106
votes
Accepted
Why is addition as fast as bit-wise operations in modern processors?
Addition is fast because CPU designers have put in the circuitry needed to make it fast. It does take significantly more gates than bitwise operations, but it is frequent enough that CPU designers ...
D.W.♦
- 141k
48
votes
Accepted
How can I multiply a binary representation by ten using logic gates?
I assume that the task is to compute $mul(10, a)= 10a$. You don't need to do multiplication. A single binary adder is enough since $$10a = 2^3a + 2a$$
meaning you add one-time left-shifted $a$ to 3-...
- 9,612
45
votes
Signed and unsigned numbers
Short version: it doesn't know. There's no way to tell.
If 1111 represents -7, then you have a sign-magnitude representation, where the first bit is the sign and ...
- 6,940
42
votes
Why is addition as fast as bit-wise operations in modern processors?
There are several aspects.
The relative cost of a bitwise operation and an addition. A naive adder will have a gate-depth which depend linearly of the width of the word. There are alternative ...
- 3,004
30
votes
Accepted
Computing exam averages in less than linear time
To compute the exact mean (no confidence interval or estimate) of each exam, you must at least observe every student's exam score. This takes $\Omega(r)$ per exam. There are $c$ exams you must do this ...
- 4,371
25
votes
Why is addition as fast as bit-wise operations in modern processors?
CPUs operate in cycles. At each cycle, something happens. Usually, an instruction takes more cycles to execute, but multiple instructions are executed at the same time, in different states.
For ...
- 508
21
votes
Represent a real number without loss of precision
There is no way to represent all real numbers without errors if each number is to have a finite representation. There are uncountably many real numbers but only countably many finite strings of 1's ...
- 80.2k
21
votes
Accepted
Confused about XORing and addition modulo $2$
The confusion here stems from a missing word. A correct statement is "The result of XORing two bits is the same as that of adding those two bits mod 2."
For example, $(0+1)\bmod 2 = 1\bmod 2 = 1=(0\...
- 14.6k
20
votes
Accepted
Represent a real number without loss of precision
It all depends what you want to do.
For example, what you show is a great way of representing rational numbers. But it still can't represent something like $\pi$ or $e$ perfectly.
In fact, many ...
- 29.1k
20
votes
Accepted
Inequality caused by float inaccuracy
In typical floating point implementations, the result of a single operation is produced as if the operation was performed with infinite precision, and then rounded to the nearest floating-point number....
- 25.2k
19
votes
Accepted
Time complexity of addition
If your algorithm uses asymptotically less than $n$ time, then it does not have enough time to read all the digits of the numbers it is adding. You are to imagine you are handling very large numbers (...
- 4,167
19
votes
Accepted
How does 0 have two values in one's complement?
In 1's complement you just invert all the bits.
Consider these 2 examples (assuming 8 bits):
$4 = 00000100$, so $-4= 11111011$
$0 = 00000000$, so $-0=11111111$.
So you have 2 ways to represent ...
- 1,615
17
votes
Accepted
Signed and unsigned numbers
The short and simple answer is: it doesn't. No modern mainstream CPU ISA works the way you think it does.
For the CPU, it's just a bit pattern. It's up to you, the programmer, to keep track of what ...
- 5,151
15
votes
Standard constructive definitions of integers, rationals, and reals?
Gilles answer is a good one, except for the paragraph on the real numbers, which is completely false, except for the fact that the real numbers are indeed a different kettle of fish. Because this sort ...
- 28.2k
13
votes
Why is addition as fast as bit-wise operations in modern processors?
Processors are clocked, so even if some instructions can clearly be done faster than others, they may well take the same number of cycles.
You'll probably find that the circuitry required to ...
- 296
12
votes
How can I compute an exponential modulo a large integer?
Modular exponentiation is a well-known algorithm. It is routinely available in libraries and languages that can manipulate large integers, including Wolfram Alpha.
When making computations modulo a ...
12
votes
Accepted
Standard constructive definitions of integers, rationals, and reals?
There are multiple ways to define a mathematical structure, depending on what properties you consider to be the definition. Between equivalent characterizations, which one you take to be the ...
12
votes
How to calculate sum of binomial coefficients efficiently?
Hint: Use Lucas's theorem.
In general, any time a programming contest problem wants you to compute something mod $p$, check for opportunities to reduce everything mod $p$ before doing any further ...
D.W.♦
- 141k
12
votes
Why is addition as fast as bit-wise operations in modern processors?
Addition is important enough to not have it wait for a carry bit to ripple through a 64-bit accumulator: the term for that is a carry-lookahead adder and they are basically part of 8-bit CPUs (and ...
- 121
11
votes
Accepted
Minimal basis for set of binary vectors using XOR
If you treat your vectors as over the field $GF(2)$ rather than over the set $\{0,1\}$, then what you ask is to find a basis for the span of a set of vectors. This is a well-studied problem in linear ...
- 270k
11
votes
How can I multiply a binary representation by ten using logic gates?
Multiplying by 10 is the same as multiplying by $(1010)_2$. To multiply a binary number $x$ by 10, we thus just have to add $x0$ and $x000$. For example, $6 \times 10 = 60$ is implemented by
$$
\begin{...
- 270k
10
votes
Accepted
Proving a language (ir)regular (standard methods have failed)
This can be proven, but you need some nontrivial tools to do it.
Start with the set S = {0,3,5,6, ...} of non-negative integers having an even number of 1's in their base-2 expansion.
It is well-...
- 301
10
votes
The math behind converting from any base to any base without going through base 10?
This is a refactoring (Python 3) of Andrej's code. While in Andrej's code numbers are represented through a list of digits (scalars), in the following code numbers are represented through a list of ...
- 201
10
votes
Why is addition as fast as bit-wise operations in modern processors?
I think you'd be hard pressed to find a processor that had addition taking more cycles than a bitwise operation. Partly because most processors must carry out at least one addition per instruction ...
- 411
10
votes
Signed and unsigned numbers
One of the great advantages of two’s-complement math, which all modern architectures use, is that the addition and subtraction instructions are exactly the same for both signed and unsigned operands.
...
- 1,121
9
votes
What is most efficient for GCD?
For numbers that are small, the binary GCD algorithm is sufficient.
GMP, a well maintained and real-world tested library, will switch to a special half GCD algorithm after passing a special threshold,...
- 543
9
votes
Why is addition as fast as bit-wise operations in modern processors?
At the gate level, you are correct that it takes more work to do addition, and thus takes longer. However, that cost is sufficiently trivial that doesn't matter.
Modern processors are clocked. You ...
- 3,085
9
votes
Accepted
Arithmetic network to compute floor of binary logarithm
It can be done with a fairly simple log-depth circuit without resorting to such hacks that only really make sense in software.
The "position of most significant set bit" function ...
- 1,813
8
votes
Accepted
Why addition algorithm is not pseudo- polynomial?
The reason is that the runtime of addition is proportional to the number of digits, not the value of the numbers, as it is for subset sum. Remember that the number of digits is the size of the input.
- 6,519
8
votes
Accepted
Optimal Algorithm for checking if a number is a multiple of three
If an algorithm that takes less than $O(n)$ time, this means that the algorithm won't be able to look at all the digits of the number. Thus you can prove a lower bound of $O(n)$ by showing that you ...
- 6,519
Only top scored, non community-wiki answers of a minimum length are eligible