src/libm/k_rem_pio2.c
author Ryan C. Gordon <icculus@icculus.org>
Tue, 26 May 2015 21:19:23 -0400
changeset 9649 d7762e30ba24
parent 8670 0c15c8a2f8c3
child 10650 b6ec7005ca15
permissions -rw-r--r--
Stack hint should look for 0, not -1, and not care about environment variables.
     1 /* @(#)k_rem_pio2.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 const char rcsid[] =
    15     "$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc Exp $";
    16 #endif
    17 
    18 /*
    19  * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
    20  * double x[],y[]; int e0,nx,prec; int ipio2[];
    21  *
    22  * __kernel_rem_pio2 return the last three digits of N with
    23  *		y = x - N*pi/2
    24  * so that |y| < pi/2.
    25  *
    26  * The method is to compute the integer (mod 8) and fraction parts of
    27  * (2/pi)*x without doing the full multiplication. In general we
    28  * skip the part of the product that are known to be a huge integer (
    29  * more accurately, = 0 mod 8 ). Thus the number of operations are
    30  * independent of the exponent of the input.
    31  *
    32  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
    33  *
    34  * Input parameters:
    35  * 	x[]	The input value (must be positive) is broken into nx
    36  *		pieces of 24-bit integers in double precision format.
    37  *		x[i] will be the i-th 24 bit of x. The scaled exponent
    38  *		of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
    39  *		match x's up to 24 bits.
    40  *
    41  *		Example of breaking a double positive z into x[0]+x[1]+x[2]:
    42  *			e0 = ilogb(z)-23
    43  *			z  = scalbn(z,-e0)
    44  *		for i = 0,1,2
    45  *			x[i] = floor(z)
    46  *			z    = (z-x[i])*2**24
    47  *
    48  *
    49  *	y[]	ouput result in an array of double precision numbers.
    50  *		The dimension of y[] is:
    51  *			24-bit  precision	1
    52  *			53-bit  precision	2
    53  *			64-bit  precision	2
    54  *			113-bit precision	3
    55  *		The actual value is the sum of them. Thus for 113-bit
    56  *		precison, one may have to do something like:
    57  *
    58  *		long double t,w,r_head, r_tail;
    59  *		t = (long double)y[2] + (long double)y[1];
    60  *		w = (long double)y[0];
    61  *		r_head = t+w;
    62  *		r_tail = w - (r_head - t);
    63  *
    64  *	e0	The exponent of x[0]
    65  *
    66  *	nx	dimension of x[]
    67  *
    68  *  	prec	an integer indicating the precision:
    69  *			0	24  bits (single)
    70  *			1	53  bits (double)
    71  *			2	64  bits (extended)
    72  *			3	113 bits (quad)
    73  *
    74  *	ipio2[]
    75  *		integer array, contains the (24*i)-th to (24*i+23)-th
    76  *		bit of 2/pi after binary point. The corresponding
    77  *		floating value is
    78  *
    79  *			ipio2[i] * 2^(-24(i+1)).
    80  *
    81  * External function:
    82  *	double scalbn(), floor();
    83  *
    84  *
    85  * Here is the description of some local variables:
    86  *
    87  * 	jk	jk+1 is the initial number of terms of ipio2[] needed
    88  *		in the computation. The recommended value is 2,3,4,
    89  *		6 for single, double, extended,and quad.
    90  *
    91  * 	jz	local integer variable indicating the number of
    92  *		terms of ipio2[] used.
    93  *
    94  *	jx	nx - 1
    95  *
    96  *	jv	index for pointing to the suitable ipio2[] for the
    97  *		computation. In general, we want
    98  *			( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
    99  *		is an integer. Thus
   100  *			e0-3-24*jv >= 0 or (e0-3)/24 >= jv
   101  *		Hence jv = max(0,(e0-3)/24).
   102  *
   103  *	jp	jp+1 is the number of terms in PIo2[] needed, jp = jk.
   104  *
   105  * 	q[]	double array with integral value, representing the
   106  *		24-bits chunk of the product of x and 2/pi.
   107  *
   108  *	q0	the corresponding exponent of q[0]. Note that the
   109  *		exponent for q[i] would be q0-24*i.
   110  *
   111  *	PIo2[]	double precision array, obtained by cutting pi/2
   112  *		into 24 bits chunks.
   113  *
   114  *	f[]	ipio2[] in floating point
   115  *
   116  *	iq[]	integer array by breaking up q[] in 24-bits chunk.
   117  *
   118  *	fq[]	final product of x*(2/pi) in fq[0],..,fq[jk]
   119  *
   120  *	ih	integer. If >0 it indicates q[] is >= 0.5, hence
   121  *		it also indicates the *sign* of the result.
   122  *
   123  */
   124 
   125 
   126 /*
   127  * Constants:
   128  * The hexadecimal values are the intended ones for the following
   129  * constants. The decimal values may be used, provided that the
   130  * compiler will convert from decimal to binary accurately enough
   131  * to produce the hexadecimal values shown.
   132  */
   133 
   134 #include "math_libm.h"
   135 #include "math_private.h"
   136 
   137 #include "SDL_assert.h"
   138 
   139 libm_hidden_proto(scalbn)
   140     libm_hidden_proto(floor)
   141 #ifdef __STDC__
   142      static const int init_jk[] = { 2, 3, 4, 6 };       /* initial value for jk */
   143 #else
   144      static int init_jk[] = { 2, 3, 4, 6 };
   145 #endif
   146 
   147 #ifdef __STDC__
   148 static const double PIo2[] = {
   149 #else
   150 static double PIo2[] = {
   151 #endif
   152     1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
   153     7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
   154     5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
   155     3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
   156     1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
   157     1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
   158     2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
   159     2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
   160 };
   161 
   162 #ifdef __STDC__
   163 static const double
   164 #else
   165 static double
   166 #endif
   167   zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07,    /* 0x41700000, 0x00000000 */
   168     twon24 = 5.96046447753906250000e-08;        /* 0x3E700000, 0x00000000 */
   169 
   170 #ifdef __STDC__
   171 int attribute_hidden
   172 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
   173                   const int32_t * ipio2)
   174 #else
   175 int attribute_hidden
   176 __kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
   177      double x[], y[];
   178      int e0, nx, prec;
   179      int32_t ipio2[];
   180 #endif
   181 {
   182     int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
   183     double z, fw, f[20], fq[20], q[20];
   184 
   185     /* initialize jk */
   186     SDL_assert((prec >= 0) && (prec < SDL_arraysize(init_jk)));
   187     jk = init_jk[prec];
   188     SDL_assert((jk >= 2) && (jk <= 6));
   189     jp = jk;
   190 
   191     /* determine jx,jv,q0, note that 3>q0 */
   192     SDL_assert(nx > 0);
   193     jx = nx - 1;
   194     jv = (e0 - 3) / 24;
   195     if (jv < 0)
   196         jv = 0;
   197     q0 = e0 - 24 * (jv + 1);
   198 
   199     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
   200     j = jv - jx;
   201     m = jx + jk;
   202     for (i = 0; i <= m; i++, j++)
   203         f[i] = (j < 0) ? zero : (double) ipio2[j];
   204 
   205     /* compute q[0],q[1],...q[jk] */
   206     for (i = 0; i <= jk; i++) {
   207         for (j = 0, fw = 0.0; j <= jx; j++)
   208             fw += x[j] * f[jx + i - j];
   209         q[i] = fw;
   210     }
   211 
   212     jz = jk;
   213   recompute:
   214     /* distill q[] into iq[] reversingly */
   215     for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
   216         fw = (double) ((int32_t) (twon24 * z));
   217         iq[i] = (int32_t) (z - two24 * fw);
   218         z = q[j - 1] + fw;
   219     }
   220 
   221     /* compute n */
   222     z = scalbn(z, q0);          /* actual value of z */
   223     z -= 8.0 * floor(z * 0.125);        /* trim off integer >= 8 */
   224     n = (int32_t) z;
   225     z -= (double) n;
   226     ih = 0;
   227     if (q0 > 0) {               /* need iq[jz-1] to determine n */
   228         i = (iq[jz - 1] >> (24 - q0));
   229         n += i;
   230         iq[jz - 1] -= i << (24 - q0);
   231         ih = iq[jz - 1] >> (23 - q0);
   232     } else if (q0 == 0)
   233         ih = iq[jz - 1] >> 23;
   234     else if (z >= 0.5)
   235         ih = 2;
   236 
   237     if (ih > 0) {               /* q > 0.5 */
   238         n += 1;
   239         carry = 0;
   240         for (i = 0; i < jz; i++) {      /* compute 1-q */
   241             j = iq[i];
   242             if (carry == 0) {
   243                 if (j != 0) {
   244                     carry = 1;
   245                     iq[i] = 0x1000000 - j;
   246                 }
   247             } else
   248                 iq[i] = 0xffffff - j;
   249         }
   250         if (q0 > 0) {           /* rare case: chance is 1 in 12 */
   251             switch (q0) {
   252             case 1:
   253                 iq[jz - 1] &= 0x7fffff;
   254                 break;
   255             case 2:
   256                 iq[jz - 1] &= 0x3fffff;
   257                 break;
   258             }
   259         }
   260         if (ih == 2) {
   261             z = one - z;
   262             if (carry != 0)
   263                 z -= scalbn(one, q0);
   264         }
   265     }
   266 
   267     /* check if recomputation is needed */
   268     if (z == zero) {
   269         j = 0;
   270         for (i = jz - 1; i >= jk; i--)
   271             j |= iq[i];
   272         if (j == 0) {           /* need recomputation */
   273             for (k = 1; iq[jk - k] == 0; k++);  /* k = no. of terms needed */
   274 
   275             for (i = jz + 1; i <= jz + k; i++) {        /* add q[jz+1] to q[jz+k] */
   276                 f[jx + i] = (double) ipio2[jv + i];
   277                 for (j = 0, fw = 0.0; j <= jx; j++)
   278                     fw += x[j] * f[jx + i - j];
   279                 q[i] = fw;
   280             }
   281             jz += k;
   282             goto recompute;
   283         }
   284     }
   285 
   286     /* chop off zero terms */
   287     if (z == 0.0) {
   288         jz -= 1;
   289         q0 -= 24;
   290         while (iq[jz] == 0) {
   291             jz--;
   292             q0 -= 24;
   293         }
   294     } else {                    /* break z into 24-bit if necessary */
   295         z = scalbn(z, -q0);
   296         if (z >= two24) {
   297             fw = (double) ((int32_t) (twon24 * z));
   298             iq[jz] = (int32_t) (z - two24 * fw);
   299             jz += 1;
   300             q0 += 24;
   301             iq[jz] = (int32_t) fw;
   302         } else
   303             iq[jz] = (int32_t) z;
   304     }
   305 
   306     /* convert integer "bit" chunk to floating-point value */
   307     fw = scalbn(one, q0);
   308     for (i = jz; i >= 0; i--) {
   309         q[i] = fw * (double) iq[i];
   310         fw *= twon24;
   311     }
   312 
   313     /* compute PIo2[0,...,jp]*q[jz,...,0] */
   314     for (i = jz; i >= 0; i--) {
   315         for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
   316             fw += PIo2[k] * q[i + k];
   317         fq[jz - i] = fw;
   318     }
   319 
   320     /* compress fq[] into y[] */
   321     switch (prec) {
   322     case 0:
   323         fw = 0.0;
   324         for (i = jz; i >= 0; i--)
   325             fw += fq[i];
   326         y[0] = (ih == 0) ? fw : -fw;
   327         break;
   328     case 1:
   329     case 2:
   330         fw = 0.0;
   331         for (i = jz; i >= 0; i--)
   332             fw += fq[i];
   333         y[0] = (ih == 0) ? fw : -fw;
   334         fw = fq[0] - fw;
   335         for (i = 1; i <= jz; i++)
   336             fw += fq[i];
   337         y[1] = (ih == 0) ? fw : -fw;
   338         break;
   339     case 3:                    /* painful */
   340         for (i = jz; i > 0; i--) {
   341             fw = fq[i - 1] + fq[i];
   342             fq[i] += fq[i - 1] - fw;
   343             fq[i - 1] = fw;
   344         }
   345         for (i = jz; i > 1; i--) {
   346             fw = fq[i - 1] + fq[i];
   347             fq[i] += fq[i - 1] - fw;
   348             fq[i - 1] = fw;
   349         }
   350         for (fw = 0.0, i = jz; i >= 2; i--)
   351             fw += fq[i];
   352         if (ih == 0) {
   353             y[0] = fq[0];
   354             y[1] = fq[1];
   355             y[2] = fw;
   356         } else {
   357             y[0] = -fq[0];
   358             y[1] = -fq[1];
   359             y[2] = -fw;
   360         }
   361     }
   362     return n & 7;
   363 }