arithmetic.c

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#include <stdint.h>
 
/** On x86, division of one 64-bit integer by another cannot be
   done with a single instruction or a short sequence.  Thus, GCC
   implements 64-bit division and remainder operations through
   function calls.  These functions are normally obtained from
   libgcc, which is automatically included by GCC in any link
   that it does.
 
   Some x86-64 machines, however, have a compiler and utilities
   that can generate 32-bit x86 code without having any of the
   necessary libraries, including libgcc.  Thus, we can make
   Pintos work on these machines by simply implementing our own
   64-bit division routines, which are the only routines from
   libgcc that Pintos requires.
 
   Completeness is another reason to include these routines.  If
   Pintos is completely self-contained, then that makes it that
   much less mysterious. */
 
/** Uses x86 DIVL instruction to divide 64-bit N by 32-bit D to
   yield a 32-bit quotient.  Returns the quotient.
   Traps with a divide error (#DE) if the quotient does not fit
   in 32 bits. */
static inline uint32_t
divl (uint64_t n, uint32_t d)
{
  uint32_t n1 = n >> 32;
  uint32_t n0 = n;
  uint32_t q, r;
 
  asm ("divl %4"
       : "=d" (r), "=a" (q)
       : "0" (n1), "1" (n0), "rm" (d));
 
  return q;
}
 
/** Returns the number of leading zero bits in X,
   which must be nonzero. */
static int
nlz (uint32_t x) 
{
  /* This technique is portable, but there are better ways to do
     it on particular systems.  With sufficiently new enough GCC,
     you can use __builtin_clz() to take advantage of GCC's
     knowledge of how to do it.  Or you can use the x86 BSR
     instruction directly. */
  int n = 0;
  if (x <= 0x0000FFFF)
    {
      n += 16;
      x <<= 16; 
    }
  if (x <= 0x00FFFFFF)
    {
      n += 8;
      x <<= 8; 
    }
  if (x <= 0x0FFFFFFF)
    {
      n += 4;
      x <<= 4;
    }
  if (x <= 0x3FFFFFFF)
    {
      n += 2;
      x <<= 2; 
    }
  if (x <= 0x7FFFFFFF)
    n++;
  return n;
}
 
/** Divides unsigned 64-bit N by unsigned 64-bit D and returns the
   quotient. */
static uint64_t
udiv64 (uint64_t n, uint64_t d)
{
  if ((d >> 32) == 0) 
    {
      /* Proof of correctness:
 
         Let n, d, b, n1, and n0 be defined as in this function.
         Let [x] be the "floor" of x.  Let T = b[n1/d].  Assume d
         nonzero.  Then:
             [n/d] = [n/d] - T + T
                   = [n/d - T] + T                         by (1) below
                   = [(b*n1 + n0)/d - T] + T               by definition of n
                   = [(b*n1 + n0)/d - dT/d] + T
                   = [(b(n1 - d[n1/d]) + n0)/d] + T
                   = [(b[n1 % d] + n0)/d] + T,             by definition of %
         which is the expression calculated below.
 
         (1) Note that for any real x, integer i: [x] + i = [x + i].
 
         To prevent divl() from trapping, [(b[n1 % d] + n0)/d] must
         be less than b.  Assume that [n1 % d] and n0 take their
         respective maximum values of d - 1 and b - 1:
                 [(b(d - 1) + (b - 1))/d] < b
             <=> [(bd - 1)/d] < b
             <=> [b - 1/d] < b
         which is a tautology.
 
         Therefore, this code is correct and will not trap. */
      uint64_t b = 1ULL << 32;
      uint32_t n1 = n >> 32;
      uint32_t n0 = n; 
      uint32_t d0 = d;
 
      return divl (b * (n1 % d0) + n0, d0) + b * (n1 / d0); 
    }
  else 
    {
      /* Based on the algorithm and proof available from
         http://www.hackersdelight.org/revisions.pdf. */
      if (n < d)
        return 0;
      else 
        {
          uint32_t d1 = d >> 32;
          int s = nlz (d1);
          uint64_t q = divl (n >> 1, (d << s) >> 32) >> (31 - s);
          return n - (q - 1) * d < d ? q - 1 : q; 
        }
    }
}
 
/** Divides unsigned 64-bit N by unsigned 64-bit D and returns the
   remainder. */
static uint32_t
umod64 (uint64_t n, uint64_t d)
{
  return n - d * udiv64 (n, d);
}
 
/** Divides signed 64-bit N by signed 64-bit D and returns the
   quotient. */
static int64_t
sdiv64 (int64_t n, int64_t d)
{
  uint64_t n_abs = n >= 0 ? (uint64_t) n : -(uint64_t) n;
  uint64_t d_abs = d >= 0 ? (uint64_t) d : -(uint64_t) d;
  uint64_t q_abs = udiv64 (n_abs, d_abs);
  return (n < 0) == (d < 0) ? (int64_t) q_abs : -(int64_t) q_abs;
}
 
/** Divides signed 64-bit N by signed 64-bit D and returns the
   remainder. */
static int32_t
smod64 (int64_t n, int64_t d)
{
  return n - d * sdiv64 (n, d);
}

/** These are the routines that GCC calls. */
 
long long __divdi3 (long long n, long long d);
long long __moddi3 (long long n, long long d);
unsigned long long __udivdi3 (unsigned long long n, unsigned long long d);
unsigned long long __umoddi3 (unsigned long long n, unsigned long long d);
 
/** Signed 64-bit division. */
long long
__divdi3 (long long n, long long d) 
{
  return sdiv64 (n, d);
}
 
/** Signed 64-bit remainder. */
long long
__moddi3 (long long n, long long d) 
{
  return smod64 (n, d);
}
 
/** Unsigned 64-bit division. */
unsigned long long
__udivdi3 (unsigned long long n, unsigned long long d) 
{
  return udiv64 (n, d);
}
 
/** Unsigned 64-bit remainder. */
unsigned long long
__umoddi3 (unsigned long long n, unsigned long long d) 
{
  return umod64 (n, d);
}