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pow(double x,double y)

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  1. /* @(#)e_pow.c 1.5 04/04/22 SMI */
  2. /*
  3.  * ====================================================
  4.  * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
  5.  *
  6.  * Permission to use, copy, modify, and distribute this
  7.  * software is freely granted, provided that this notice
  8.  * is preserved.
  9.  * ====================================================
  10.  */
  11.  
  12. #ifndef lint
  13. static char rcsid[] = "$FreeBSD: src/lib/msun/src/e_pow.c,v 1.11 2005/02/04 18:26:06 das Exp $";
  14. #endif
  15.  
  16. /* pow(x,y) return x**y
  17.  *
  18.  *            n
  19.  * Method:  Let x =  2   * (1+f)
  20.  *  1. Compute and return log2(x) in two pieces:
  21.  *      log2(x) = w1 + w2,
  22.  *     where w1 has 53-24 = 29 bit trailing zeros.
  23.  *  2. Perform y*log2(x) = n+y' by simulating muti-precision
  24.  *     arithmetic, where |y'|<=0.5.
  25.  *  3. Return x**y = 2**n*exp(y'*log2)
  26.  *
  27.  * Special cases:
  28.  *  1.  (anything) ** 0  is 1
  29.  *  2.  (anything) ** 1  is itself
  30.  *  3.  (anything) ** NAN is NAN
  31.  *  4.  NAN ** (anything except 0) is NAN
  32.  *  5.  +-(|x| > 1) **  +INF is +INF
  33.  *  6.  +-(|x| > 1) **  -INF is +0
  34.  *  7.  +-(|x| < 1) **  +INF is +0
  35.  *  8.  +-(|x| < 1) **  -INF is +INF
  36.  *  9.  +-1         ** +-INF is NAN
  37.  *  10. +0 ** (+anything except 0, NAN)               is +0
  38.  *  11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  39.  *  12. +0 ** (-anything except 0, NAN)               is +INF
  40.  *  13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  41.  *  14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  42.  *  15. +INF ** (+anything except 0,NAN) is +INF
  43.  *  16. +INF ** (-anything except 0,NAN) is +0
  44.  *  17. -INF ** (anything)  = -0 ** (-anything)
  45.  *  18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  46.  *  19. (-anything except 0 and inf) ** (non-integer) is NAN
  47.  *
  48.  * Accuracy:
  49.  *  pow(x,y) returns x**y nearly rounded. In particular
  50.  *          pow(integer,integer)
  51.  *  always returns the correct integer provided it is
  52.  *  representable.
  53.  *
  54.  * Constants :
  55.  * The hexadecimal values are the intended ones for the following
  56.  * constants. The decimal values may be used, provided that the
  57.  * compiler will convert from decimal to binary accurately enough
  58.  * to produce the hexadecimal values shown.
  59.  */
  60.  
  61. #include "math.h"
  62. #include "math_private.h"
  63.  
  64. static const double
  65. bp[] = {1.0, 1.5,},
  66. dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  67. dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  68. zero    =  0.0,
  69. one =  1.0,
  70. two =  2.0,
  71. two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
  72. huge    =  1.0e300,
  73. tiny    =  1.0e-300,
  74.     /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  75. L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  76. L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  77. L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  78. L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  79. L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
  80. L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
  81. P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  82. P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  83. P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  84. P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  85. P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
  86. lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
  87. lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
  88. lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
  89. ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
  90. cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
  91. cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
  92. cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
  93. ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
  94. ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
  95. ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
  96.  
  97. double
  98. pow(double x, double y)
  99. {
  100.     double z,ax,z_h,z_l,p_h,p_l;
  101.     double y1,t1,t2,r,s,t,u,v,w;
  102.     int32_t i,j,k,yisint,n;
  103.     int32_t hx,hy,ix,iy;
  104.     u_int32_t lx,ly;
  105.  
  106.     EXTRACT_WORDS(hx,lx,x);
  107.     EXTRACT_WORDS(hy,ly,y);
  108.     ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
  109.  
  110.     /* y==zero: x**0 = 1 */
  111.     if((iy|ly)==0) return one;  
  112.  
  113.     /* +-NaN return x+y */
  114.     if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
  115.        iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
  116.         return x+y;
  117.  
  118.     /* determine if y is an odd int when x < 0
  119.      * yisint = 0   ... y is not an integer
  120.      * yisint = 1   ... y is an odd int
  121.      * yisint = 2   ... y is an even int
  122.      */
  123.     yisint  = 0;
  124.     if(hx<0) { 
  125.         if(iy>=0x43400000) yisint = 2; /* even integer y */
  126.         else if(iy>=0x3ff00000) {
  127.         k = (iy>>20)-0x3ff;    /* exponent */
  128.         if(k>20) {
  129.             j = ly>>(52-k);
  130.             if((j<<(52-k))==ly) yisint = 2-(j&1);
  131.         } else if(ly==0) {
  132.             j = iy>>(20-k);
  133.             if((j<<(20-k))==iy) yisint = 2-(j&1);
  134.         }
  135.         }      
  136.     }
  137.  
  138.     /* special value of y */
  139.     if(ly==0) {    
  140.         if (iy==0x7ff00000) {   /* y is +-inf */
  141.             if(((ix-0x3ff00000)|lx)==0)
  142.             return  y - y;  /* inf**+-1 is NaN */
  143.             else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
  144.             return (hy>=0)? y: zero;
  145.             else            /* (|x|<1)**-,+inf = inf,0 */
  146.             return (hy<0)?-y: zero;
  147.         }
  148.         if(iy==0x3ff00000) {    /* y is  +-1 */
  149.         if(hy<0) return one/x; else return x;
  150.         }
  151.         if(hy==0x40000000) return x*x; /* y is  2 */
  152.         if(hy==0x3fe00000) {    /* y is  0.5 */
  153.         if(hx>=0)   /* x >= +0 */
  154.         return sqrt(x);
  155.         }
  156.     }
  157.  
  158.     ax   = fabs(x);
  159.     /* special value of x */
  160.     if(lx==0) {
  161.         if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
  162.         z = ax;         /*x is +-0,+-inf,+-1*/
  163.         if(hy<0) z = one/z; /* z = (1/|x|) */
  164.         if(hx<0) {
  165.             if(((ix-0x3ff00000)|yisint)==0) {
  166.             z = (z-z)/(z-z); /* (-1)**non-int is NaN */
  167.             } else if(yisint==1)
  168.             z = -z;     /* (x<0)**odd = -(|x|**odd) */
  169.         }
  170.         return z;
  171.         }
  172.     }
  173.    
  174.     /* CYGNUS LOCAL + fdlibm-5.3 fix: This used to be
  175.     n = (hx>>31)+1;
  176.        but ANSI C says a right shift of a signed negative quantity is
  177.        implementation defined.  */
  178.     n = ((u_int32_t)hx>>31)-1;
  179.  
  180.     /* (x<0)**(non-int) is NaN */
  181.     if((n|yisint)==0) return (x-x)/(x-x);
  182.  
  183.     s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
  184.     if((n|(yisint-1))==0) s = -one;/* (-ve)**(odd int) */
  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)? s*huge*huge:s*tiny*tiny;
  194.         if(ix>0x3ff00000) return (hy>0)? s*huge*huge:s*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 = ax-one;     /* 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 ss,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 ss = 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.         ss = u*v;
  223.         s_h = ss;
  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 = ss*ss;
  232.         r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
  233.         r += s_l*(s_h+ss);
  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 = ss*(1+...) */
  239.         u = s_h*t_h;
  240.         v = s_l*t_h+t_l*ss;
  241.     /* 2/(3log2)*(ss+...) */
  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) = (ss+..)*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.     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
  255.     y1  = y;
  256.     SET_LOW_WORD(y1,0);
  257.     p_l = (y-y1)*t1+y*t2;
  258.     p_h = y1*t1;
  259.     z = p_l+p_h;
  260.     EXTRACT_WORDS(j,i,z);
  261.     if (j>=0x40900000) {                /* z >= 1024 */
  262.         if(((j-0x40900000)|i)!=0)           /* if z > 1024 */
  263.         return s*huge*huge;         /* overflow */
  264.         else {
  265.         if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
  266.         }
  267.     } else if((j&0x7fffffff)>=0x4090cc00 ) {    /* z <= -1075 */
  268.         if(((j-0xc090cc00)|i)!=0)       /* z < -1075 */
  269.         return s*tiny*tiny;     /* underflow */
  270.         else {
  271.         if(p_l<=z-p_h) return s*tiny*tiny;  /* underflow */
  272.         }
  273.     }
  274.     /*
  275.      * compute 2**(p_h+p_l)
  276.      */
  277.     i = j&0x7fffffff;
  278.     k = (i>>20)-0x3ff;
  279.     n = 0;
  280.     if(i>0x3fe00000) {      /* if |z| > 0.5, set n = [z+0.5] */
  281.         n = j+(0x00100000>>(k+1));
  282.         k = ((n&0x7fffffff)>>20)-0x3ff; /* new k for n */
  283.         t = zero;
  284.         SET_HIGH_WORD(t,n&~(0x000fffff>>k));
  285.         n = ((n&0x000fffff)|0x00100000)>>(20-k);
  286.         if(j<0) n = -n;
  287.         p_h -= t;
  288.     }
  289.     t = p_l+p_h;
  290.     SET_LOW_WORD(t,0);
  291.     u = t*lg2_h;
  292.     v = (p_l-(t-p_h))*lg2+t*lg2_l;
  293.     z = u+v;
  294.     w = v-(z-u);
  295.     t  = z*z;
  296.     t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
  297.     r  = (z*t1)/(t1-two)-(w+z*w);
  298.     z  = one-(r-z);
  299.     GET_HIGH_WORD(j,z);
  300.     j += (n<<20);
  301.     if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
  302.     else SET_HIGH_WORD(z,j);
  303.     return s*z;
  304. }
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