initial check-in, version 0.5.0v0.5.0

1 files changed, 300 insertions, 0 deletions

diff --git a/src/math/k_rem_pio2.c b/src/math/k_rem_pio2.c
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+++ b/src/math/k_rem_pio2.c
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+
+/* @(#)k_rem_pio2.c 1.3 95/01/18 */
+/*
+ * ====================================================
+ * 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.
+ */
+
+#include <math.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 __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;
+}