src/video/e_pow.h
author Ozkan Sezer <sezeroz@gmail.com>
Sat, 14 Sep 2019 17:00:50 +0300
branchSDL-1.2
changeset 13077 cd46eb772bee
parent 1424 7a610f25c12f
child 1662 782fd950bd46
child 1895 c121d94672cb
permissions -rw-r--r--
make SDL_qsort.c to actually link (missing SDL_stdinc.h include)

also sync its comments with version 1.15 from mainstream
     1 /* @(#)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"*/
    63 #include "math_private.h"
    64 
    65 #ifdef __STDC__
    66 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 	/* poly coefs for (3/2)*(log(x)-2s-2/3*s**3 */
    74 L1  =  5.99999999999994648725e-01, /* 0x3FE33333, 0x33333303 */
    75 L2  =  4.28571428578550184252e-01, /* 0x3FDB6DB6, 0xDB6FABFF */
    76 L3  =  3.33333329818377432918e-01, /* 0x3FD55555, 0x518F264D */
    77 L4  =  2.72728123808534006489e-01, /* 0x3FD17460, 0xA91D4101 */
    78 L5  =  2.30660745775561754067e-01, /* 0x3FCD864A, 0x93C9DB65 */
    79 L6  =  2.06975017800338417784e-01, /* 0x3FCA7E28, 0x4A454EEF */
    80 P1   =  1.66666666666666019037e-01, /* 0x3FC55555, 0x5555553E */
    81 P2   = -2.77777777770155933842e-03, /* 0xBF66C16C, 0x16BEBD93 */
    82 P3   =  6.61375632143793436117e-05, /* 0x3F11566A, 0xAF25DE2C */
    83 P4   = -1.65339022054652515390e-06, /* 0xBEBBBD41, 0xC5D26BF1 */
    84 P5   =  4.13813679705723846039e-08, /* 0x3E663769, 0x72BEA4D0 */
    85 lg2  =  6.93147180559945286227e-01, /* 0x3FE62E42, 0xFEFA39EF */
    86 lg2_h  =  6.93147182464599609375e-01, /* 0x3FE62E43, 0x00000000 */
    87 lg2_l  = -1.90465429995776804525e-09, /* 0xBE205C61, 0x0CA86C39 */
    88 ovt =  8.0085662595372944372e-0017, /* -(1024-log2(ovfl+.5ulp)) */
    89 cp    =  9.61796693925975554329e-01, /* 0x3FEEC709, 0xDC3A03FD =2/(3ln2) */
    90 cp_h  =  9.61796700954437255859e-01, /* 0x3FEEC709, 0xE0000000 =(float)cp */
    91 cp_l  = -7.02846165095275826516e-09, /* 0xBE3E2FE0, 0x145B01F5 =tail of cp_h*/
    92 ivln2    =  1.44269504088896338700e+00, /* 0x3FF71547, 0x652B82FE =1/ln2 */
    93 ivln2_h  =  1.44269502162933349609e+00, /* 0x3FF71547, 0x60000000 =24b 1/ln2*/
    94 ivln2_l  =  1.92596299112661746887e-08; /* 0x3E54AE0B, 0xF85DDF44 =1/ln2 tail*/
    95 
    96 #ifdef __STDC__
    97 	double __ieee754_pow(double x, double y)
    98 #else
    99 	double __ieee754_pow(x,y)
   100 	double x, y;
   101 #endif
   102 {
   103 	double z,ax,z_h,z_l,p_h,p_l;
   104 	double y1,t1,t2,r,s,t,u,v,w;
   105 	int32_t i,j,k,yisint,n;
   106 	int32_t hx,hy,ix,iy;
   107 	u_int32_t lx,ly;
   108 
   109 	EXTRACT_WORDS(hx,lx,x);
   110 	EXTRACT_WORDS(hy,ly,y);
   111 	ix = hx&0x7fffffff;  iy = hy&0x7fffffff;
   112 
   113     /* y==zero: x**0 = 1 */
   114 	if((iy|ly)==0) return one;
   115 
   116     /* +-NaN return x+y */
   117 	if(ix > 0x7ff00000 || ((ix==0x7ff00000)&&(lx!=0)) ||
   118 	   iy > 0x7ff00000 || ((iy==0x7ff00000)&&(ly!=0)))
   119 		return x+y;
   120 
   121     /* determine if y is an odd int when x < 0
   122      * yisint = 0	... y is not an integer
   123      * yisint = 1	... y is an odd int
   124      * yisint = 2	... y is an even int
   125      */
   126 	yisint  = 0;
   127 	if(hx<0) {
   128 	    if(iy>=0x43400000) yisint = 2; /* even integer y */
   129 	    else if(iy>=0x3ff00000) {
   130 		k = (iy>>20)-0x3ff;	   /* exponent */
   131 		if(k>20) {
   132 		    j = ly>>(52-k);
   133 		    if((u_int32_t)(j<<(52-k))==ly) yisint = 2-(j&1);
   134 		} else if(ly==0) {
   135 		    j = iy>>(20-k);
   136 		    if((j<<(20-k))==iy) yisint = 2-(j&1);
   137 		}
   138 	    }
   139 	}
   140 
   141     /* special value of y */
   142 	if(ly==0) {
   143 	    if (iy==0x7ff00000) {	/* y is +-inf */
   144 	        if(((ix-0x3ff00000)|lx)==0)
   145 		    return  y - y;	/* inf**+-1 is NaN */
   146 	        else if (ix >= 0x3ff00000)/* (|x|>1)**+-inf = inf,0 */
   147 		    return (hy>=0)? y: zero;
   148 	        else			/* (|x|<1)**-,+inf = inf,0 */
   149 		    return (hy<0)?-y: zero;
   150 	    }
   151 	    if(iy==0x3ff00000) {	/* y is  +-1 */
   152 		if(hy<0) return one/x; else return x;
   153 	    }
   154 	    if(hy==0x40000000) return x*x; /* y is  2 */
   155 	    if(hy==0x3fe00000) {	/* y is  0.5 */
   156 		if(hx>=0)	/* x >= +0 */
   157 		return __ieee754_sqrt(x);
   158 	    }
   159 	}
   160 
   161 	ax   = x < 0 ? -x : x; /*fabs(x);*/
   162     /* special value of x */
   163 	if(lx==0) {
   164 	    if(ix==0x7ff00000||ix==0||ix==0x3ff00000){
   165 		z = ax;			/*x is +-0,+-inf,+-1*/
   166 		if(hy<0) z = one/z;	/* z = (1/|x|) */
   167 		if(hx<0) {
   168 		    if(((ix-0x3ff00000)|yisint)==0) {
   169 			z = (z-z)/(z-z); /* (-1)**non-int is NaN */
   170 		    } else if(yisint==1)
   171 			z = -z;		/* (x<0)**odd = -(|x|**odd) */
   172 		}
   173 		return z;
   174 	    }
   175 	}
   176 
   177     /* (x<0)**(non-int) is NaN */
   178 	if(((((u_int32_t)hx>>31)-1)|yisint)==0) return (x-x)/(x-x);
   179 
   180     /* |y| is huge */
   181 	if(iy>0x41e00000) { /* if |y| > 2**31 */
   182 	    if(iy>0x43f00000){	/* if |y| > 2**64, must o/uflow */
   183 		if(ix<=0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
   184 		if(ix>=0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
   185 	    }
   186 	/* over/underflow if x is not close to one */
   187 	    if(ix<0x3fefffff) return (hy<0)? huge*huge:tiny*tiny;
   188 	    if(ix>0x3ff00000) return (hy>0)? huge*huge:tiny*tiny;
   189 	/* now |1-x| is tiny <= 2**-20, suffice to compute
   190 	   log(x) by x-x^2/2+x^3/3-x^4/4 */
   191 	    t = x-1;		/* t has 20 trailing zeros */
   192 	    w = (t*t)*(0.5-t*(0.3333333333333333333333-t*0.25));
   193 	    u = ivln2_h*t;	/* ivln2_h has 21 sig. bits */
   194 	    v = t*ivln2_l-w*ivln2;
   195 	    t1 = u+v;
   196 	    SET_LOW_WORD(t1,0);
   197 	    t2 = v-(t1-u);
   198 	} else {
   199 	    double s2,s_h,s_l,t_h,t_l;
   200 	    n = 0;
   201 	/* take care subnormal number */
   202 	    if(ix<0x00100000)
   203 		{ax *= two53; n -= 53; GET_HIGH_WORD(ix,ax); }
   204 	    n  += ((ix)>>20)-0x3ff;
   205 	    j  = ix&0x000fffff;
   206 	/* determine interval */
   207 	    ix = j|0x3ff00000;		/* normalize ix */
   208 	    if(j<=0x3988E) k=0;		/* |x|<sqrt(3/2) */
   209 	    else if(j<0xBB67A) k=1;	/* |x|<sqrt(3)   */
   210 	    else {k=0;n+=1;ix -= 0x00100000;}
   211 	    SET_HIGH_WORD(ax,ix);
   212 
   213 	/* compute s = s_h+s_l = (x-1)/(x+1) or (x-1.5)/(x+1.5) */
   214 	    u = ax-bp[k];		/* bp[0]=1.0, bp[1]=1.5 */
   215 	    v = one/(ax+bp[k]);
   216 	    s = u*v;
   217 	    s_h = s;
   218 	    SET_LOW_WORD(s_h,0);
   219 	/* t_h=ax+bp[k] High */
   220 	    t_h = zero;
   221 	    SET_HIGH_WORD(t_h,((ix>>1)|0x20000000)+0x00080000+(k<<18));
   222 	    t_l = ax - (t_h-bp[k]);
   223 	    s_l = v*((u-s_h*t_h)-s_h*t_l);
   224 	/* compute log(ax) */
   225 	    s2 = s*s;
   226 	    r = s2*s2*(L1+s2*(L2+s2*(L3+s2*(L4+s2*(L5+s2*L6)))));
   227 	    r += s_l*(s_h+s);
   228 	    s2  = s_h*s_h;
   229 	    t_h = 3.0+s2+r;
   230 	    SET_LOW_WORD(t_h,0);
   231 	    t_l = r-((t_h-3.0)-s2);
   232 	/* u+v = s*(1+...) */
   233 	    u = s_h*t_h;
   234 	    v = s_l*t_h+t_l*s;
   235 	/* 2/(3log2)*(s+...) */
   236 	    p_h = u+v;
   237 	    SET_LOW_WORD(p_h,0);
   238 	    p_l = v-(p_h-u);
   239 	    z_h = cp_h*p_h;		/* cp_h+cp_l = 2/(3*log2) */
   240 	    z_l = cp_l*p_h+p_l*cp+dp_l[k];
   241 	/* log2(ax) = (s+..)*2/(3*log2) = n + dp_h + z_h + z_l */
   242 	    t = (double)n;
   243 	    t1 = (((z_h+z_l)+dp_h[k])+t);
   244 	    SET_LOW_WORD(t1,0);
   245 	    t2 = z_l-(((t1-t)-dp_h[k])-z_h);
   246 	}
   247 
   248 	s = one; /* s (sign of result -ve**odd) = -1 else = 1 */
   249 	if(((((u_int32_t)hx>>31)-1)|(yisint-1))==0)
   250 	    s = -one;/* (-ve)**(odd int) */
   251 
   252     /* split up y into y1+y2 and compute (y1+y2)*(t1+t2) */
   253 	y1  = y;
   254 	SET_LOW_WORD(y1,0);
   255 	p_l = (y-y1)*t1+y*t2;
   256 	p_h = y1*t1;
   257 	z = p_l+p_h;
   258 	EXTRACT_WORDS(j,i,z);
   259 	if (j>=0x40900000) {				/* z >= 1024 */
   260 	    if(((j-0x40900000)|i)!=0)			/* if z > 1024 */
   261 		return s*huge*huge;			/* overflow */
   262 	    else {
   263 		if(p_l+ovt>z-p_h) return s*huge*huge;	/* overflow */
   264 	    }
   265 	} else if((j&0x7fffffff)>=0x4090cc00 ) {	/* z <= -1075 */
   266 	    if(((j-0xc090cc00)|i)!=0) 		/* z < -1075 */
   267 		return s*tiny*tiny;		/* underflow */
   268 	    else {
   269 		if(p_l<=z-p_h) return s*tiny*tiny;	/* underflow */
   270 	    }
   271 	}
   272     /*
   273      * compute 2**(p_h+p_l)
   274      */
   275 	i = j&0x7fffffff;
   276 	k = (i>>20)-0x3ff;
   277 	n = 0;
   278 	if(i>0x3fe00000) {		/* if |z| > 0.5, set n = [z+0.5] */
   279 	    n = j+(0x00100000>>(k+1));
   280 	    k = ((n&0x7fffffff)>>20)-0x3ff;	/* new k for n */
   281 	    t = zero;
   282 	    SET_HIGH_WORD(t,n&~(0x000fffff>>k));
   283 	    n = ((n&0x000fffff)|0x00100000)>>(20-k);
   284 	    if(j<0) n = -n;
   285 	    p_h -= t;
   286 	}
   287 	t = p_l+p_h;
   288 	SET_LOW_WORD(t,0);
   289 	u = t*lg2_h;
   290 	v = (p_l-(t-p_h))*lg2+t*lg2_l;
   291 	z = u+v;
   292 	w = v-(z-u);
   293 	t  = z*z;
   294 	t1  = z - t*(P1+t*(P2+t*(P3+t*(P4+t*P5))));
   295 	r  = (z*t1)/(t1-two)-(w+z*w);
   296 	z  = one-(r-z);
   297 	GET_HIGH_WORD(j,z);
   298 	j += (n<<20);
   299 	if((j>>20)<=0) z = SDL_NAME(scalbn)(z,n);	/* subnormal output */
   300 	else SET_HIGH_WORD(z,j);
   301 	return s*z;
   302 }