/*

* ====================================================

* 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 "math_libm.h"

#include "math_private.h"

static const int init_jk[] = {2,3,4,6}; /* initial value for jk */

static const double PIo2[] = {

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 */

};

static const double

zero = 0.0,

one = 1.0,

two24 = 1.67772160000000000000e+07, /* 0x41700000, 0x00000000 */

twon24 = 5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */

int attribute_hidden __kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec, const int32_t *ipio2)

{

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;

}