src/libm/k_rem_pio2.c
author Ryan C. Gordon <icculus@icculus.org>
Mon, 21 May 2018 12:00:21 -0400
changeset 11994 8e094f91ddab
parent 11838 5ef6e4e70103
child 12068 ce88faaf8bd2
permissions -rw-r--r--
thread: fixed compiler warnings on non-Linux systems that use pthread.

(static function rtkit_setpriority was unused, moved it in with rest of
__LINUX__ section.)
slouken@2757
     1
/*
slouken@2757
     2
 * ====================================================
slouken@2757
     3
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
slouken@2757
     4
 *
slouken@2757
     5
 * Developed at SunPro, a Sun Microsystems, Inc. business.
slouken@2757
     6
 * Permission to use, copy, modify, and distribute this
slouken@2757
     7
 * software is freely granted, provided that this notice
slouken@2757
     8
 * is preserved.
slouken@2757
     9
 * ====================================================
slouken@2757
    10
 */
slouken@2757
    11
slouken@2757
    12
/*
slouken@2757
    13
 * __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
slouken@2757
    14
 * double x[],y[]; int e0,nx,prec; int ipio2[];
slouken@2757
    15
 *
slouken@2757
    16
 * __kernel_rem_pio2 return the last three digits of N with
slouken@2757
    17
 *		y = x - N*pi/2
slouken@2757
    18
 * so that |y| < pi/2.
slouken@2757
    19
 *
slouken@2757
    20
 * The method is to compute the integer (mod 8) and fraction parts of
slouken@2757
    21
 * (2/pi)*x without doing the full multiplication. In general we
slouken@2757
    22
 * skip the part of the product that are known to be a huge integer (
slouken@2757
    23
 * more accurately, = 0 mod 8 ). Thus the number of operations are
slouken@2757
    24
 * independent of the exponent of the input.
slouken@2757
    25
 *
slouken@2757
    26
 * (2/pi) is represented by an array of 24-bit integers in ipio2[].
slouken@2757
    27
 *
slouken@2757
    28
 * Input parameters:
slouken@2757
    29
 * 	x[]	The input value (must be positive) is broken into nx
slouken@2757
    30
 *		pieces of 24-bit integers in double precision format.
slouken@2757
    31
 *		x[i] will be the i-th 24 bit of x. The scaled exponent
slouken@2757
    32
 *		of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
slouken@2757
    33
 *		match x's up to 24 bits.
slouken@2757
    34
 *
slouken@2757
    35
 *		Example of breaking a double positive z into x[0]+x[1]+x[2]:
slouken@2757
    36
 *			e0 = ilogb(z)-23
slouken@2757
    37
 *			z  = scalbn(z,-e0)
slouken@2757
    38
 *		for i = 0,1,2
slouken@2757
    39
 *			x[i] = floor(z)
slouken@2757
    40
 *			z    = (z-x[i])*2**24
slouken@2757
    41
 *
slouken@2757
    42
 *
slouken@2757
    43
 *	y[]	ouput result in an array of double precision numbers.
slouken@2757
    44
 *		The dimension of y[] is:
slouken@2757
    45
 *			24-bit  precision	1
slouken@2757
    46
 *			53-bit  precision	2
slouken@2757
    47
 *			64-bit  precision	2
slouken@2757
    48
 *			113-bit precision	3
slouken@2757
    49
 *		The actual value is the sum of them. Thus for 113-bit
slouken@2757
    50
 *		precison, one may have to do something like:
slouken@2757
    51
 *
slouken@2757
    52
 *		long double t,w,r_head, r_tail;
slouken@2757
    53
 *		t = (long double)y[2] + (long double)y[1];
slouken@2757
    54
 *		w = (long double)y[0];
slouken@2757
    55
 *		r_head = t+w;
slouken@2757
    56
 *		r_tail = w - (r_head - t);
slouken@2757
    57
 *
slouken@2757
    58
 *	e0	The exponent of x[0]
slouken@2757
    59
 *
slouken@2757
    60
 *	nx	dimension of x[]
slouken@2757
    61
 *
slouken@2757
    62
 *  	prec	an integer indicating the precision:
slouken@2757
    63
 *			0	24  bits (single)
slouken@2757
    64
 *			1	53  bits (double)
slouken@2757
    65
 *			2	64  bits (extended)
slouken@2757
    66
 *			3	113 bits (quad)
slouken@2757
    67
 *
slouken@2757
    68
 *	ipio2[]
slouken@2757
    69
 *		integer array, contains the (24*i)-th to (24*i+23)-th
slouken@2757
    70
 *		bit of 2/pi after binary point. The corresponding
slouken@2757
    71
 *		floating value is
slouken@2757
    72
 *
slouken@2757
    73
 *			ipio2[i] * 2^(-24(i+1)).
slouken@2757
    74
 *
slouken@2757
    75
 * External function:
slouken@2757
    76
 *	double scalbn(), floor();
slouken@2757
    77
 *
slouken@2757
    78
 *
slouken@2757
    79
 * Here is the description of some local variables:
slouken@2757
    80
 *
slouken@2757
    81
 * 	jk	jk+1 is the initial number of terms of ipio2[] needed
slouken@2757
    82
 *		in the computation. The recommended value is 2,3,4,
slouken@2757
    83
 *		6 for single, double, extended,and quad.
slouken@2757
    84
 *
slouken@2757
    85
 * 	jz	local integer variable indicating the number of
slouken@2757
    86
 *		terms of ipio2[] used.
slouken@2757
    87
 *
slouken@2757
    88
 *	jx	nx - 1
slouken@2757
    89
 *
slouken@2757
    90
 *	jv	index for pointing to the suitable ipio2[] for the
slouken@2757
    91
 *		computation. In general, we want
slouken@2757
    92
 *			( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
slouken@2757
    93
 *		is an integer. Thus
slouken@2757
    94
 *			e0-3-24*jv >= 0 or (e0-3)/24 >= jv
slouken@2757
    95
 *		Hence jv = max(0,(e0-3)/24).
slouken@2757
    96
 *
slouken@2757
    97
 *	jp	jp+1 is the number of terms in PIo2[] needed, jp = jk.
slouken@2757
    98
 *
slouken@2757
    99
 * 	q[]	double array with integral value, representing the
slouken@2757
   100
 *		24-bits chunk of the product of x and 2/pi.
slouken@2757
   101
 *
slouken@2757
   102
 *	q0	the corresponding exponent of q[0]. Note that the
slouken@2757
   103
 *		exponent for q[i] would be q0-24*i.
slouken@2757
   104
 *
slouken@2757
   105
 *	PIo2[]	double precision array, obtained by cutting pi/2
slouken@2757
   106
 *		into 24 bits chunks.
slouken@2757
   107
 *
slouken@2757
   108
 *	f[]	ipio2[] in floating point
slouken@2757
   109
 *
slouken@2757
   110
 *	iq[]	integer array by breaking up q[] in 24-bits chunk.
slouken@2757
   111
 *
slouken@2757
   112
 *	fq[]	final product of x*(2/pi) in fq[0],..,fq[jk]
slouken@2757
   113
 *
slouken@2757
   114
 *	ih	integer. If >0 it indicates q[] is >= 0.5, hence
slouken@2757
   115
 *		it also indicates the *sign* of the result.
slouken@2757
   116
 *
slouken@2757
   117
 */
slouken@2757
   118
slouken@2757
   119
slouken@2757
   120
/*
slouken@2757
   121
 * Constants:
slouken@2757
   122
 * The hexadecimal values are the intended ones for the following
slouken@2757
   123
 * constants. The decimal values may be used, provided that the
slouken@2757
   124
 * compiler will convert from decimal to binary accurately enough
slouken@2757
   125
 * to produce the hexadecimal values shown.
slouken@2757
   126
 */
slouken@2757
   127
slouken@6044
   128
#include "math_libm.h"
slouken@2757
   129
#include "math_private.h"
slouken@2757
   130
slouken@11683
   131
static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
icculus@8670
   132
slouken@2757
   133
static const double PIo2[] = {
slouken@11683
   134
  1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
slouken@11683
   135
  7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
slouken@11683
   136
  5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
slouken@11683
   137
  3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
slouken@11683
   138
  1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
slouken@11683
   139
  1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
slouken@11683
   140
  2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
slouken@11683
   141
  2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
slouken@2757
   142
};
slouken@2757
   143
slouken@2757
   144
static const double
slouken@11683
   145
zero   = 0.0,
slouken@11683
   146
one    = 1.0,
slouken@11683
   147
two24   =  1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
slouken@11683
   148
twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
slouken@2757
   149
slouken@11683
   150
int attribute_hidden __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)
slouken@2757
   151
{
slouken@11683
   152
	int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
slouken@11683
   153
	double z,fw,f[20],fq[20],q[20];
slouken@2757
   154
slouken@11683
   155
    /* initialize jk*/
slouken@11683
   156
	jk = init_jk[prec];
slouken@11683
   157
	jp = jk;
slouken@2757
   158
slouken@2757
   159
    /* determine jx,jv,q0, note that 3>q0 */
slouken@11683
   160
	jx =  nx-1;
slouken@11683
   161
	jv = (e0-3)/24; if(jv<0) jv=0;
slouken@11683
   162
	q0 =  e0-24*(jv+1);
slouken@2757
   163
slouken@2757
   164
    /* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
slouken@11683
   165
	j = jv-jx; m = jx+jk;
slouken@11683
   166
	for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
slouken@2757
   167
slouken@2757
   168
    /* compute q[0],q[1],...q[jk] */
slouken@11683
   169
	for (i=0;i<=jk;i++) {
slouken@11838
   170
	    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
slouken@11838
   171
	    q[i] = fw;
slouken@11683
   172
	}
slouken@2757
   173
slouken@11683
   174
	jz = jk;
slouken@11683
   175
recompute:
slouken@2757
   176
    /* distill q[] into iq[] reversingly */
slouken@11683
   177
	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
slouken@11683
   178
	    fw    =  (double)((int32_t)(twon24* z));
slouken@11683
   179
	    iq[i] =  (int32_t)(z-two24*fw);
slouken@11683
   180
	    z     =  q[j-1]+fw;
slouken@11683
   181
	}
slouken@2757
   182
slouken@2757
   183
    /* compute n */
slouken@11683
   184
	z  = scalbn(z,q0);		/* actual value of z */
slouken@11683
   185
	z -= 8.0*floor(z*0.125);		/* trim off integer >= 8 */
slouken@11683
   186
	n  = (int32_t) z;
slouken@11683
   187
	z -= (double)n;
slouken@11683
   188
	ih = 0;
slouken@11683
   189
	if(q0>0) {	/* need iq[jz-1] to determine n */
slouken@11683
   190
	    i  = (iq[jz-1]>>(24-q0)); n += i;
slouken@11683
   191
	    iq[jz-1] -= i<<(24-q0);
slouken@11683
   192
	    ih = iq[jz-1]>>(23-q0);
slouken@11683
   193
	}
slouken@11683
   194
	else if(q0==0) ih = iq[jz-1]>>23;
slouken@11683
   195
	else if(z>=0.5) ih=2;
slouken@2757
   196
slouken@11683
   197
	if(ih>0) {	/* q > 0.5 */
slouken@11683
   198
	    n += 1; carry = 0;
slouken@11683
   199
	    for(i=0;i<jz ;i++) {	/* compute 1-q */
slouken@11683
   200
		j = iq[i];
slouken@11683
   201
		if(carry==0) {
slouken@11683
   202
		    if(j!=0) {
slouken@11683
   203
			carry = 1; iq[i] = 0x1000000- j;
slouken@11683
   204
		    }
slouken@11683
   205
		} else  iq[i] = 0xffffff - j;
slouken@11683
   206
	    }
slouken@11683
   207
	    if(q0>0) {		/* rare case: chance is 1 in 12 */
slouken@11683
   208
	        switch(q0) {
slouken@11683
   209
	        case 1:
slouken@11683
   210
	    	   iq[jz-1] &= 0x7fffff; break;
slouken@11683
   211
	    	case 2:
slouken@11683
   212
	    	   iq[jz-1] &= 0x3fffff; break;
slouken@11683
   213
	        }
slouken@11683
   214
	    }
slouken@11683
   215
	    if(ih==2) {
slouken@11683
   216
		z = one - z;
slouken@11683
   217
		if(carry!=0) z -= scalbn(one,q0);
slouken@11683
   218
	    }
slouken@11683
   219
	}
slouken@2757
   220
slouken@2757
   221
    /* check if recomputation is needed */
slouken@11683
   222
	if(z==zero) {
slouken@11683
   223
	    j = 0;
slouken@11683
   224
	    for (i=jz-1;i>=jk;i--) j |= iq[i];
slouken@11683
   225
	    if(j==0) { /* need recomputation */
slouken@11683
   226
		for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */
slouken@2757
   227
slouken@11683
   228
		for(i=jz+1;i<=jz+k;i++) {   /* add q[jz+1] to q[jz+k] */
slouken@11683
   229
		    f[jx+i] = (double) ipio2[jv+i];
slouken@11683
   230
		    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
slouken@11683
   231
		    q[i] = fw;
slouken@11683
   232
		}
slouken@11683
   233
		jz += k;
slouken@11683
   234
		goto recompute;
slouken@11683
   235
	    }
slouken@11683
   236
	}
slouken@2757
   237
slouken@2757
   238
    /* chop off zero terms */
slouken@11683
   239
	if(z==0.0) {
slouken@11683
   240
	    jz -= 1; q0 -= 24;
slouken@11683
   241
	    while(iq[jz]==0) { jz--; q0-=24;}
slouken@11683
   242
	} else { /* break z into 24-bit if necessary */
slouken@11683
   243
	    z = scalbn(z,-q0);
slouken@11683
   244
	    if(z>=two24) {
slouken@11683
   245
		fw = (double)((int32_t)(twon24*z));
slouken@11683
   246
		iq[jz] = (int32_t)(z-two24*fw);
slouken@11683
   247
		jz += 1; q0 += 24;
slouken@11683
   248
		iq[jz] = (int32_t) fw;
slouken@11683
   249
	    } else iq[jz] = (int32_t) z ;
slouken@11683
   250
	}
slouken@2757
   251
slouken@2757
   252
    /* convert integer "bit" chunk to floating-point value */
slouken@11683
   253
	fw = scalbn(one,q0);
slouken@11683
   254
	for(i=jz;i>=0;i--) {
slouken@11683
   255
	    q[i] = fw*(double)iq[i]; fw*=twon24;
slouken@11683
   256
	}
slouken@2757
   257
slouken@2757
   258
    /* compute PIo2[0,...,jp]*q[jz,...,0] */
slouken@11683
   259
	for(i=jz;i>=0;i--) {
slouken@11683
   260
	    for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
slouken@11683
   261
	    fq[jz-i] = fw;
slouken@11683
   262
	}
slouken@2757
   263
slouken@2757
   264
    /* compress fq[] into y[] */
slouken@11683
   265
	switch(prec) {
slouken@11683
   266
	    case 0:
slouken@11683
   267
		fw = 0.0;
slouken@11683
   268
		for (i=jz;i>=0;i--) fw += fq[i];
slouken@11683
   269
		y[0] = (ih==0)? fw: -fw;
slouken@11683
   270
		break;
slouken@11683
   271
	    case 1:
slouken@11683
   272
	    case 2:
slouken@11683
   273
		fw = 0.0;
slouken@11683
   274
		for (i=jz;i>=0;i--) fw += fq[i];
slouken@11683
   275
		y[0] = (ih==0)? fw: -fw;
slouken@11683
   276
		fw = fq[0]-fw;
slouken@11683
   277
		for (i=1;i<=jz;i++) fw += fq[i];
slouken@11683
   278
		y[1] = (ih==0)? fw: -fw;
slouken@11683
   279
		break;
slouken@11683
   280
	    case 3:	/* painful */
slouken@11683
   281
		for (i=jz;i>0;i--) {
slouken@11683
   282
		    fw      = fq[i-1]+fq[i];
slouken@11683
   283
		    fq[i]  += fq[i-1]-fw;
slouken@11683
   284
		    fq[i-1] = fw;
slouken@11683
   285
		}
slouken@11683
   286
		for (i=jz;i>1;i--) {
slouken@11683
   287
		    fw      = fq[i-1]+fq[i];
slouken@11683
   288
		    fq[i]  += fq[i-1]-fw;
slouken@11683
   289
		    fq[i-1] = fw;
slouken@11683
   290
		}
slouken@11683
   291
		for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
slouken@11683
   292
		if(ih==0) {
slouken@11683
   293
		    y[0] =  fq[0]; y[1] =  fq[1]; y[2] =  fw;
slouken@11683
   294
		} else {
slouken@11683
   295
		    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
slouken@11683
   296
		}
slouken@11683
   297
	}
slouken@11683
   298
	return n&7;
slouken@2757
   299
}