ix);
218
 
219
        /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
220
            u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
221
            v = one/(ax+bp[k]);
222
            s = u*v;
223
            s_h = s;
224
            SET_LOW_WORD(s_h,0);
225
        /* t_h=ax+bp[k] High */
226
            t_h = zero;
227
            SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
228
            t_l = ax - (t_h-bp[k]);
229
            s_l = v*((u-s_h*t_h)-s_h*t_l);
230
        /* compute log(ax) */
231
            s2 = s*s;
232
            r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
233
            r += s_l*(s_h+s);
234
            s2  = s_h*s_h;
235
            t_h = 3.0+s2+r;
236
            SET_LOW_WORD(t_h,0);
237
            t_l = r-((t_h-3.0)-s2);
238
        /* u+v = s*(1+...) */
239
            u = s_h*t_h;
240
            v = s_l*t_h+t_l*s;
241
        /* 2/(3log2)*(s+...) */
242
            p_h = u+v;
243
            SET_LOW_WORD(p_h,0);
244
            p_l = v-(p_h-u);
245
            z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
246
            z_l = cp_l*p_h+p_l*cp+dp_l[k];
247
        /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
248
            t = (double)n;
249
            t1 = (((z_h+z_l)+dp_h[k])+t);
250
            SET_LOW_WORD(t1,0);
251
            t2 = z_l-(((t1-t)-dp_h[k])-z_h);
252
        }
253
 
254
        s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
255
        if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
256
            s = -one;/* (-ve)**(odd int) */
257
 
258
    /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
259
        y1  = y;
260
        SET_LOW_WORD(y1,0);
261
        p_l = (y-y1)*t1+y*t2;
262
        p_h = y1*t1;
263
        z = p_l+p_h;
264
        EXTRACT_WORDS(j,i,z);
265
        if (j>=0x40900000) {                            /* z >= 1024 */
266
            if(((j-0x40900000)|i)!=0)                    /* if z > 1024 */
267
                return s*huge*huge;                     /* overflow */
268
            else {
269
                if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
270
            }
271
        } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
272
            if(((j-0xc090cc00)|i)!=0)            /* z < -1075 */
273
                return s*tiny*tiny;             /* underflow */
274
            else {
275
                if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
276
            }
277
        }
278
    /*
279
     * compute 2**(p_h+p_l)
280
     */
281
        i = j&0x7fffffff;
282
        k = (i>>20)-0x3ff;
283
        n = 0;
284
        if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
285
            n = j+(0x00100000>>(k+1));
286
            k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
287
            t = zero;
288
            SET_HIGH_WORD(t,n&~(0x000fffff>>k));
289
            n = ((n&0x000fffff)|0x00100000)>>(20-k);
290
            if(j<0) n = -n;
291
            p_h -= t;
292
        }
293
        t = p_l+p_h;
294
        SET_LOW_WORD(t,0);
295
        u = t*lg2_h;
296
        v = (p_l-(t-p_h))*lg2+t*lg2_l;
297
        z = u+v;
298
        w = v-(z-u);
299
        t  = z*z;
300
        t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
301
        r  = (z*t1)/(t1-two)-(w+z*w);
302
        z  = one-(r-z);
303
        GET_HIGH_WORD(j,z);
304
        j += (n<<20);
305
        if((j>>20)<=0) z = scalbn(z,n);  /* subnormal output */
306
        else SET_HIGH_WORD(z,j);
307
        return s*z;
308
}

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