mirror of
git://gcc.gnu.org/git/gcc.git
synced 2024-12-19 09:49:11 +08:00
321 lines
8.4 KiB
C
321 lines
8.4 KiB
C
|
|
/* @(#)k_rem_pio2.c 5.1 93/09/24 */
|
|
/*
|
|
* ====================================================
|
|
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
|
|
*
|
|
* Developed at SunPro, a Sun Microsystems, Inc. business.
|
|
* Permission to use, copy, modify, and distribute this
|
|
* software is freely granted, provided that this notice
|
|
* is preserved.
|
|
* ====================================================
|
|
*/
|
|
|
|
/*
|
|
* __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
|
|
* double x[],y[]; int e0,nx,prec; int ipio2[];
|
|
*
|
|
* __kernel_rem_pio2 return the last three digits of N with
|
|
* y = x - N*pi/2
|
|
* so that |y| < pi/2.
|
|
*
|
|
* The method is to compute the integer (mod 8) and fraction parts of
|
|
* (2/pi)*x without doing the full multiplication. In general we
|
|
* skip the part of the product that are known to be a huge integer (
|
|
* more accurately, = 0 mod 8 ). Thus the number of operations are
|
|
* independent of the exponent of the input.
|
|
*
|
|
* (2/pi) is represented by an array of 24-bit integers in ipio2[].
|
|
*
|
|
* Input parameters:
|
|
* x[] The input value (must be positive) is broken into nx
|
|
* pieces of 24-bit integers in double precision format.
|
|
* x[i] will be the i-th 24 bit of x. The scaled exponent
|
|
* of x[0] is given in input parameter e0 (i.e., x[0]*2^e0
|
|
* match x's up to 24 bits.
|
|
*
|
|
* Example of breaking a double positive z into x[0]+x[1]+x[2]:
|
|
* e0 = ilogb(z)-23
|
|
* z = scalbn(z,-e0)
|
|
* for i = 0,1,2
|
|
* x[i] = floor(z)
|
|
* z = (z-x[i])*2**24
|
|
*
|
|
*
|
|
* y[] ouput result in an array of double precision numbers.
|
|
* The dimension of y[] is:
|
|
* 24-bit precision 1
|
|
* 53-bit precision 2
|
|
* 64-bit precision 2
|
|
* 113-bit precision 3
|
|
* The actual value is the sum of them. Thus for 113-bit
|
|
* precison, one may have to do something like:
|
|
*
|
|
* long double t,w,r_head, r_tail;
|
|
* t = (long double)y[2] + (long double)y[1];
|
|
* w = (long double)y[0];
|
|
* r_head = t+w;
|
|
* r_tail = w - (r_head - t);
|
|
*
|
|
* e0 The exponent of x[0]
|
|
*
|
|
* nx dimension of x[]
|
|
*
|
|
* prec an integer indicating the precision:
|
|
* 0 24 bits (single)
|
|
* 1 53 bits (double)
|
|
* 2 64 bits (extended)
|
|
* 3 113 bits (quad)
|
|
*
|
|
* ipio2[]
|
|
* integer array, contains the (24*i)-th to (24*i+23)-th
|
|
* bit of 2/pi after binary point. The corresponding
|
|
* floating value is
|
|
*
|
|
* ipio2[i] * 2^(-24(i+1)).
|
|
*
|
|
* External function:
|
|
* double scalbn(), floor();
|
|
*
|
|
*
|
|
* Here is the description of some local variables:
|
|
*
|
|
* jk jk+1 is the initial number of terms of ipio2[] needed
|
|
* in the computation. The recommended value is 2,3,4,
|
|
* 6 for single, double, extended,and quad.
|
|
*
|
|
* jz local integer variable indicating the number of
|
|
* terms of ipio2[] used.
|
|
*
|
|
* jx nx - 1
|
|
*
|
|
* jv index for pointing to the suitable ipio2[] for the
|
|
* computation. In general, we want
|
|
* ( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
|
|
* is an integer. Thus
|
|
* e0-3-24*jv >= 0 or (e0-3)/24 >= jv
|
|
* Hence jv = max(0,(e0-3)/24).
|
|
*
|
|
* jp jp+1 is the number of terms in PIo2[] needed, jp = jk.
|
|
*
|
|
* q[] double array with integral value, representing the
|
|
* 24-bits chunk of the product of x and 2/pi.
|
|
*
|
|
* q0 the corresponding exponent of q[0]. Note that the
|
|
* exponent for q[i] would be q0-24*i.
|
|
*
|
|
* PIo2[] double precision array, obtained by cutting pi/2
|
|
* into 24 bits chunks.
|
|
*
|
|
* f[] ipio2[] in floating point
|
|
*
|
|
* iq[] integer array by breaking up q[] in 24-bits chunk.
|
|
*
|
|
* fq[] final product of x*(2/pi) in fq[0],..,fq[jk]
|
|
*
|
|
* ih integer. If >0 it indicates q[] is >= 0.5, hence
|
|
* it also indicates the *sign* of the result.
|
|
*
|
|
*/
|
|
|
|
|
|
/*
|
|
* Constants:
|
|
* The hexadecimal values are the intended ones for the following
|
|
* constants. The decimal values may be used, provided that the
|
|
* compiler will convert from decimal to binary accurately enough
|
|
* to produce the hexadecimal values shown.
|
|
*/
|
|
|
|
#include "fdlibm.h"
|
|
|
|
#ifndef _DOUBLE_IS_32BITS
|
|
|
|
#ifdef __STDC__
|
|
static const int init_jk[] = {2,3,4,6}; /* initial value for jk */
|
|
#else
|
|
static int init_jk[] = {2,3,4,6};
|
|
#endif
|
|
|
|
#ifdef __STDC__
|
|
static const double PIo2[] = {
|
|
#else
|
|
static double PIo2[] = {
|
|
#endif
|
|
1.57079625129699707031e+00, /* 0x3FF921FB, 0x40000000 */
|
|
7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
|
|
5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
|
|
3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
|
|
1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
|
|
1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
|
|
2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
|
|
2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
|
|
};
|
|
|
|
#ifdef __STDC__
|
|
static const double
|
|
#else
|
|
static double
|
|
#endif
|
|
zero = 0.0,
|
|
one = 1.0,
|
|
two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */
|
|
twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
|
|
|
|
#ifdef __STDC__
|
|
int __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const __int32_t *ipio2)
|
|
#else
|
|
int __kernel_rem_pio2(x,y,e0,nx,prec,ipio2)
|
|
double x[], y[]; int e0,nx,prec; __int32_t ipio2[];
|
|
#endif
|
|
{
|
|
__int32_t jz,jx,jv,jp,jk,carry,n,iq[20],i,j,k,m,q0,ih;
|
|
double z,fw,f[20],fq[20],q[20];
|
|
|
|
/* initialize jk*/
|
|
jk = init_jk[prec];
|
|
jp = jk;
|
|
|
|
/* determine jx,jv,q0, note that 3>q0 */
|
|
jx = nx-1;
|
|
jv = (e0-3)/24; if(jv<0) jv=0;
|
|
q0 = e0-24*(jv+1);
|
|
|
|
/* set up f[0] to f[jx+jk] where f[jx+jk] = ipio2[jv+jk] */
|
|
j = jv-jx; m = jx+jk;
|
|
for(i=0;i<=m;i++,j++) f[i] = (j<0)? zero : (double) ipio2[j];
|
|
|
|
/* compute q[0],q[1],...q[jk] */
|
|
for (i=0;i<=jk;i++) {
|
|
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;
|
|
}
|
|
|
|
jz = jk;
|
|
recompute:
|
|
/* distill q[] into iq[] reversingly */
|
|
for(i=0,j=jz,z=q[jz];j>0;i++,j--) {
|
|
fw = (double)((__int32_t)(twon24* z));
|
|
iq[i] = (__int32_t)(z-two24*fw);
|
|
z = q[j-1]+fw;
|
|
}
|
|
|
|
/* compute n */
|
|
z = scalbn(z,(int)q0); /* actual value of z */
|
|
z -= 8.0*floor(z*0.125); /* trim off integer >= 8 */
|
|
n = (__int32_t) z;
|
|
z -= (double)n;
|
|
ih = 0;
|
|
if(q0>0) { /* need iq[jz-1] to determine n */
|
|
i = (iq[jz-1]>>(24-q0)); n += i;
|
|
iq[jz-1] -= i<<(24-q0);
|
|
ih = iq[jz-1]>>(23-q0);
|
|
}
|
|
else if(q0==0) ih = iq[jz-1]>>23;
|
|
else if(z>=0.5) ih=2;
|
|
|
|
if(ih>0) { /* q > 0.5 */
|
|
n += 1; carry = 0;
|
|
for(i=0;i<jz ;i++) { /* compute 1-q */
|
|
j = iq[i];
|
|
if(carry==0) {
|
|
if(j!=0) {
|
|
carry = 1; iq[i] = 0x1000000- j;
|
|
}
|
|
} else iq[i] = 0xffffff - j;
|
|
}
|
|
if(q0>0) { /* rare case: chance is 1 in 12 */
|
|
switch(q0) {
|
|
case 1:
|
|
iq[jz-1] &= 0x7fffff; break;
|
|
case 2:
|
|
iq[jz-1] &= 0x3fffff; break;
|
|
}
|
|
}
|
|
if(ih==2) {
|
|
z = one - z;
|
|
if(carry!=0) z -= scalbn(one,(int)q0);
|
|
}
|
|
}
|
|
|
|
/* check if recomputation is needed */
|
|
if(z==zero) {
|
|
j = 0;
|
|
for (i=jz-1;i>=jk;i--) j |= iq[i];
|
|
if(j==0) { /* need recomputation */
|
|
for(k=1;iq[jk-k]==0;k++); /* k = no. of terms needed */
|
|
|
|
for(i=jz+1;i<=jz+k;i++) { /* add q[jz+1] to q[jz+k] */
|
|
f[jx+i] = (double) ipio2[jv+i];
|
|
for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];
|
|
q[i] = fw;
|
|
}
|
|
jz += k;
|
|
goto recompute;
|
|
}
|
|
}
|
|
|
|
/* chop off zero terms */
|
|
if(z==0.0) {
|
|
jz -= 1; q0 -= 24;
|
|
while(iq[jz]==0) { jz--; q0-=24;}
|
|
} else { /* break z into 24-bit if necessary */
|
|
z = scalbn(z,-(int)q0);
|
|
if(z>=two24) {
|
|
fw = (double)((__int32_t)(twon24*z));
|
|
iq[jz] = (__int32_t)(z-two24*fw);
|
|
jz += 1; q0 += 24;
|
|
iq[jz] = (__int32_t) fw;
|
|
} else iq[jz] = (__int32_t) z ;
|
|
}
|
|
|
|
/* convert integer "bit" chunk to floating-point value */
|
|
fw = scalbn(one,(int)q0);
|
|
for(i=jz;i>=0;i--) {
|
|
q[i] = fw*(double)iq[i]; fw*=twon24;
|
|
}
|
|
|
|
/* compute PIo2[0,...,jp]*q[jz,...,0] */
|
|
for(i=jz;i>=0;i--) {
|
|
for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];
|
|
fq[jz-i] = fw;
|
|
}
|
|
|
|
/* compress fq[] into y[] */
|
|
switch(prec) {
|
|
case 0:
|
|
fw = 0.0;
|
|
for (i=jz;i>=0;i--) fw += fq[i];
|
|
y[0] = (ih==0)? fw: -fw;
|
|
break;
|
|
case 1:
|
|
case 2:
|
|
fw = 0.0;
|
|
for (i=jz;i>=0;i--) fw += fq[i];
|
|
y[0] = (ih==0)? fw: -fw;
|
|
fw = fq[0]-fw;
|
|
for (i=1;i<=jz;i++) fw += fq[i];
|
|
y[1] = (ih==0)? fw: -fw;
|
|
break;
|
|
case 3: /* painful */
|
|
for (i=jz;i>0;i--) {
|
|
fw = fq[i-1]+fq[i];
|
|
fq[i] += fq[i-1]-fw;
|
|
fq[i-1] = fw;
|
|
}
|
|
for (i=jz;i>1;i--) {
|
|
fw = fq[i-1]+fq[i];
|
|
fq[i] += fq[i-1]-fw;
|
|
fq[i-1] = fw;
|
|
}
|
|
for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];
|
|
if(ih==0) {
|
|
y[0] = fq[0]; y[1] = fq[1]; y[2] = fw;
|
|
} else {
|
|
y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;
|
|
}
|
|
}
|
|
return n&7;
|
|
}
|
|
|
|
#endif /* defined(_DOUBLE_IS_32BITS) */
|