--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/libm/k_rem_pio2.c	Mon Sep 15 06:46:23 2008 +0000
@@ -0,0 +1,358 @@
+/* @(#)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 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.h"
+#include "math_private.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 */
+    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;
+}