18  *		y = x - N*pi/2
27  * (2/pi) is represented by an array of 24-bit integers in ipio2[].
31  *		pieces of 24-bit integers in double precision format.
32  *		x[i] will be the i-th 24 bit of x. The scaled exponent 
37  *			e0 = ilogb(z)-23
38  *			z  = scalbn(z,-e0)
41  *			z    = (z-x[i])*2**24
44  *	y[]	output result in an array of double precision numbers.
46  *			24-bit  precision	1
47  *			53-bit  precision	2
48  *			64-bit  precision	2
49  *			113-bit precision	3
50  *		The actual value is the sum of them. Thus for 113-bit
51  *		precision, one may have to do something like:
57  *		r_tail = w - (r_head - t);
64  *  	prec	an integer indicating the precision:
68  *			3	113 bits (quad)
78  *		for jk is 3,4,4,6 for single, double, extended, and quad.
88  *	jx	nx - 1
92  *			( 2^e0*x[0] * ipio2[jv-1]*2^(-24jv) )/8
94  *			e0-3-24*jv >= 0 or (e0-3)/24 >= jv
95  *		Hence jv = max(0,(e0-3)/24).
100  *		24-bits chunk of the product of x and 2/pi.
103  *		exponent for q[i] would be q0-24*i.
105  *	PIo2[]	double precision array, obtained by cutting pi/2
110  *	iq[]	integer array by breaking up q[] in 24-bits chunk.
138  *		integer array, contains the (24*i)-th to (24*i+23)-th 
142  *			ipio2[i] * 2^(-24(i+1)).
144  * NB: This table must have at least (e0-3)/24 + jk terms.
145  *     For quad precision (e0 <= 16360, jk = 6), this is 686.
274   7.54978941586159635335e-08, /* 0x3E74442D, 0x00000000 */
275   5.39030252995776476554e-15, /* 0x3CF84698, 0x80000000 */
276   3.28200341580791294123e-22, /* 0x3B78CC51, 0x60000000 */
277   1.27065575308067607349e-29, /* 0x39F01B83, 0x80000000 */
278   1.22933308981111328932e-36, /* 0x387A2520, 0x40000000 */
279   2.73370053816464559624e-44, /* 0x36E38222, 0x80000000 */
280   2.16741683877804819444e-51, /* 0x3569F31D, 0x00000000 */
287 twon24  =  5.96046447753906250000e-08; /* 0x3E700000, 0x00000000 */
300 	jx =  nx-1;  in __kernel_rem_pio2()
301 	jv = (e0-3)/24; if(jv<0) jv=0;  in __kernel_rem_pio2()
302 	q0 =  e0-24*(jv+1);  in __kernel_rem_pio2()
305 	j = jv-jx; m = jx+jk;  in __kernel_rem_pio2()
310 	    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j]; q[i] = fw;  in __kernel_rem_pio2()
316 	for(i=0,j=jz,z=q[jz];j>0;i++,j--) {  in __kernel_rem_pio2()
318 	    iq[i] =  (int32_t)(z-two24*fw);  in __kernel_rem_pio2()
319 	    z     =  q[j-1]+fw;  in __kernel_rem_pio2()
324 	z -= 8.0*floor(z*0.125);		/* trim off integer >= 8 */  in __kernel_rem_pio2()
326 	z -= (double)n;  in __kernel_rem_pio2()
328 	if(q0>0) {	/* need iq[jz-1] to determine n */  in __kernel_rem_pio2()
329 	    i  = (iq[jz-1]>>(24-q0)); n += i;  in __kernel_rem_pio2()
330 	    iq[jz-1] -= i<<(24-q0);  in __kernel_rem_pio2()
331 	    ih = iq[jz-1]>>(23-q0);  in __kernel_rem_pio2()
333 	else if(q0==0) ih = iq[jz-1]>>23;  in __kernel_rem_pio2()
338 	    for(i=0;i<jz ;i++) {	/* compute 1-q */  in __kernel_rem_pio2()
342 			carry = 1; iq[i] = 0x1000000- j;  in __kernel_rem_pio2()
344 		} else  iq[i] = 0xffffff - j;  in __kernel_rem_pio2()
349 	    	   iq[jz-1] &= 0x7fffff; break;  in __kernel_rem_pio2()
351 	    	   iq[jz-1] &= 0x3fffff; break;  in __kernel_rem_pio2()
355 		z = one - z;  in __kernel_rem_pio2()
356 		if(carry!=0) z -= scalbn(one,q0);  in __kernel_rem_pio2()
363 	    for (i=jz-1;i>=jk;i--) j |= iq[i];  in __kernel_rem_pio2()
365 		for(k=1;iq[jk-k]==0;k++);   /* k = no. of terms needed */  in __kernel_rem_pio2()
369 		    for(j=0,fw=0.0;j<=jx;j++) fw += x[j]*f[jx+i-j];  in __kernel_rem_pio2()
379 	    jz -= 1; q0 -= 24;  in __kernel_rem_pio2()
380 	    while(iq[jz]==0) { jz--; q0-=24;}  in __kernel_rem_pio2()
381 	} else { /* break z into 24-bit if necessary */  in __kernel_rem_pio2()
382 	    z = scalbn(z,-q0);  in __kernel_rem_pio2()
385 		iq[jz] = (int32_t)(z-two24*fw);  in __kernel_rem_pio2()
391     /* convert integer "bit" chunk to floating-point value */  in __kernel_rem_pio2()
393 	for(i=jz;i>=0;i--) {  in __kernel_rem_pio2()
398 	for(i=jz;i>=0;i--) {  in __kernel_rem_pio2()
399 	    for(fw=0.0,k=0;k<=jp&&k<=jz-i;k++) fw += PIo2[k]*q[i+k];  in __kernel_rem_pio2()
400 	    fq[jz-i] = fw;  in __kernel_rem_pio2()
407 		for (i=jz;i>=0;i--) fw += fq[i];  in __kernel_rem_pio2()
408 		y[0] = (ih==0)? fw: -fw;   in __kernel_rem_pio2()
413 		for (i=jz;i>=0;i--) fw += fq[i];   in __kernel_rem_pio2()
415 		y[0] = (ih==0)? fw: -fw;   in __kernel_rem_pio2()
416 		fw = fq[0]-fw;  in __kernel_rem_pio2()
418 		y[1] = (ih==0)? fw: -fw;   in __kernel_rem_pio2()
421 		for (i=jz;i>0;i--) {  in __kernel_rem_pio2()
422 		    fw      = fq[i-1]+fq[i];   in __kernel_rem_pio2()
423 		    fq[i]  += fq[i-1]-fw;  in __kernel_rem_pio2()
424 		    fq[i-1] = fw;  in __kernel_rem_pio2()
426 		for (i=jz;i>1;i--) {  in __kernel_rem_pio2()
427 		    fw      = fq[i-1]+fq[i];   in __kernel_rem_pio2()
428 		    fq[i]  += fq[i-1]-fw;  in __kernel_rem_pio2()
429 		    fq[i-1] = fw;  in __kernel_rem_pio2()
431 		for (fw=0.0,i=jz;i>=2;i--) fw += fq[i];   in __kernel_rem_pio2()
435 		    y[0] = -fq[0]; y[1] = -fq[1]; y[2] = -fw;  in __kernel_rem_pio2()