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/* @(#)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.
* ====================================================
*/
#ifndef lint
static char rcsid
[] = "$\Id: k_rem_pio2.c,v 1.2 1995/05/30 05:48:57 rgrimes Exp $";
#endif
/*
* __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 "math.h"
#include "math_private.h"
#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
,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
,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
,-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
,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;
}