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| //
// ====================================================
// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
//
// Developed at SunSoft, 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.
//
var init_jk = [2,3,4,6]; // initial value for jk
var PIo2 = new Float64Array(
[
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
]);
var zero = 0.0;
var one = 1.0;
var two24 = Math.pow(2, 24);
var twon24 = Math.pow(2, -24);
// Compute x*2^n using exponent manipulation instead of exponentiation
// or multiplication.
function kernel_rem_pio2(x, y, e0, nx, prec, ipio2)
{
var jz,jx,jv,jp,jk,carry,n;
var iq = new Int32Array(20);
var i,j,k,m,q0,ih;
var z,fw;
var f = new Float64Array(20);
var fq = new Float64Array(20);
var q = new Float64Array(20);
/* istanbul ignore if */
Iif (verbose > 0) {
console.log("P-H: x = " + x);
console.log("e0 = " + e0);
console.log("nx = " + nx);
console.log("prec = " + prec);
}
//console.log("ipio2 = " + ipio2);
/* initialize jk*/
jk = init_jk[prec];
jp = jk;
/* determine jx,jv,q0, note that 3>q0 */
jx = nx - 1;
jv = Math.floor((e0 - 3) / 24);
/* istanbul ignore if */
Iif (verbose > 0)
console.log("jv = " + jv);
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;
/* istanbul ignore if */
Iif (verbose > 0)
console.log("Setup f: j, m = " + j + ", " + m);
for (i = 0; i <= m; i++, j++)
f[i] = (j < 0) ? zero : ipio2[j];
/* istanbul ignore if */
Iif (verbose > 0) {
console.log("Post setup f: j, m = " + j + ", " + m);
console.log(" f = " + f);
}
/* 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;
}
/* istanbul ignore if */
Iif (verbose > 0) {
console.log("f = " + f);
console.log("q = " + q);
}
jz = jk;
var doRecompute = true;
recompute:
while (doRecompute) {
/* distill q[] into iq[] reversingly */
for (i = 0, j = jz, z = q[jz]; j > 0; i++, j--) {
//fw = (double)((int)(twon24 * z));
//iq[i] = (int)(z - two24 * fw);
fw = Math.floor(twon24 * z);
iq[i] = Math.floor(z - two24 * fw);
z = q[j - 1] + fw;
}
/* compute n */
z = scalbn(z, q0); /* actual value of z */
z -= 8.0 * Math.floor(z * 0.125); /* trim off integer >= 8 */
//n = (int) z;
n = Math.floor(z);
z -= 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;
Eif (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] = ipio2[jv + i];
for (j = 0, fw = 0.0; j <= jx; j++)
fw += x[j] * f[jx + i - j];
q[i] = fw;
}
jz += k;
/* istanbul ignore if */
Iif (verbose > 0)
console.log("Doing recomputation! jz = " + jz);
continue recompute;
}
}
doRecompute = false;
}
/* 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)((int)(twon24 * z));
//iq[jz] = (int)(z - two24 * fw);
fw = Math.floor(twon24 * z);
iq[jz] = Math.floor(z - two24 * fw);
jz += 1;
q0 += 24;
//iq[jz] = (int) fw;
iq[jz] = Math.floor(fw);
} else {
//iq[jz] = (int) z;
iq[jz] = Math.floor(z);
}
}
/* convert integer "bit" chunk to floating-point value */
fw = scalbn(one, q0);
for (i = jz; i >= 0; i--) {
q[i] = fw * 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;
}
/* istanbul ignore if */
Iif (verbose > 0)
console.log("PIo2 comp " + fq);
/* 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;
}
*/
}
/* istanbul ignore if */
Iif (verbose > 0)
console.log ("Return n = " + n + ", y = " + y);
return n & 7;
}
|