/* @(#)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.

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

*/

#if defined(LIBM_SCCS) && !defined(lint)

static const char rcsid[] =

"$NetBSD: k_rem_pio2.c,v 1.7 1995/05/10 20:46:25 jtc 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_libm.h"

#include "math_private.h"

#include "SDL_assert.h"

libm_hidden_proto(scalbn)

libm_hidden_proto(floor)

#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 attribute_hidden

__kernel_rem_pio2(double *x, double *y, int e0, int nx, int prec,

const int32_t * ipio2)

#else

int attribute_hidden

__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 */

SDL_assert((prec >= 0) && (prec < SDL_arraysize(init_jk)));

jk = init_jk[prec];

SDL_assert((jk >= 2) && (jk <= 6));

jp = jk;

/* determine jx,jv,q0, note that 3>q0 */

SDL_assert(nx > 0);

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++) {

const int32_t idx = jx + i - j;

SDL_assert(idx >= 0);

SDL_assert(idx < 20);

SDL_assert(idx <= m);

fw += x[j] * f[idx];

}

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;

}