src/libm/k_rem_pio2.c
author Sam Lantinga
Sat, 19 Sep 2009 13:29:40 +0000
changeset 3280 00cace2d9080
parent 3162 dc1eb82ffdaa
child 6044 35448a5ea044
permissions -rw-r--r--
Merged a cleaned up version of Jiang's code changes from Google Summer of Code 2009
     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.h"
   135 #include "math_private.h"
   136 
   137 libm_hidden_proto(scalbn)
   138     libm_hidden_proto(floor)
   139 #ifdef __STDC__
   140      static const int init_jk[] = { 2, 3, 4, 6 };       /* initial value for jk */
   141 #else
   142      static int init_jk[] = { 2, 3, 4, 6 };
   143 #endif
   144 
   145 #ifdef __STDC__
   146 static const double PIo2[] = {
   147 #else
   148 static double PIo2[] = {
   149 #endif
   150     1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
   151     7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
   152     5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
   153     3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
   154     1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
   155     1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
   156     2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
   157     2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
   158 };
   159 
   160 #ifdef __STDC__
   161 static const double
   162 #else
   163 static double
   164 #endif
   165   zero = 0.0, one = 1.0, two24 = 1.67772160000000000000e+07,    /* 0x41700000, 0x00000000 */
   166     twon24 = 5.96046447753906250000e-08;        /* 0x3E700000, 0x00000000 */
   167 
   168 #ifdef __STDC__
   169 int attribute_hidden
   170 __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,
   171                   const int32_t * ipio2)
   172 #else
   173 int attribute_hidden
   174 __kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
   175      double x[], y[];
   176      int e0, nx, prec;
   177      int32_t ipio2[];
   178 #endif
   179 {
   180     int32_t jz, jx, jv, jp, jk, carry, n, iq[20], i, j, k, m, q0, ih;
   181     double z, fw, f[20], fq[20], q[20];
   182 
   183     /* initialize jk */
   184     jk = init_jk[prec];
   185     jp = jk;
   186 
   187     /* determine jx,jv,q0, note that 3>q0 */
   188     jx = nx - 1;
   189     jv = (e0 - 3) / 24;
   190     if (jv < 0)
   191         jv = 0;
   192     q0 = e0 - 24 * (jv + 1);
   193 
   194     /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
   195     j = jv - jx;
   196     m = jx + jk;
   197     for (i = 0; i <= m; i++, j++)
   198         f[i] = (j < 0) ? zero : (double) ipio2[j];
   199 
   200     /* compute q[0],q[1],...q[jk] */
   201     for (i = 0; i <= jk; i++) {
   202         for (j = 0, fw = 0.0; j <= jx; j++)
   203             fw += x[j] * f[jx + i - j];
   204         q[i] = fw;
   205     }
   206 
   207     jz = jk;
   208   recompute:
   209     /* distill q[] into iq[] reversingly */
   210     for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
   211         fw = (double) ((int32_t) (twon24 * z));
   212         iq[i] = (int32_t) (z - two24 * fw);
   213         z = q[j - 1] + fw;
   214     }
   215 
   216     /* compute n */
   217     z = scalbn(z, q0);          /* actual value of z */
   218     z -= 8.0 * floor(z * 0.125);        /* trim off integer >= 8 */
   219     n = (int32_t) z;
   220     z -= (double) n;
   221     ih = 0;
   222     if (q0 > 0) {               /* need iq[jz-1] to determine n */
   223         i = (iq[jz - 1] >> (24 - q0));
   224         n += i;
   225         iq[jz - 1] -= i << (24 - q0);
   226         ih = iq[jz - 1] >> (23 - q0);
   227     } else if (q0 == 0)
   228         ih = iq[jz - 1] >> 23;
   229     else if (z >= 0.5)
   230         ih = 2;
   231 
   232     if (ih > 0) {               /* q > 0.5 */
   233         n += 1;
   234         carry = 0;
   235         for (i = 0; i < jz; i++) {      /* compute 1-q */
   236             j = iq[i];
   237             if (carry == 0) {
   238                 if (j != 0) {
   239                     carry = 1;
   240                     iq[i] = 0x1000000 - j;
   241                 }
   242             } else
   243                 iq[i] = 0xffffff - j;
   244         }
   245         if (q0 > 0) {           /* rare case: chance is 1 in 12 */
   246             switch (q0) {
   247             case 1:
   248                 iq[jz - 1] &= 0x7fffff;
   249                 break;
   250             case 2:
   251                 iq[jz - 1] &= 0x3fffff;
   252                 break;
   253             }
   254         }
   255         if (ih == 2) {
   256             z = one - z;
   257             if (carry != 0)
   258                 z -= scalbn(one, q0);
   259         }
   260     }
   261 
   262     /* check if recomputation is needed */
   263     if (z == zero) {
   264         j = 0;
   265         for (i = jz - 1; i >= jk; i--)
   266             j |= iq[i];
   267         if (j == 0) {           /* need recomputation */
   268             for (k = 1; iq[jk - k] == 0; k++);  /* k = no. of terms needed */
   269 
   270             for (i = jz + 1; i <= jz + k; i++) {        /* add q[jz+1] to q[jz+k] */
   271                 f[jx + i] = (double) ipio2[jv + i];
   272                 for (j = 0, fw = 0.0; j <= jx; j++)
   273                     fw += x[j] * f[jx + i - j];
   274                 q[i] = fw;
   275             }
   276             jz += k;
   277             goto recompute;
   278         }
   279     }
   280 
   281     /* chop off zero terms */
   282     if (z == 0.0) {
   283         jz -= 1;
   284         q0 -= 24;
   285         while (iq[jz] == 0) {
   286             jz--;
   287             q0 -= 24;
   288         }
   289     } else {                    /* break z into 24-bit if necessary */
   290         z = scalbn(z, -q0);
   291         if (z >= two24) {
   292             fw = (double) ((int32_t) (twon24 * z));
   293             iq[jz] = (int32_t) (z - two24 * fw);
   294             jz += 1;
   295             q0 += 24;
   296             iq[jz] = (int32_t) fw;
   297         } else
   298             iq[jz] = (int32_t) z;
   299     }
   300 
   301     /* convert integer "bit" chunk to floating-point value */
   302     fw = scalbn(one, q0);
   303     for (i = jz; i >= 0; i--) {
   304         q[i] = fw * (double) iq[i];
   305         fw *= twon24;
   306     }
   307 
   308     /* compute PIo2[0,...,jp]*q[jz,...,0] */
   309     for (i = jz; i >= 0; i--) {
   310         for (fw = 0.0, k = 0; k <= jp && k <= jz - i; k++)
   311             fw += PIo2[k] * q[i + k];
   312         fq[jz - i] = fw;
   313     }
   314 
   315     /* compress fq[] into y[] */
   316     switch (prec) {
   317     case 0:
   318         fw = 0.0;
   319         for (i = jz; i >= 0; i--)
   320             fw += fq[i];
   321         y[0] = (ih == 0) ? fw : -fw;
   322         break;
   323     case 1:
   324     case 2:
   325         fw = 0.0;
   326         for (i = jz; i >= 0; i--)
   327             fw += fq[i];
   328         y[0] = (ih == 0) ? fw : -fw;
   329         fw = fq[0] - fw;
   330         for (i = 1; i <= jz; i++)
   331             fw += fq[i];
   332         y[1] = (ih == 0) ? fw : -fw;
   333         break;
   334     case 3:                    /* painful */
   335         for (i = jz; i > 0; i--) {
   336             fw = fq[i - 1] + fq[i];
   337             fq[i] += fq[i - 1] - fw;
   338             fq[i - 1] = fw;
   339         }
   340         for (i = jz; i > 1; i--) {
   341             fw = fq[i - 1] + fq[i];
   342             fq[i] += fq[i - 1] - fw;
   343             fq[i - 1] = fw;
   344         }
   345         for (fw = 0.0, i = jz; i >= 2; i--)
   346             fw += fq[i];
   347         if (ih == 0) {
   348             y[0] = fq[0];
   349             y[1] = fq[1];
   350             y[2] = fw;
   351         } else {
   352             y[0] = -fq[0];
   353             y[1] = -fq[1];
   354             y[2] = -fw;
   355         }
   356     }
   357     return n & 7;
   358 }