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  1. /* Copyright (C) 1994 DJ Delorie, see COPYING.DJ for details */
  2. /* @(#)e_pow.c 5.1 93/09/24 */
  3. /*
  4.  * ====================================================
  5.  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
  6.  *
  7.  * Developed at SunPro, a Sun Microsystems, Inc. business.
  8.  * Permission to use, copy, modify, and distribute this
  9.  * software is freely granted, provided that this notice
  10.  * is preserved.
  11.  * ====================================================
  12.  */
  13.  
  14. #if defined(LIBM_SCCS) && !defined(lint)
  15. static char rcsid[] = "$Id: e_pow.c,v 1.6 1994/09/13 00:40:33 jtc Exp $";
  16. #endif
  17.  
  18. /* __ieee754_pow(x,y) return x**y
  19.  *
  20.  *                    n
  21.  * Method:  Let x =  2   * (1+f)
  22.  *      1. Compute and return log2(x) in two pieces:
  23.  *              log2(x) = w1 + w2,
  24.  *         where w1 has 53-24 = 29 bit trailing zeros.
  25.  *      2. Perform y*log2(x) = n+y' by simulating muti-precision
  26.  *         arithmetic, where |y'|<=0.5.
  27.  *      3. Return x**y = 2**n*exp(y'*log2)
  28.  *
  29.  * Special cases:
  30.  *      1.  (anything) ** 0  is 1
  31.  *      2.  (anything) ** 1  is itself
  32.  *      3.  (anything) ** NAN is NAN
  33.  *      4.  NAN ** (anything except 0) is NAN
  34.  *      5.  +-(|x| > 1) **  +INF is +INF
  35.  *      6.  +-(|x| > 1) **  -INF is +0
  36.  *      7.  +-(|x| < 1) **  +INF is +0
  37.  *      8.  +-(|x| < 1) **  -INF is +INF
  38.  *      9.  +-1         ** +-INF is NAN
  39.  *      10. +0 ** (+anything except 0, NAN)               is +0
  40.  *      11. -0 ** (+anything except 0, NAN, odd integer)  is +0
  41.  *      12. +0 ** (-anything except 0, NAN)               is +INF
  42.  *      13. -0 ** (-anything except 0, NAN, odd integer)  is +INF
  43.  *      14. -0 ** (odd integer) = -( +0 ** (odd integer) )
  44.  *      15. +INF ** (+anything except 0,NAN) is +INF
  45.  *      16. +INF ** (-anything except 0,NAN) is +0
  46.  *      17. -INF ** (anything)  = -0 ** (-anything)
  47.  *      18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer)
  48.  *      19. (-anything except 0 and inf) ** (non-integer) is NAN
  49.  *
  50.  * Accuracy:
  51.  *      pow(x,y) returns x**y nearly rounded. In particular
  52.  *                      pow(integer,integer)
  53.  *      always returns the correct integer provided it is
  54.  *      representable.
  55.  *
  56.  * Constants :
  57.  * The hexadecimal values are the intended ones for the following
  58.  * constants. The decimal values may be used, provided that the
  59.  * compiler will convert from decimal to binary accurately enough
  60.  * to produce the hexadecimal values shown.
  61.  */
  62.  
  63. #include "math.h"
  64. #include "math_private.h"
  65.  
  66. #ifdef __STDC__
  67. static const double
  68. #else
  69. static double
  70. #endif
  71. bp[] = {1.0, 1.5,},
  72. dp_h[] = { 0.0, 5.84962487220764160156e-01,}, /* 0x3FE2B803, 0x40000000 */
  73. dp_l[] = { 0.0, 1.35003920212974897128e-08,}, /* 0x3E4CFDEB, 0x43CFD006 */
  74. zero    =  0.0,
  75. one     =  1.0,
  76. two     =  2.0,
  77. two53   =  9007199254740992.0,  /* 0x43400000, 0x00000000 */
  78. huge    =  1.0e300,
  79. tiny    =  1.0e-300,
  80.         /* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
  81. L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
  82. L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
  83. L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
  84. L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
  85. L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
  86. L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
  87. P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
  88. P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
  89. P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
  90. P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
  91. P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
  92. lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
  93. lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
  94. lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
  95. ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
  96. cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
  97. cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
  98. cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
  99. ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
  100. ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
  101. ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
  102.  
  103. #ifdef __STDC__
  104.         double __ieee754_pow(double x, double y)
  105. #else
  106.         double __ieee754_pow(x,y)
  107.         double x, y;
  108. #endif
  109. {
  110.         double z,ax,z_h,z_l,p_h,p_l;
  111.         double y1,t1,t2,r,s,t,u,v,w;
  112.         int32_t i,j,k,yisint,n;
  113.         int32_t hx,hy,ix,iy;
  114.         u_int32_t lx,ly;
  115.  
  116.         EXTRACT_WORDS(hx,lx,x);
  117.         EXTRACT_WORDS(hy,ly,y);
  118.         ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
  119.  
  120.     /* y==zero: x**0 = 1 */
  121.         if((iy|ly)==0) return one;      
  122.  
  123.     /* +-NaN return x+y */
  124.         if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
  125.            iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
  126.                 return x+y;    
  127.  
  128.     /* determine if y is an odd int when x < 0
  129.      * yisint = 0       ... y is not an integer
  130.      * yisint = 1       ... y is an odd int
  131.      * yisint = 2       ... y is an even int
  132.      */
  133.         yisint  = 0;
  134.         if(hx<0) {     
  135.             if(iy>=0x43400000) yisint = 2; /* even integer y */
  136.             else if(iy>=0x3ff00000) {
  137.                 k = (iy>>20)-0x3ff;        /* exponent */
  138.                 if(k>20) {
  139.                     j = ly>>(52-k);
  140.                     if((j<<(52-k))==ly) yisint = 2-(j&1);
  141.                 } else if(ly==0) {
  142.                     j = iy>>(20-k);
  143.                     if((j<<(20-k))==iy) yisint = 2-(j&1);
  144.                 }
  145.             }          
  146.         }
  147.  
  148.     /* special value of y */
  149.         if(ly==0) {    
  150.             if (iy==0x7ff00000) {       /* y is +-inf */
  151.                 if(((ix-0x3ff00000)|lx)==0)
  152.                     return  y - y;      /* inf**+-1 is NaN */
  153.                 else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
  154.                     return (hy>=0)? y: zero;
  155.                 else                    /* (|x|<1)**-,+inf = inf,0 */
  156.                     return (hy<0)?-y: zero;
  157.             }
  158.             if(iy==0x3ff00000) {        /* y is  +-1 */
  159.                 if(hy<0) return one/x; else return x;
  160.             }
  161.             if(hy==0x40000000) return x*x; /* y is  2 */
  162.             if(hy==0x3fe00000) {        /* y is  0.5 */
  163.                 if(hx>=0)       /* x >= +0 */
  164.                 return sqrt(x);
  165.             }
  166.         }
  167.  
  168.         ax   = fabs(x);
  169.     /* special value of x */
  170.         if(lx==0) {
  171.             if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
  172.                 z = ax;                 /*x is +-0,+-inf,+-1*/
  173.                 if(hy<0) z = one/z;     /* z = (1/|x|) */
  174.                 if(hx<0) {
  175.                     if(((ix-0x3ff00000)|yisint)==0) {
  176.                         z = (z-z)/(z-z); /* (-1)**non-int is NaN */
  177.                     } else if(yisint==1)
  178.                         z = -z;         /* (x<0)**odd = -(|x|**odd) */
  179.                 }
  180.                 return z;
  181.             }
  182.         }
  183.    
  184.     /* (x<0)**(non-int) is NaN */
  185.         if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
  186.  
  187.     /* |y| is huge */
  188.         if(iy>0x41e00000) { /* if |y| > 2**31 */
  189.             if(iy>0x43f00000){  /* if |y| > 2**64, must o/uflow */
  190.                 if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
  191.                 if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
  192.             }
  193.         /* over/underflow if x is not close to one */
  194.             if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
  195.             if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
  196.         /* now |1-x| is tiny <= 2**-20, suffice to compute
  197.            log(x) by x-x^2/2+x^3/3-x^4/4 */
  198.             t = x-1;            /* t has 20 trailing zeros */
  199.             w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
  200.             u = ivln2_h*t;      /* ivln2_h has 21 sig. bits */
  201.             v = t*ivln2_l-w*ivln2;
  202.             t1 = u+v;
  203.             SET_LOW_WORD(t1,0);
  204.             t2 = v-(t1-u);
  205.         } else {
  206.             double s2,s_h,s_l,t_h,t_l;
  207.             n = 0;
  208.         /* take care subnormal number */
  209.             if(ix<0x00100000)
  210.                 {ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
  211.             n  += ((ix)>>20)-0x3ff;
  212.             j  = ix&0x000fffff;
  213.         /* determine interval */
  214.             ix = j|0x3ff00000;          /* normalize ix */
  215.             if(j<=0x3988E) k=0;         /* |x|<sqrt(3/2) */
  216.             else if(j<0xBB67A) k=1;     /* |x|<sqrt(3)   */
  217.             else {k=0;n+=1;ix -= 0x00100000;}
  218.             SET_HIGH_WORD(ax,ix);
  219.  
  220.         /* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
  221.             u = ax-bp[k];               /* bp[0]=1.0, bp[1]=1.5 */
  222.             v = one/(ax+bp[k]);
  223.             s = u*v;
  224.             s_h = s;
  225.             SET_LOW_WORD(s_h,0);
  226.         /* t_h=ax+bp[k] High */
  227.             t_h = zero;
  228.             SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
  229.             t_l = ax - (t_h-bp[k]);
  230.             s_l = v*((u-s_h*t_h)-s_h*t_l);
  231.         /* compute log(ax) */
  232.             s2 = s*s;
  233.             r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
  234.             r += s_l*(s_h+s);
  235.             s2  = s_h*s_h;
  236.             t_h = 3.0+s2+r;
  237.             SET_LOW_WORD(t_h,0);
  238.             t_l = r-((t_h-3.0)-s2);
  239.         /* u+v = s*(1+...) */
  240.             u = s_h*t_h;
  241.             v = s_l*t_h+t_l*s;
  242.         /* 2/(3log2)*(s+...) */
  243.             p_h = u+v;
  244.             SET_LOW_WORD(p_h,0);
  245.             p_l = v-(p_h-u);
  246.             z_h = cp_h*p_h;             /* cp_h+cp_l = 2/(3*log2) */
  247.             z_l = cp_l*p_h+p_l*cp+dp_l[k];
  248.         /* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
  249.             t = (double)n;
  250.             t1 = (((z_h+z_l)+dp_h[k])+t);
  251.             SET_LOW_WORD(t1,0);
  252.             t2 = z_l-(((t1-t)-dp_h[k])-z_h);
  253.         }
  254.  
  255.         s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
  256.         if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
  257.             s = -one;/* (-ve)**(odd int) */
  258.  
  259.     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
  260.         y1  = y;
  261.         SET_LOW_WORD(y1,0);
  262.         p_l = (y-y1)*t1+y*t2;
  263.         p_h = y1*t1;
  264.         z = p_l+p_h;
  265.         EXTRACT_WORDS(j,i,z);
  266.         if (j>=0x40900000) {                            /* z >= 1024 */
  267.             if(((j-0x40900000)|i)!=0)                   /* if z > 1024 */
  268.                 return s*huge*huge;                     /* overflow */
  269.             else {
  270.                 if(p_l+ovt>z-p_h) return s*huge*huge;   /* overflow */
  271.             }
  272.         } else if((j&0x7fffffff)>=0x4090cc00 ) {        /* z <= -1075 */
  273.             if(((j-0xc090cc00)|i)!=0)           /* z < -1075 */
  274.                 return s*tiny*tiny;             /* underflow */
  275.             else {
  276.                 if(p_l<=z-p_h) return s*tiny*tiny;      /* underflow */
  277.             }
  278.         }
  279.     /*
  280.      * compute 2**(p_h+p_l)
  281.      */
  282.         i = j&0x7fffffff;
  283.         k = (i>>20)-0x3ff;
  284.         n = 0;
  285.         if(i>0x3fe00000) {              /* if |z| > 0.5, set n = [z+0.5] */
  286.             n = j+(0x00100000>>(k+1));
  287.             k = ((n&0x7fffffff)>>20)-0x3ff;     /* new k for n */
  288.             t = zero;
  289.             SET_HIGH_WORD(t,n&~(0x000fffff>>k));
  290.             n = ((n&0x000fffff)|0x00100000)>>(20-k);
  291.             if(j<0) n = -n;
  292.             p_h -= t;
  293.         }
  294.         t = p_l+p_h;
  295.         SET_LOW_WORD(t,0);
  296.         u = t*lg2_h;
  297.         v = (p_l-(t-p_h))*lg2+t*lg2_l;
  298.         z = u+v;
  299.         w = v-(z-u);
  300.         t  = z*z;
  301.         t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
  302.         r  = (z*t1)/(t1-two)-(w+z*w);
  303.         z  = one-(r-z);
  304.         GET_HIGH_WORD(j,z);
  305.         j += (n<<20);
  306.         if((j>>20)<=0) z = scalbn(z,n); /* subnormal output */
  307.         else SET_HIGH_WORD(z,j);
  308.         return s*z;
  309. }
  310.