CS302 Lecture Notes - Bit Operations


Topcoder problems that you can solve with bit arithmetic


I'm assuming that bit operations are review for you, but I'm also assuming that you've had very little practice programming with them. This lecture is to show you a little programming with bit operations, and to give you a program (ba_helper.cpp) that can help you practice bit operations and hexadecimal.

The standard bit operations AND, OR and XOR work on bits, and I'm assume that you learned this in CS130, if not in high school:

Computers implement instructions that allow you to do these operations on multi-bit words, where the instructions work on each set of corresponding bits of the operands. That sentence may be hard to parse, but what happens is: So, for example, if you want to do 5 AND 9, what you do is convert them to binary and do the operation on each bit.
5 = 0101
9 = 1001
    ----
    0001 -- The AND of each bit gives you 0001 in binary, which is 1.
If we're doing XOR, then you take the XOR of each bit:
5 = 0101
9 = 1001
    ----
    1100 -- The XOR of each bit gives you 1100 in binary, which is 12.
Other bit operations are NOT (which flips each bit), and the shift operations: So, for example, if you left-shift 3 by two digits (and then let's just assume our numbers are 8 bits), then you turn 00000011 into 00011000, which equals 24. If you right-shift 24 by three digits, you get three. If you right-shift 7 by two digits, then you turn 00000111 into 00000001, which equals one.

In C and C++, the following are the bit arithmetic operators:

AND: this is a single ampersand: &
OR: this is a single vertical bar: |
XOR: this is a single carat: ^
NOT: this is a single tilde: ~
Left-shift: this is two less-than signs: <<
Right-shift: this is two greater-than signs: >>


ba_helper.cpp

I've written the program ba_helper.cpp to help you practice bit arithmetic and hex. You enter operations in the form:

number operator number.

The numbers can be represented in any of three ways:

  1. Standard decimal, up to 264-1 in value.
  2. Hexadecimal, preceded by "0x" and up to 16 digits.
  3. Binary, preceded by "B" and up to 64 digits.
The operations are AND, OR, XOR, LS, RS and ANDNOT. The program does the operations, and then prints the inputs and the results in all three representations. (By the way, ANDNOT does a AND (NOT b).)

Here are some examples:

UNIX> g++ -o ba_helper -std=c++98 ba_helper.cpp
UNIX> ba_helper
When entering numbers, you can enter:
  A normal decimal number as big as 2^{64}-1.
  A number in hex up to 16 digits, starting with 0x.
  A number in binary up to 64 digits, starting with B.
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
5 AND 9

Operator: AND
A:                            5   0x0000000000000005  0000000000000000000000000000000000000000000000000000000000000101
B:                            9   0x0000000000000009  0000000000000000000000000000000000000000000000000000000000001001
C:                            1   0x0000000000000001  0000000000000000000000000000000000000000000000000000000000000001

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
B101 XOR 0x9

Operator: XOR
A:                            5   0x0000000000000005  0000000000000000000000000000000000000000000000000000000000000101
B:                            9   0x0000000000000009  0000000000000000000000000000000000000000000000000000000000001001
C:                           12   0x000000000000000c  0000000000000000000000000000000000000000000000000000000000001100

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
837261 OR 276591827

Operator: OR
A:                       837261   0x00000000000cc68d  0000000000000000000000000000000000000000000011001100011010001101
B:                    276591827   0x00000000107c74d3  0000000000000000000000000000000000010000011111000111010011010011
C:                    276625119   0x00000000107cf6df  0000000000000000000000000000000000010000011111001111011011011111

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
<CNTL-D>
UNIX> 
The first two examples are explained above. The last one is kind of a pain, but I'm hoping that you see that looking at the hex is a nice way to solve the problem. With hex, each digit corresponds to four bits. So, you can iterate through the hex digits and solve the OR problem for each of those. Start with the right-most one: 0xd is 1101 and 0x3 is 0011. So (0xd OR 0x3) is equal to 1111 - 0xf. Moving left: 0x8 is 1000 and 0xd is 1101, so their OR is 0xd. And so on. Yes, looking at the bits is easier, but you can do it directly from the hex after a little practice.

Left-shifts and right-shifts are easy if the shifting value is a multiple of 4. When that happens, you can divide by four and shift the hex. For example, in the first call, since we are left-shifting the bits by eight, that's the same as left-shifting the hex by 8/4 = 2. And in the second call, that's the same as right-shifting the hex by 16/4 = 4:

UNIX> ba_helper
When entering numbers, you can enter:
  A normal decimal number as big as 2^{64}-1.
  A number in hex up to 16 digits, starting with 0x.
  A number in binary up to 64 digits, starting with B.
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x83726987264 LS 8

Operator: LS
A:                9032963748452   0x0000083726987264  0000000000000000000010000011011100100110100110000111001001100100
B:                            8   0x0000000000000008  0000000000000000000000000000000000000000000000000000000000001000
C:             2312438719603712   0x0008372698726400  0000000000001000001101110010011010011000011100100110010000000000

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0xabcdef123454 RS 16

Operator: RS
A:              188900967593044   0x0000abcdef123454  0000000000000000101010111100110111101111000100100011010001010100
B:                           16   0x0000000000000010  0000000000000000000000000000000000000000000000000000000000010000
C:                   2882400018   0x00000000abcdef12  0000000000000000000000000000000010101011110011011110111100010010

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
<CNTL-D>
UNIX> 
Left-shift and right-shift become a pain when the number of digits is not a multiple of four. Then the best thing to do is convert the hex to binary, do the bit shift, and then partition the binary digits into groups of four, and convert back into hex. Here's an example. Suppose you want to do:

0x8a7e6c8 LS 7

First, convert the hex to binary, digit by digit:

8    a    7    e    6    c    8   
1000 1010 0111 1110 0110 1100 1000

Now, add seven 0's to the right side, and get rid of the spaces. In VI, you can do that with ":s/ //g".

1000 1010 0111 1110 0110 1100 1000 0000000
10001010011111100110110010000000000       

Now, add a zero to the beginning, so that the number of digits is a multiple of four, and then group the digits in groups of four. In VI, you can do that with ":s/\(....\)/\1 /g". And then you can go back to hex:

010001010011111100110110010000000000        
0100 0101 0011 1111 0011 0110 0100 0000 0000
4    5    3    f    3    6    4    0    0   

So the answer is 0x453f36400:

UNIX> ba_helper 
When entering numbers, you can enter:
  A normal decimal number as big as 2^{64}-1.
  A number in hex up to 16 digits, starting with 0x.
  A number in binary up to 64 digits, starting with B.
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x8a7e6c8 LS 7

Operator: LS
A:                    145221320   0x0000000008a7e6c8  0000000000000000000000000000000000001000101001111110011011001000
B:                            7   0x0000000000000007  0000000000000000000000000000000000000000000000000000000000000111
C:                  18588328960   0x0000000453f36400  0000000000000000000000000000010001010011111100110110010000000000

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
<CNTL-D>
UNIX> 
The nice thing about ba_helper is that you can use it to practice your bit arithmetic. Just try random hex numbers for the inputs, and do all of the operations.

Common things that we do with bit operations

Here are a few common things that we do with bit operations. When you do them enough times, they become second nature to you, but before that, you have to think about them a little bit.

In this discussion, I'm going to talk about "bit x of a number." When I say that, I mean the x-th bit from the right side of the binary representation of a number. For example, with the number 12, which is (1100) in binary, bits 0 and 1 are equal to zero, and bits 2 and 3 are equal to one. If the number is a 64-bit number, then bits 4 through 63 are also zero.

Setting a bit

To make sure that bit x is set in a number, you take the OR of the number with one, left-shifted by x. Basically, the left shift moves the one into bit position x, and then the OR makes sure that it is set in the given number.

In C/C++, to set bit x in number v, you do:

   v |= (1ULL << x);

That operator is "OR-Equals", which is like "+=" and "*=", on with OR instead of addition or multiplication. It's a good idea to put the left-shift in parentheses, because operator precedence is a little odd with bit arithmetic. The "ULL" is something you need if you are dealing with 64-bit numbers (like unsigned long long). The ULL tells the compiler to treat the number one as a 64-bit number, and that way it knows to do a bit shift on 64-bit numbers. If you don't do "ULL", then it will treat one as an integer, and then, for example (1 << 32) will equal zero, because in a 32-bit number, this shifts the one all the way off the number.

Here's an example using ba_helper that shows how you make sure that bit 6 is set in two numbers: 0x83, where the bit is not set already, and 0x64, where the bit is set already:

UNIX> ba_helper
When entering numbers, you can enter:
  A normal decimal number as big as 2^{64}-1.
  A number in hex up to 16 digits, starting with 0x.
  A number in binary up to 64 digits, starting with B.
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
1 LS 6

Operator: LS
A:                            1   0x0000000000000001  0000000000000000000000000000000000000000000000000000000000000001
B:                            6   0x0000000000000006  0000000000000000000000000000000000000000000000000000000000000110
C:                           64   0x0000000000000040  0000000000000000000000000000000000000000000000000000000001000000
                                                                                                               ^
                                           As you can see, this creates a number where only bit six is set     |
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x83 OR 0x40 

Operator: OR
A:                          131   0x0000000000000083  0000000000000000000000000000000000000000000000000000000010000011
B:                           64   0x0000000000000040  0000000000000000000000000000000000000000000000000000000001000000
C:                          195   0x00000000000000c3  0000000000000000000000000000000000000000000000000000000011000011
                                                                                                               ^
                                           Now, bit six is set in C                                            |

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x64 OR 0x40

Operator: OR
A:                          100   0x0000000000000064  0000000000000000000000000000000000000000000000000000000001100100
B:                           64   0x0000000000000040  0000000000000000000000000000000000000000000000000000000001000000
C:                          100   0x0000000000000064  0000000000000000000000000000000000000000000000000000000001100100
                                                                                                               ^
                                           In this example, bit 6 was already set in A, so C equals A.         |

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
<CNTL-D>
UNIX> 

Clearing a bit

To make sure that bit x is not set in a number, you take the AND of the number with the NOT of (one left-shifted by x). The left-shift sets the one bit, and the NOT flips all of the bits, so that every bit is one, except for bit x. Now, when you AND this with the number, it makes sure that all of the numbers bits remain the same, with the exception of bit x, which is cleared. In C/C++, to clear bit x in number v, you do:

   v &= (~(1ULL << x));

In the example below, we are going to clear the 6th bit of 0x64 (where it is set), and 0x83 (where it is not). As above, (1 << 6) equals 0x40, so we do AND-NOT with 0x40: so

UNIX> ba_helper
When entering numbers, you can enter:
  A normal decimal number as big as 2^{64}-1.
  A number in hex up to 16 digits, starting with 0x.
  A number in binary up to 64 digits, starting with B.
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
UNIX> 0x64 ANDNOT 0x40

Operator: ANDNOT
A:                          100   0x0000000000000064  0000000000000000000000000000000000000000000000000000000001100100
B:                           64   0x0000000000000040  0000000000000000000000000000000000000000000000000000000001000000
C:                           36   0x0000000000000024  0000000000000000000000000000000000000000000000000000000000100100
                                                                                                               ^
                                           In this example, we clear bit 6:                                    |
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
UNIX> 0x83 ANDNOT 0x40

Operator: ANDNOT
A:                          131   0x0000000000000083  0000000000000000000000000000000000000000000000000000000010000011
B:                           64   0x0000000000000040  0000000000000000000000000000000000000000000000000000000001000000
C:                          131   0x0000000000000083  0000000000000000000000000000000000000000000000000000000010000011
                                                                                                               ^
                                           In this example, bit 6 is already cleared:                          |
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
<CNTL-D>
UNIX> 
To see this a little more clearly, let's use (NOT 0x40). That is equal to 0xffffffffffffffbf. You can see the clearing of bit six a little more clearly here, I think:
UNIX> ba_helper
When entering numbers, you can enter:
  A normal decimal number as big as 2^{64}-1.
  A number in hex up to 16 digits, starting with 0x.
  A number in binary up to 64 digits, starting with B.
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x64 AND 0xffffffffffffffbf

Operator: AND
A:                          100   0x0000000000000064  0000000000000000000000000000000000000000000000000000000001100100
B:         18446744073709551551   0xffffffffffffffbf  1111111111111111111111111111111111111111111111111111111110111111
C:                           36   0x0000000000000024  0000000000000000000000000000000000000000000000000000000000100100
                                                                                                               ^
                                           In this example, we clear bit 6:                                    |
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x83 AND 0xffffffffffffffbf

Operator: AND
A:                          131   0x0000000000000083  0000000000000000000000000000000000000000000000000000000010000011
B:         18446744073709551551   0xffffffffffffffbf  1111111111111111111111111111111111111111111111111111111110111111
C:                          131   0x0000000000000083  0000000000000000000000000000000000000000000000000000000010000011
                                                                                                               ^
                                           In this example, bit 6 is already cleared:                          |
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
<CNTL-D>
UNIX> 

Checking to see if bit x is set

To check if bit x is set in number v, you do:

   if (v & (1ULL << x)) ...

If the bit isn't set, then the AND will result in all zero bits, which is false. If it is set, then the AND will equal (1 << x), which is not equal to zero. Boolean expressions that are not equal to zero are true.

Extracting the lowest x bits of a number

To do this, you create a "mask", which is a number where bits 0 through x-1 are set, and the rest are not. You then perform an AND of the mask with the number. The way you create the mask is that you subtract one from (1 << x). So, the C/C++ is:

   extracted_bits = v & ( (1ULL << x) - 1);

You may want to do some examples to convince yourself. Recall that (1 << 6) is 0x40, which equals 64. That means that 63, which equals 0x3f, will have bits 0 through 5 set, and the rest clear. That is our mask. In this example, we AND that with 0x827364, which extracts the lowest six bits from the number:

UNIX> ba_helper
When entering numbers, you can enter:
  A normal decimal number as big as 2^{64}-1.
  A number in hex up to 16 digits, starting with 0x.
  A number in binary up to 64 digits, starting with B.
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x827364 AND 0x3f  

Operator: AND
A:                      8549220   0x0000000000827364  0000000000000000000000000000000000000000100000100111001101100100
B:                           63   0x000000000000003f  0000000000000000000000000000000000000000000000000000000000111111
C:                           36   0x0000000000000024  0000000000000000000000000000000000000000000000000000000000100100
                                                                                                                ^^^^^^
                                           Here we are extracting the lowest six bits of 0x827364:              ||||||
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
UNIX> 
<CNTL-D>
UNIX> 

Extracting bits x through y of a number

You do this in two parts. First, you extract bits 0 through y using the technique above. Then you right shift by x. In other words:

   extracted_bits = ( ( v & ( (1ULL << (y+1)) - 1) ) >> x);

Let's use an example of extracting bits 2 through 5 of 0x827364. As in our previous examples, (1 << (5+1)) equals 0x40, so ( (1ULL << (y+1)) - 1) equals 0x3f. The last example above shows that 0x827364 AND 0x3f is 0x24 (100100), and our final action is to right shift that by two bits, to get (1001) or 9. Let's just show this in the original number 0x827364:

A:                      8549220   0x0000000000827364  0000000000000000000000000000000000000000100000100111001101100100
                                                                                                                ^^^^
                                           Below we are extracting bits 2 through 5, which are 1001.             ||||
And below, we'll show the two operations:
Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x827364 AND 0x3f

Operator: AND
A:                      8549220   0x0000000000827364  0000000000000000000000000000000000000000100000100111001101100100
B:                           63   0x000000000000003f  0000000000000000000000000000000000000000000000000000000000111111
C:                           36   0x0000000000000024  0000000000000000000000000000000000000000000000000000000000100100

Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:
0x24 RS 2

Operator: RS
A:                           36   0x0000000000000024  0000000000000000000000000000000000000000000000000000000000100100
B:                            2   0x0000000000000002  0000000000000000000000000000000000000000000000000000000000000010
C:                            9   0x0000000000000009  0000000000000000000000000000000000000000000000000000000000001001

Representing sets with bits

If you need to represent sets of integers, where the integers are small, you can use the bits of a number. For example, suppose that you are representing sets of integers from 0 to 63. Then you can use a long long to represent the set. If bit x is set in the long long, then the number x is in the set represented by the long long.

For example, you can represent the set { 1, 3, 6, 7 } with the number 0xca. In binary, that number is 11001010, which, as you can see, has bits 1, 3, 6 and 7 set. When you represent sets in this way, you use the methods above to set a bit (adding an element to a set), clear a bit (removing an element from a set) and testing to see if a bit is set (testing to see if an element is in the set).

This is often much faster than using a set to represent the set. That is because a set is implemented with a balanced binary tree data structure, which uses a lot of memory for each element. With bit arithmetic, all it takes is one int or one long long. That is a big savings! With bits, you can implement set intersection with a simple binary AND, set union with a simple binary OR, and take the complement of a set with NOT. How cool is that?


ba_helper.cpp

It's not a bad idea to look over the entire program. I'm going to break the program into parts to talk about them. We start with the Number class, where I hold a number, and two string representations:

/* This is how we're holding a number.  The class facilitates printing out the number. */

class Number {
  public:
    unsigned long long d;      /* The number. */
    string hex;                /* Its representation in hex (16 hex digits with a 0x in front). */
    string binary;             /* Its representation in binary */
    string To_String();        /* This creates a bigger string, which is kind of formatted. */
};

Because I want to deal with 64-bit integers, I use the type unsigned long long. The "unsigned" part means that it goes from 0 to 264-1. I keep two string representations, and I have a method called To_String() which prints out all three representations of the number, formatted. When I create a Number, I make sure to set both string representations at that time.

Let's look at To_String():

/* This returns a "formatted" string for a number. */

string Number::To_String()
{
  char buf[200];
  string s;

  sprintf(buf, "%21llu   %s  %s", d, hex.c_str(), binary.c_str());
  s = buf;
  return s;
}

The only thing here that is remotely subtle is the "%21llu" -- this is a way of specifying that you want to print out an unsigned long long, padded to 21 spaces, right justified.

Now, I've written two procedures that create Number instances. The first is called number_from_ull(), and it creates an instance of Number from an unsigned long long. It calls new to create the instance and sets the d field. Next, it sets the hex string using sprintf(), with a format string of "0x%016llx". That says to start with "0x", then print the unsigned long long as a 16-digit hex number with leading zeros.

Finally, it creates the binary string by running through the digits from 0 to 63, checking to see if the digit is set in v using the same technique as I describe above in "Checking to see if bit x is set", and if a bit is set, its corresponding character in the string is set to '1':

Number *number_from_ull(unsigned long long v)
{
  Number *n;
  int i;
  char buf[200];
 
  /* Create the Number class instance, and set the strings. */

  n = new Number;
  n->d = v;

  /* Set the hexadecimal using sprintf. */

  sprintf(buf, "0x%016llx", v);
  n->hex = buf;

  /* For the binary, examine each bit by doing AND with one 
     left-shifted the proper number of bits. "1ULL" forces
     the compiler to treat one as an unsigned long long.  
     Otherwise, if you shift it more than 31 bits, it will
     treat one as an integer, and turn it into zero.  */

  n->binary.resize(64, '0');
  for (i = 0; i < 64; i++) if (v & (1ULL << i)) n->binary[64-i-1] = '1';

  return n;
}

The second procedure that creates Number instances is number_from_string(), and it creates a number from a string that is in any of the three formats described above. It does this by converting the string to an unsigned long long named v, and then calling number_from_ull() on v.

I'm showing the code below up to the point where the procedure reads the number from a binary string beginning with 'B':

/* This creates a number from a string, which is either decimal,
   hexadecimal (starting with 0x), or binary (starting with B). 
   It creates all of the string representations. */

Number *number_from_string(string &s)
{
  unsigned long long v;
  unsigned long long i;
  int b;
  Number *n;
  char buf[100];

  v = 0;

  if (s.size() == 0) return NULL;

  /* Convert from binary if the string begins with 'B' */

  if (s[0] == 'B') {
    if (s.size() == 1) return NULL;
    if (s.size() > 65) return NULL;
    for (i = 0; i < s.size()-1; i++) {
      b = s[s.size()-i-1];
      if (b != '0' && b != '1') return NULL;
      if (b == '1') v |= (1ULL << i);     /* Set bit i, if the corresponding character is '1' */
    }

Take a look at the for loop -- that loops through the digits, where i is the number of the digit. In the binary string, digit 0 is the last digit of the string, so it is s[s.size()-1]. Digit 1 is the digit before that one, so it is s[s.size()-2]. And so on -- this is why we set b to be s[s.size()-i-1].

When b is equal to '1', that means that bit i should be set, and I set it exactly as described above in "Setting a bit":

if (b == '1') v |= (1ULL << i);     

In the for loop, I stop at i < s.size()-1 instead of s.size(), because I want to ignore the initial 'B' character.

Now, the next block of code reads the string if it is specified in hex, and if not, it tries to read it in decimal. This code is pretty straightforward, except we use "%llx" to read an unsigned long long in hex, and "%llu" to read an unsigned long long as a decimal. At the end, it calls number_from_ull().

  /* Convert from hex if the string begins with "0x" */

  } else if (s.substr(0, 2) == "0x") {
    if (s.size() == 2 || s.size() > 18) return NULL;
    if (sscanf(s.c_str(), "0x%llx", &v) != 1) return NULL;

  /* Attempt to convert from decimal. */

  } else {
    if (sscanf(s.c_str(), "%llu", &v) != 1) return NULL;
  }

  return number_from_ull(v);
}

Finally, this last code block implements the main(), which reads the user input and prints the output. I don't say anything more about this code except for what's in the comments -- this is straightforward code, but it's good code for you to read, because I think it is laid out well, and is easy to read, despite the fact that it handles input errors pretty cleanly:

int main()
{
  Number *A, *B, *C;
  string sa, sb, sop;
  int error;

  printf("When entering numbers, you can enter:\n");
  printf("  A normal decimal number as big as 2^{64}-1.\n");
  printf("  A number in hex up to 16 digits, starting with 0x.\n");
  printf("  A number in binary up to 64 digits, starting with B.\n");

  while (1) {
    error = 0;
    C = NULL;

    /* Grab A, B and the operator. */

    printf("Enter a problem: number AND|OR|XOR|LS|RS|ANDNOT number:\n");
    fflush(stdout);

    if (! (cin >> sa >> sop >> sb)) exit(1);

    /* Convert A and B to instances of the Number class, and error check. */

    A = number_from_string(sa);
    B = number_from_string(sb);
    if (A == NULL) {
      printf("Bad format for the first number.\n");
      error = 1;
    } 
    if (B == NULL) {
      printf("Bad format for the second number.\n");
      error = 1;
    }
    
    /* Do the operation if we haven't had an error so far. */

    if (error == 0) {
      if (sop == "AND") {
        C = number_from_ull(A->d & B->d);
      } else if (sop == "OR") {
        C = number_from_ull(A->d | B->d);
      } else if (sop == "XOR") {
        C = number_from_ull(A->d ^ B->d);
      } else if (sop == "LS") {
        C = number_from_ull(A->d << B->d);
      } else if (sop == "RS") {
        C = number_from_ull(A->d >> B->d);
      } else if (sop == "ANDNOT") {
        C = number_from_ull(A->d & (~B->d));
      } else {
        printf("Bad operator.\n");
        error = 1;
      }
    }
    
    /* If everything was successful, print the results. */

    if (error == 0) {
      printf("\n");
      printf("Operator: %s\n", sop.c_str());
      printf("A:        %s\n", A->To_String().c_str());
      printf("B:        %s\n", B->To_String().c_str());
      printf("C:        %s\n", C->To_String().c_str());
      printf("\n");
    }

    /* Free up memory: Call delete on anything that you created
       with new. */

    if (A != NULL) delete A;
    if (B != NULL) delete B;
    if (C != NULL) delete C;
  
  }

  exit(0);
}