When understanding how primitive operators such as +, -, * and / are implemented in C, I found the following snippet from an interesting answer.
// replaces the + operator
int add(int x, int y) {
while(x) {
int t = (x & y) <<1;
y ^= x;
x = t;
}
return y;
}
It seems that this function demonstrates how + actually works in the background. However, it's too confusing for me to understand it. I believed that such operations are done using assembly directives generated by the compiler for a long time!
Is the + operator implemented as the code posted on MOST implementations? Does this take advantage of two's complement or other implementation-dependent features?
To be pedantic, the C specification does not specify how addition is implemented.
But to be realistic, the + operator on integer types smaller than or equal to the word size of your CPU get translated directly into an addition instruction for the CPU, and larger integer types get translated into multiple addition instructions with some extra bits to handle overflow.
The CPU internally uses logic circuits to implement the addition, and does not use loops, bitshifts, or anything that has a close resemblance to how C works.
When you add two bits, following is the result: (truth table)
a | b | sum (a^b) | carry bit (a&b) (goes to next)
--+---+-----------+--------------------------------
0 | 0 | 0 | 0
0 | 1 | 1 | 0
1 | 0 | 1 | 0
1 | 1 | 0 | 1
So if you do bitwise xor, you can get the sum without carry.
And if you do bitwise and you can get the carry bits.
Extending this observation for multibit numbers a and b
a+b = sum_without_carry(a, b) + carry_bits(a, b) shifted by 1 bit left
= a^b + ((a&b) << 1)
Once b is 0:
a+0 = a
So algorithm boils down to:
Add(a, b)
if b == 0
return a;
else
carry_bits = a & b;
sum_bits = a ^ b;
return Add(sum_bits, carry_bits << 1);
If you get rid of recursion and convert it to a loop
Add(a, b)
while(b != 0) {
carry_bits = a & b;
sum_bits = a ^ b;
a = sum_bits;
b = carrry_bits << 1; // In next loop, add carry bits to a
}
return a;
With above algorithm in mind explanation from code should be simpler:
int t = (x & y) << 1;
Carry bits. Carry bit is 1 if 1 bit to the right in both operands is 1.
y ^= x; // x is used now
Addition without carry (Carry bits ignored)
x = t;
Reuse x to set it to carry
while(x)
Repeat while there are more carry bits
A recursive implementation (easier to understand) would be:
int add(int x, int y) {
return (y == 0) ? x : add(x ^ y, (x&y) << 1);
}
Seems that this function demonstrates how + actually works in the
background
No. Usually (almost always) integer addition translates to machine instruction add. This just demonstrate an alternate implementation using bitwise xor and and.
Seems that this function demonstrates how + actually works in the background
No. This is translated to the native add machine instruction, which is actually using the hardware adder, in the ALU.
If you're wondering how does the computer add, here is a basic adder.
Everything in the computer is done using logic gates, which are mostly made of transistors. The full adder has half-adders in it.
For a basic tutorial on logic gates, and adders, see this. The video is extremely helpful, though long.
In that video, a basic half-adder is shown. If you want a brief description, this is it:
The half adder add's two bits given. The possible combinations are:
Add 0 and 0 = 0
Add 1 and 0 = 1
Add 1 and 1 = 10 (binary)
So now how does the half adder work? Well, it is made up of three logic gates, the and, xor and the nand. The nand gives a positive current if both the inputs are negative, so that means this solves the case of 0 and 0. The xor gives a positive output one of the input is positive, and the other negative, so that means that it solves the problem of 1 and 0. The and gives a positive output only if both the inputs are positive, so that solves the problem of 1 and 1. So basically, we have now got our half-adder. But we still can only add bits.
Now we make our full-adder. A full adder consists of calling the half-adder again and again. Now this has a carry. When we add 1 and 1, we get a carry 1. So what the full-adder does is, it takes the carry from the half-adder, stores it, and passes it as another argument to the half-adder.
If you're confused how can you pass the carry, you basically first add the bits using the half-adder, and then add the sum and the carry. So now you've added the carry, with the two bits. So you do this again and again, till the bits you have to add are over, and then you get your result.
Surprised? This is how it actually happens. It looks like a long process, but the computer does it in fractions of a nanosecond, or to be more specific, in half a clock cycle. Sometimes it is performed even in a single clock cycle. Basically, the computer has the ALU (a major part of the CPU), memory, buses, etc..
If you want to learn computer hardware, from logic gates, memory and the ALU, and simulate a computer, you can see this course, from which I learnt all this: Build a Modern Computer from First Principles
It's free if you do not want an e-certificate. The part two of the course is coming up in spring this year
C uses an abstract machine to describe what C code does. So how it works is not specified. There are C "compilers" that actually compile C into a scripting language, for example.
But, in most C implementations, + between two integers smaller than the machine integer size will be translated into an assembly instruction (after many steps). The assembly instruction will be translated into machine code and embedded within your executable. Assembly is a language "one step removed" from machine code, intended to be easier to read than a bunch of packed binary.
That machine code (after many steps) is then interpreted by the target hardware platform, where it is interpreted by the instruction decoder on the CPU. This instruction decoder takes the instruction, and translates it into signals to send along "control lines". These signals route data from registers and memory through the CPU, where the values are added together often in an arithmetic logic unit.
The arithmetic logic unit might have separate adders and multipliers, or might mix them together.
The arithmetic logic unit has a bunch of transistors that perform the addition operation, then produce the output. Said output is routed via the signals generated from the instruction decoder, and stored in memory or registers.
The layout of said transistors in both the arithmetic logic unit and instruction decoder (as well as parts I have glossed over) is etched into the chip at the plant. The etching pattern is often produced by compiling a hardware description language, which takes an abstraction of what is connected to what and how they operate and generates transistors and interconnect lines.
The hardware description language can contain shifts and loops that don't describe things happening in time (like one after another) but rather in space -- it describes the connections between different parts of hardware. Said code may look very vaguely like the code you posted above.
The above glosses over many parts and layers and contains inaccuracies. This is both from my own incompetence (I have written both hardware and compilers, but am an expert in neither) and because full details would take a career or two, and not a SO post.
Here is a SO post about an 8-bit adder. Here is a non-SO post, where you'll note some of the adders just use operator+ in the HDL! (The HDL itself understands + and generates the lower level adder code for you).
Almost any modern processor that can run compiled C code will have builtin support for integer addition. The code you posted is a clever way to perform integer addition without executing an integer add opcode, but it is not how integer addition is normally performed. In fact, the function linkage probably uses some form of integer addition to adjust the stack pointer.
The code you posted relies on the observation that when adding x and y, you can decompose it into the bits they have in common and the bits that are unique to one of x or y.
The expression x & y (bitwise AND) gives the bits common to x and y. The expression x ^ y (bitwise exclusive OR) gives the bits that are unique to one of x or y.
The sum x + y can be rewritten as the sum of two times the bits they have in common (since both x and y contribute those bits) plus the bits that are unique to x or y.
(x & y) << 1 is twice the bits they have in common (the left shift by 1 effectively multiplies by two).
x ^ y is the bits that are unique to one of x or y.
So if we replace x by the first value and y by the second, the sum should be unchanged. You can think of the first value as the carries of the bitwise additions, and the second as the low-order bit of the bitwise additions.
This process continues until x is zero, at which point y holds the sum.
The code that you found tries to explain how very primitive computer hardware might implement an "add" instruction. I say "might" because I can guarantee that this method isn't used by any CPU, and I'll explain why.
In normal life, you use decimal numbers and you have learned how to add them: To add two numbers, you add the lowest two digits. If the result is less than 10, you write down the result and proceed to the next digit position. If the result is 10 or more, you write down the result minus 10, proceed to the next digit, buy you remember to add 1 more. For example: 23 + 37, you add 3+7 = 10, you write down 0 and remember to add 1 more for the next position. At the 10s position, you add (2+3) + 1 = 6 and write that down. Result is 60.
You can do the exact same thing with binary numbers. The difference is that the only digits are 0 and 1, so the only possible sums are 0, 1, 2. For a 32 bit number, you would handle one digit position after the other. And that is how really primitive computer hardware would do it.
This code works differently. You know the sum of two binary digits is 2 if both digits are 1. So if both digits are 1 then you would add 1 more at the next binary position and write down 0. That's what the calculation of t does: It finds all places where both binary digits are 1 (that's the &) and moves them to the next digit position (<< 1). Then it does the addition: 0+0 = 0, 0+1 = 1, 1+0 = 1, 1+1 is 2, but we write down 0. That's what the excludive or operator does.
But all the 1's that you had to handle in the next digit position haven't been handled. They still need to be added. That's why the code does a loop: In the next iteration, all the extra 1's are added.
Why does no processor do it that way? Because it's a loop, and processors don't like loops, and it is slow. It's slow, because in the worst case, 32 iterations are needed: If you add 1 to the number 0xffffffff (32 1-bits), then the first iteration clears bit 0 of y and sets x to 2. The second iteration clears bit 1 of y and sets x to 4. And so on. It takes 32 iterations to get the result. However, each iteration has to process all bits of x and y, which takes a lot of hardware.
A primitive processor would do things just as quick in the way you do decimal arithmetic, from the lowest position to the highest. It also takes 32 steps, but each step processes only two bits plus one value from the previous bit position, so it is much easier to implement. And even in a primitive computer, one can afford to do this without having to implement loops.
A modern, fast and complex CPU will use a "conditional sum adder". Especially if the number of bits is high, for example a 64 bit adder, it saves a lot of time.
A 64 bit adder consists of two parts: First, a 32 bit adder for the lowest 32 bit. That 32 bit adder produces a sum, and a "carry" (an indicator that a 1 must be added to the next bit position). Second, two 32 bit adders for the higher 32 bits: One adds x + y, the other adds x + y + 1. All three adders work in parallel. Then when the first adder has produced its carry, the CPU just picks which one of the two results x + y or x + y + 1 is the correct one, and you have the complete result. So a 64 bit adder only takes a tiny bit longer than a 32 bit adder, not twice as long.
The 32 bit adder parts are again implemented as conditional sum adders, using multiple 16 bit adders, and the 16 bit adders are conditional sum adders, and so on.
My question is: Is the + operator implemented as the code posted on MOST implementations?
Let's answer the actual question. All operators are implemented by the compiler as some internal data structure that eventually gets translated into code after some transformations. You can't say what code will be generated by a single addition because almost no real world compiler generates code for individual statements.
The compiler is free to generate any code as long as it behaves as if the actual operations were performed according to the standard. But what actually happens can be something completely different.
A simple example:
static int
foo(int a, int b)
{
return a + b;
}
[...]
int a = foo(1, 17);
int b = foo(x, x);
some_other_function(a, b);
There's no need to generate any addition instructions here. It's perfectly legal for the compiler to translate this into:
some_other_function(18, x * 2);
Or maybe the compiler notices that you call the function foo a few times in a row and that it is a simple arithmetic and it will generate vector instructions for it. Or that the result of the addition is used for array indexing later and the lea instruction will be used.
You simply can't talk about how an operator is implemented because it is almost never used alone.
In case a breakdown of the code helps anyone else, take the example x=2, y=6:
x isn't zero, so commence adding to y:
while(2) {
x & y = 2 because
x: 0 0 1 0 //2
y: 0 1 1 0 //6
x&y: 0 0 1 0 //2
2 <<1 = 4 because << 1 shifts all bits to the left:
x&y: 0 0 1 0 //2
(x&y) <<1: 0 1 0 0 //4
In summary, stash that result, 4, in t with
int t = (x & y) <<1;
Now apply the bitwise XOR y^=x:
x: 0 0 1 0 //2
y: 0 1 1 0 //6
y^=x: 0 1 0 0 //4
So x=2, y=4. Finally, sum t+y by resetting x=t and going back to the beginning of the while loop:
x = t;
When t=0 (or, at the beginning of the loop, x=0), finish with
return y;
Just out of interest, on the Atmega328P processor, with the avr-g++ compiler, the following code implements adding one by subtracting -1 :
volatile char x;
int main ()
{
x = x + 1;
}
Generated code:
00000090 <main>:
volatile char x;
int main ()
{
x = x + 1;
90: 80 91 00 01 lds r24, 0x0100
94: 8f 5f subi r24, 0xFF ; 255
96: 80 93 00 01 sts 0x0100, r24
}
9a: 80 e0 ldi r24, 0x00 ; 0
9c: 90 e0 ldi r25, 0x00 ; 0
9e: 08 95 ret
Notice in particular that the add is done by the subi instruction (subtract constant from register) where 0xFF is effectively -1 in this case.
Also of interest is that this particular processor does not have a addi instruction, which implies that the designers thought that doing a subtract of the complement would be adequately handled by the compiler-writers.
Does this take advantage of two's complement or other implementation-dependent features?
It would probably be fair to say that compiler-writers would attempt to implement the wanted effect (adding one number to another) in the most efficient way possible for that particularly architecture. If that requires subtracting the complement, so be it.
So I have been told that this can be done and that bitwise operations and masks can be very useful but I must be missing something in how they work.
I am trying to calculate whether a number, say x, is a multiple of y. If x is a multiple of y great end of story, otherwise I want to increase x to reach the closest multiple of y that is greater than x (so that all of x fits in the result). I have just started learning C and am having difficulty understanding some of these tasks.
Here is what I have tried but when I input numbers such as 5, 9, or 24 I get the following respectively: 0, 4, 4.
if(x&(y-1)){ //if not 0 then multiple of y
x = x&~(y-1) + y;
}
Any explanations, examples of the math that is occurring behind the scenes, are greatly appreciated.
EDIT: So to clarify, I somewhat understand the shifting of bits to get whether an item is a multiple. (As was explained in a reply 10100 is a multiple of 101 as it is just shifted over). If I have the number 16, which is 10000, its complement is 01111. How would I use this complement to see if an item is a multiple of 16? Also can someone give a numerical explanation of the code given above? Showing this may help me understand why it does not work. Once I understand why it does not work I will be able to problem solve on my own I believe.
Why would you even think about using bit-wise operations for this? They certainly have their place but this isn't it.
A better method is to simply use something like:
unsigned multGreaterOrEqual(unsigned x, unsigned y) {
if ((x % y) == 0)
return x;
return (x / y + 1) * y;
}
In the trivial cases, every number that is an even multiple of a power of 2 is just shifted to the left (this doesn't apply when possibly altering the sign bit)
For example
10100
is 4 times
101
and
10100
is 2 time
1010
As for other multiples, they would have to be found by combining the outputs of two shifts. You might want to look up some primitive means of computer division, where division looks roughly like
x = a / b
implemented like
buffer = a
while a is bigger than b; do
yes: subtract a from b
add 1 to x
done
faster routines try to figure out higher level place values first, skipping lots of subtractions. All of these routine can be done bitwise; but it is a big pain. In the ALU these routines are done bitwise. Might want to look up a digital logic design book for more ideas.
Ok, so I have discovered what the error was in my code and since the majority say that it is impossible to calculate whether a number is a multiple of another number using masks I figured I would share what I have learned.
It is possible! - if you are using the correct data types that is.
The code given above works if y is declared as a constant unsigned long as x which was being passed in was also an unsigned long. The key point is not the long or constant part but that the number is unsigned. This sign bit causes miscalculation as the first place in the number indicates sign and when performing bitwise operations signs can get muddled.
So here is my code if we are looking for multiples of 16:
const unsigned long y = 16; //declared globally in my case
Then an unsigned long is passed to the function which runs the following code:
if(x&(y-1)){ //if not 0 then multiple of y
x = x&~(y-1) + y;
}
x will now be the size of the nearest multiple of 16.
In the book I am reading to learn C "The C programming language" in chapter 2.
The book is explaining Bitwise operations and it has a function that shows how many bits are in an integer.
The following is the function...
int Bitcount(unsigned x){
int b;
for(b = 0; x != 0; x >>=1){
if(x & 01){
b++
}
}
return b;
}
Then an exercise is given to us stating exactly this.
"In a two's complement number system, x &= (x-1) deletes the rightmost 1-bit in x;
Explain why. Use this observation to write a faster version of Bitcount".
The problem is I really cannot understand how "x &= (x-1)" would work? can someone explain this to me? or send me to a resource that could better help me understand?
I have been trying to figure this out but I really can't.
Thank you for any help that you may give.
also if there is anything wrong with my question this is my first post so please help me make my questions better.
X and X-1 cannot both have their rightmost bit set to 1, because in the binary system numbers ending in 0 and 1 alternate - so X & (X-1) is guaranteed to be a number whose rightmost bit set to 0 as AND only evaluates to true if both terms are true. Maybe the confusion stems from what Andrew W said, here a bitwise AND is used (which ANDs each bit individually)?
EDIT: now, as Inspired points out, this is only part of the answer as the original problem specifies that the rightmost 1-bit will get reset. As Andrew W already answered the correct version in detail, I'm not going to repeat his explanation here but I refer to his answer.
It is equivalent to x = x & (x-1) Here, the & is a bitwise and, not a logical and.
So here's what happens:
1) The expression on the right will be evaluated first, and that value will be stored in x.
2) Suppose x = 01001011 in binary (this isn't the case, since more than 8 bits will be used to represent x, but pretend it is for this example). Then x-1 = 01001010.
3) compute the bitwise and:
01001011 &
01001010 =
01001010
which deleted the rightmost one bit.
now suppose number didn't end with a 1 bit:
Say: x = 01001100, the (x-1) = 01001011
compute the bitwise and:
01001100 &
01001011 =
01001000
again removing the rightmost 1.
Good book by the way. I hope you enjoy it!
Let's take a closer look at the rightmost 1 bit in x: suppose x = AAAAAA10000..0, where AAAAAA are arbitrary bits. Then x-1 = AAAAAA01111..1. Bitwise AND of these two expressions gives AAAAAA00000..0. This is how it resets the rightmost non-zero bit.
The problem is I really cannot understand how "x &= (x-1)" would work?
Binary number is positional the same way as decimal number. When we increase the number we carry a bit to the left, when we decrease we borrow from the left the same way we do with decimals. So in case x-01 we borrow the first 1-bit from the right while others being set to 1-bit:
10101000
- 00000001
--------
10100111
which is inversion of those bits till the first 1-bit. And as stated before by others ~y & y = 0 that is why this method can be used to count 1-bits as proposed by the book to make the method faster comparing to bits shifting.
using only bitwise operators (|, &, ~, ^, >>, <<), is it possible to replace the != below?
// ...
if(a != b){
// Some code
}
/// ...
this is mainly out of self interest, since I saw how to do it with == but not !=.
if(a ^ b) {
//some code
}
should work.
You can also use your preferred method for == and add ^ 0xFFFFFFFF behind it (with the right amount of Fs to match the length of the datatype). This negates the value (same as ! in front of it).
a != b means that there is at least one different bit in the bit representations of a and b. The XOR bit operator returns 1 if both input bit operands are different, 0 otherwise.
So, you can apply a XOR operation to a and b and check if the result is not equal to zero.
A bitwise version of the '!=' test could look something like:
if((a - b) | (b - a)) {
/* code... */
}
which ORs the two subtractions. If the two numbers are the same, the result will be 0. However, if they differ (aka, the '!=' operator) then the result will be 1.
Note: The above snippet will only work with integers (and those integers should probably be unsigned).
If you want to simulate the '==' operator, however, check out Fabian Giesen's answer in Replacing "==" with bitwise operators
x ^ y isn't always sufficient. Use !!(x ^ y). Values expecting a one bit return value will not work with x ^ y since it leaves a remainder that could be greater than just 1.
Yes, using this:
if (a ^ b) { }
"~" is equaled to NOT so that should work. example would be "a & ~b".
So I see that this question has already been asked, however the answers were a little vague and unhelpful. Okay, I need to implement a c expression using only "& ^ ~ ! + | >> <<"
The expression needs to resemble: a ? b : c
So, from what I've been able to tell, the expression needs to look something like:
return (a & b) | (~a & c)
This works when a = 0, because anding it with b will give zero, and then the or expression will return the right side, (~a & c) which works because ~0 gives all ones, and anding c with all ones returns c.
However, this doesn't work when a > 0. Can someone try to explain why this is, or how to fix it?
I would convert a to a boolean using !!a, to get 0 or 1. x = !!a.
Then I'd negate that in two's complement. Since you don't have unary minus available, you use the definition of 2's complement negation: invert the bits, then add one: y = ~x + 1. That will give either all bits clear, or all bits set.
Then I'd and that directly with one variable y & b, its inverse with the other: ~y & c. That will give a 0 for one of the expressions, and the original variable for the other. When we or those together, the zero will have no effect, so we'll get the original variable, unchanged.
In other words, you need a to have all bits set to 0, if a is false (i.e. 0), and have all bits set to 1, if a is true (i.e. a > 0).
For the former case, the work is already done for you; for the latter -- try to work out result of the expression ~!1.