pi_agm.c

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** -----------
** 30 13 62 67
**
** We need only 3 multiplications, plus a few additions.  And of course,
** additions are a lot simpler and faster than multiplications, so we
** end up ahead.
**
** The way it is written requires that the size to be a power of two.
** That's why the program itself requires the size to be a power of two.
** There is no actual requirement in the method, it's just how I did it
** because I would be working close to powers of two anyway, and it was
** easier.
*/

void FastMul(Short *prod, Short *a, Short *b, Indexer Len)
{
      Indexer x, HLen;
      int SignA, SignB;
      Short *offset;
      Short *NextProd;

      /*
      ** This is the threshold where it's faster to do it the ordinary way
      ** On my computer, it's 16.  Yours may be different.
      */

      if (Len <= 16 )
      {
            Mul(prod, a, b, Len);
            return;
      }

      HLen = Len/2;
      NextProd = prod + 2*Len;

      SignA = Sub(prod, a, a + HLen, HLen);
      if (SignA)
      {
            SignA = 1;
            Negate(prod, HLen);
      }
      SignB = Sub(prod + HLen, b + HLen, b, HLen);
      if (SignB)
      {
            SignB = 1;
            Negate(prod + HLen, HLen);
      }

      FastMul(NextProd, prod, prod + HLen, HLen);
      for (x = 0; x < 2 * Len; x++)
            prod[x] = 0;
      offset = prod + HLen;
      if (SignA == SignB)
            DoCarries(prod, offset - 1, Add(offset, offset, NextProd, Len));
      else  DoBorrows(prod, offset - 1, Sub(offset, offset, NextProd, Len));

      FastMul(NextProd, a, b, HLen);
      offset = prod + HLen;
      DoCarries(prod, offset - 1, Add(offset, offset, NextProd, Len));
      Add(prod, prod, NextProd, Len);

      FastMul(NextProd, a + HLen, b + HLen, HLen);
      offset = prod + HLen;
      DoCarries(prod, offset - 1, Add(offset, offset, NextProd, Len));
      offset = prod + Len;
      DoCarries(prod, offset - 1, Add(offset, offset, NextProd, Len));
}

/*
** Compute the reciprocal of 'a'.
** x[k+1] = x[k]*(2-a*x[k])
*/

void Reciprocal(Short *r, Short *n, Indexer Len)
{
      Indexer x, SubLen;
      int iter;
      double t;

      /* Estimate our first reciprocal */

      for (x = 0; x < Len; x++)
            r[x] = 0;
      t = n[0] + n[1]*1.0e-4 + n[2]*1.0e-8;
      if (t == 0.0)
            t = 1;
      t = 1.0/t;
      r[0] = t;
      t = (t - floor(t))*10000.0;
      r[1] = t;
      t = (t - floor(t))*10000.0;
      r[2] = t;

      iter = log(SIZE)/log(2.0) + 1;
      SubLen = 1;
      while (iter--)
      {
            SubLen *= 2;
            if (SubLen > SIZE)
                  SubLen = SIZE;
            FastMul(MulWork, n, r, SubLen);
            Round2x(Work1, MulWork, SubLen);

            MulWork[0] = 2;
            for (x = 1; x < Len; x++)
                  MulWork[x] = 0;
            Sub(Work1, MulWork, Work1, SubLen);

            FastMul(MulWork, r, Work1, SubLen);
            Round2x(r, MulWork, SubLen);
      }
}

/*
** Computes the reciprocal of the square root of 'a'
** y[k+1] = y[k]*(3-a*y[k]^2)/2
**
**
** We can save quite a bit of time by using part of our previous
** results!  Since the number is converging onto a specific value,
** at least part of our previous result will be correct, so we
** can skip a bit of work.
*/

void RSqrt(Short *r, Short *n, Indexer Len, Indexer SubLen)
{
      Indexer x;
      int iter;
      double t;

      /* Estimate our first reciprocal square root */

      if (SubLen < 2 )
      {
            for (x = 0; x < Len; x++)
                  r[x] = 0;
            t = n[0] + n[1]*1.0e-4 + n[2]*1.0e-8;
            if (t == 0.0)
                  t = 1;
            t = 1.0/sqrt(t);
            r[0] = t;
            t = (t - floor(t))*10000.0;
            r[1] = t;
            t = (t - floor(t))*10000.0;
            r[2] = t;
      }

      /*
      ** PrintShort2("\n  ~R: ", r, SIZE);
      */

      if (SubLen > SIZE)
            SubLen = SIZE;
      iter = SubLen;
      while (iter <= SIZE*2)
      {
            FastMul(MulWork, r, r, SubLen);
            Round2x(Work1, MulWork, SubLen);
            FastMul(MulWork, n, Work1, SubLen);
            Round2x(Work1, MulWork, SubLen);

            MulWork[0] = 3;
            for (x = 1; x < SubLen; x++)
                  MulWork[x] = 0;
            Sub(Work1, MulWork, Work1, SubLen);

            FastMul(MulWork, r, Work1, SubLen);
            Round2x(r, MulWork, SubLen);
            DivBy2(r, SubLen);

            /*
            printf("%3d", SubLen);
            PrintShort2("R: ", r, SubLen);
            printf("%3d", SubLen);
            PrintShort2("R: ", r, Len);
            */

            SubLen *= 2;
            if (SubLen > SIZE)
                  SubLen = SIZE;
            iter *= 2;
      }
}

/*
** d = a/b by computing the reciprocal of b and then multiplying
** that by a.  It's faster this way because the normal digit by
** digit method has N^2 growth, and this method will have the same
** growth as our multiplication method.
*/

void Divide(Short *d, Short *a, Short *b, Indexer Len)
{
      Reciprocal(Work2, b, Len);
      FastMul(MulWork, a, Work2, Len);
      Round2x(d, MulWork, Len);
}

/*
** r = sqrt(n) by computing the reciprocal of the square root of
** n and then multiplying that by n
*/

void Sqrt(Short *r, Short *n, Indexer Len, Indexer SubLen)
{
      RSqrt(Work2, n, Len, SubLen);
      FastMul(MulWork, n, Work2, Len);
      Round2x(r, MulWork, Len);
}

void usage(void)
{
      puts("This program requires the number of digits/4 to calculate");
      puts("This number must be a power of two.");
      exit(-1);
}

int main(int argc,char *argv[])
{
      Indexer x;
      Indexer SubLen;
      double Pow2, tempd, carryd;
      int AGMIter;
      int NeededIter;
      clock_t T0, T1, T2, T3;

      if (argc < 2)
            usage();

      SIZE = atoi(argv[1]);
      if (!ispow2(SIZE))
            usage();

      a = (Short *)malloc(sizeof(Short)*SIZE);
      b = (Short *)malloc(sizeof(Short)*SIZE);
      c = (Short *)malloc(sizeof(Short)*SIZE);
      Work1 = (Short *)malloc(sizeof(Short)*SIZE);
      Work2 = (Short *)malloc(sizeof(Short)*SIZE);
      Work3 = (Short *)malloc(sizeof(Short)*SIZE);
      MulWork = (Short *)malloc(sizeof(Short)*SIZE*4);

      if ((a ==NULL) || (b == NULL) || (c == NULL) || (Work1 == NULL) ||
          (Work2 == NULL) || (MulWork == NULL))
      {
            printf("Unable to allocate memory\n");exit(1);
      }

      NeededIter = log(SIZE)/log(2.0) + 1;
      Pow2 = 4.0;
      AGMIter = NeededIter + 1;
      SubLen = 1;

      T0 = clock();

      for (x = 0; x < SIZE; x++)
            a[x] = b[x] = c[x] = Work3[x] = 0;
      a[0] = 1;
      Work3[0] = 2;
      RSqrt(b, Work3, SIZE, 1);
      c[0] = 1;

      T1 = clock();

      /*
      printf("AGMIter %d\n", AGMIter);
      PrintShort2("a AGM: ", a, SIZE);
      PrintShort2("b AGM: ", b, SIZE);
      PrintShort2("C sum: ", c, SIZE);
      */

      while (AGMIter--)
      {
            Sub(Work1, a, b, SIZE);                /* w = (a-b)/2      */
            DivBy2(Work1, SIZE);
            FastMul(MulWork, Work1, Work1, SIZE);  /* m = w*w          */
            Round2x(Work1, MulWork, SIZE);

            carryd = 0.0;                         /* m = m* w^(J+1)   */
            for (x = SIZE - 1; x >= 0; x--)
            {
                  tempd = Pow2*Work1[x] + carryd;
                  carryd = floor(tempd / 10000.0);
                  Work1[x] = tempd - carryd*10000.0;
            }
            Pow2 *= 2.0;
            Sub(c, c, Work1, SIZE);                /* c = c - m        */

            /* Save some time on last iter */

            if (AGMIter != 0)
                  FastMul(MulWork, a, b, SIZE);    /* m = a*b          */
            Add(a, a, b, SIZE);                    /* a = (a+b)/2      */
            DivBy2(a, SIZE);
            if (AGMIter != 0)
            {
                  Round2x(Work3, MulWork, SIZE);
                  Sqrt(b, Work3, SIZE, SubLen); /* b = sqrt(a*b) */
            }
            /*
            printf("AGMIter %d\n", AGMIter);
            PrintShort2("a AGM: ", a, SIZE);
            PrintShort2("b AGM: ", b, SIZE);
            PrintShort2("C sum: ", c, SIZE);
            */
            SubLen *= 2;if (SubLen > SIZE) SubLen = SIZE;
      }

      T2 = clock();

      FastMul(MulWork, a, a, SIZE);
      Round2x(a, MulWork, SIZE);
      carryd = 0.0;
      for (x = SIZE - 1; x >= 0; x--)
      {
            tempd = 4.0*a[x] + carryd;
            carryd = floor(tempd / 10000.0);
            a[x] = tempd - carryd*10000.0;
      }

      Divide(b, a, c, SIZE);
      T3 = clock();

      PrintShort("\nPI = ", b, SIZE);

      printf("\nTotal Execution time: %12.4lf\n",
		 (double)(T3 - T0) / CLOCKS_PER_SEC);
      printf("Setup time          : %12.4lf\n",
		 (double)(T1 - T0) / CLOCKS_PER_SEC);
      printf("AGM Calculation time: %12.4lf\n",
		 (double)(T2 - T1) / CLOCKS_PER_SEC);
      printf("Post calc time      : %12.4lf\n",
		 (double)(T3 - T2) / CLOCKS_PER_SEC);

      return 0;
}