Questions tagged [arithmetic]

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Multiplication mod 2 without extra registers

For an arbitrary bitstring $(x_1, x_2,\ldots, x_n)$ and an $n\times n$ invertible binary matrix $M$ (fixed ahead of time), I would like to construct a circuit $C$ acting on these $n$ bits whose output ...
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12 views

How does the structure of a Luk-Vuillemin multiplier differ from a Wallace or Dadda multiplier?

I've read that Luk-Vuillemin multipliers are similar to Wallace multipliers but partition their inputs in a different way. How exactly does this partitioning work, how does it change the structure of ...
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1answer
24 views

Embedded arithmetic set expressions

In set builder notation, we can represent the set: $$\{ 2, 7\}$$ as: $$\{ x | x=2 \vee x=7 \}$$. Therefore, the PA arithmetic predicate: $$φ(x) := x=2 \vee x=7$$ is capable of representing this set....
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33 views

What is the strongest arithmetic theory decidable by a DFA, DPDA or PDA?

It is known that WS1S can be decided by a DFA. Is this the strongest arithmetic theory decidable by a DFA? What happens when the automata class is extended to include DPDAs or PDAs?
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2answers
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Do we need to check for mantissa overflow in floating point multiplication?

We do check for the mantisas overflow in floating point addition e.g. If we are adding $8.02 \times 10^3 + 9.01 \times 10^3 =17.03 \times 10^3$ i.e we get an overflow, so we shift the number right ...
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1answer
42 views

Why do we need/use operator precedence for Arithmetic operators?

Why do we use operator precedence rule for Arithmatic operators? Can't we simply just do the operation in a linear manner from left to right or vice-versa and deal with the operator that comes first. ...
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0answers
152 views

Replacing 0x1021 polynomial with 0x8005 in this CRC-16 code

I have some highly optimized code for a CRC-16 implementation. It focuses on speed rather than flexibility, and as a result, it is hard-coded to model the specific unreflected polynomial ...
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0answers
112 views

Advantages/Disadvantages of adaptive contexual arithmetic coding

What are the advantages and drawbacks of considering ever longer block lengths or context lengths, if one was to work with estimated probabilities(measuring on the fly: aka "adaptive") rather than ...
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0answers
17 views

How do you compute and compare the delays between a (4:2) compressor and (3,2) counter carry save tree?

My question: How is Table 6.8 shown below computed for different operands? For example, for 3 operands how did they compute: Number of levels using (3,2) = 1 Number of levels using (4;2) = 1 ...
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0answers
44 views

Quick calculation for $x^y \bmod 2^d$

I need to calculate $x^y \bmod 2^d$ in $O(d)$ summations/bitwise operations and $1$ multiplication by $y$. $x$ is restricted to be odd, $d\geq 3$. $a$-bit arithmetic (for any $a$) is allowed, as this ...
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2answers
181 views

Minimum number of bits to represent negative number

Minimum number of bits required to represent $(+32)_{base10}$ and $(-32)_{base10}$ in signed two's compliment form? My attempt: 32 = 0100000 ( 1st zero - sign bit as positive) So to represent +32 ...
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2answers
55 views

How to convert 010111 to decimal using only odd numbered positions while still applying the correct positional exponent?

I am trying to understand what this question is asking in terms of conversion. I know how to convert binary to decimal and vice versa, but the method you need to use has me confused. I need to ...
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1answer
22 views

What's the least signifcant bit of a mantissa system?

If Mantissa is a 1-dot-M fixed-point number whose most significant bit is always 1 then, how is the least significant bit calculated? I know the least and most significant bit of the mantissa ...
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55 views

Who invented the adder, full-adder, half-adder?

I didn't find, in the digital design books, who invented the adders. The same person invented the half-adder and the full-adder? What's the oldest publication on digital arithmetic design?
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1answer
76 views

How can I calculate the maximum sum/product of sequence?

I am looking for an algorithm in $O(N^2)$ that finds the maximum value that be obtained from a sequence of real numbers greater than 0 (e.g. $\{ 1, 2, 3 , 4\}$) by inserting a plus ($+$) or ...
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11 views

Is Decimal (correctly-rounded arbitrary precision decimal floating point arithmetic) fixed-point, floating-point or something else?

The Decimal data type I am referring to is GNU MPFR(https://en.wikipedia.org/wiki/GNU_MPFR), or libmpdec (http://www.bytereef.org/mpdecimal/doc/libmpdec/index.html). I have been searching for ...
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4answers
70 views

How does a computer compute negative(-) and positive(+) Infinity?

If we divide (1.0/0.0) we will get +Infinity and if we divide (-1.0/0.0) we will get -Infinity. How does a computer calculate this value internally?
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32 views

Least computationally expensive bitwise addition

I am familiar with the oft-cited method of bitwise addition using XOR and left shift for the carry, applied recursively. I was wondering if this is the least computationally expensive way to achieve ...
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12 views

Number of bits for scalar product signal

let's consider two vectors $v_1[i]$, $v_2[i]$, for $i = 0, 1, ... , N$. Suppose you want to calculate the scalar product between them: it is simply the sum of all $v_1[i] * v_2[i]$ terms $(i=0,...,N)$...
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52 views

Can we represent $\sqrt{2}$ exactly even with infinite bits in mantissa [closed]

Can we represent $\sqrt{2}$ exactly even with infinite bits in mantissa in floating point notation or otherwise. We actually have to prove this is not possible. But why can't we if we have infinite ...
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1answer
137 views

Why does arithmetic left shift of negative number leads to positive number?

According to this Wikipedia article, when arithmetic left shift operation is applied to a signed number, the number is multiplied by 2. But there are certain situations where a negative number becomes ...
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1answer
255 views

What is the time complexity of a binary multiplication using Karatsuba Algorithm?

My apologies if the question sounds naive, but I'm trying wrap my head around the idea of time complexity. In general, the Karatsuba Multiplication is said to have a time complexity of ...
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1answer
210 views

Binary representation of 129 when using 8bits for two's complement?

I have this confusion regarding binary representation of decimal value 129 (or even 128). If 8 bits are used to represent numbers when doing the two's complement, then we know that '00000000' to '...
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27 views

Floating point substraction

if $x=1.0e38=1.0 * 10^{38}$ and $y=3.0$ i want to find $ (x-x)+y $ and $(x+y)-x$ i think the value of (x-x)+y will be just substract $x-x=0 + y=3.0 = 3.0$ but how can i perfom addition of different ...
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10 views

Why the multiplier and quotients can not store in the ACC?

When I study the Arithmetic Unit, there is the below information: there I have some questions: Why the multiplier must store in the MQ and the product must divide ...
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1answer
56 views

How to represent calculable real numbers?

Suppose I want to do arithmetic without any loss of precision. Floats and doubles are inappropriate. I want to use dynamic memory allocations to store any real number obtained after a finite amount of ...
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2answers
137 views

How does the bitlength of the divisor affect the running-time complexity of division algorithms?

Wikipedia lists $O(M(n))$ as the best complexity (out of the algorithms listed) for division on two $n$-digit numbers, where $M(n)$ is the complexity of the multiplication algorithm of choice. This is ...
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1answer
63 views

Prove that the next multiple of 4 is obtained using the next formula

I was reading an assembly procedure that needed to align addresses on 4 bytes boundary for performance reasons so it has used the next statement that i formulated as a theorem to be proven. Let $s$ ...
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2answers
44 views

Running time complexity of finding maximal power of divisor that divides natural number

Given $n \in \mathbb{N}$, a divisor $p\vert n$, I would like to efficiently find $e\in\mathbb{N}$ with $p^e \vert n$, and $e$ maximal with this property. I will assume that multiplication/division of ...
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5answers
3k views

Signed and unsigned numbers

How would the ALU in a microprocessor differentiate between a signed number, -7 that is denoted by 1111 and an unsigned number 15, also denoted by 1111?
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5answers
1k views

Number of FLOPs (floating point operations) for exponentiation

What is the number of floating point operations needed to perform exponentiation (power of)? Assuming multiplication of two floats use one FLOP, the number of operations for $x^n$ will be $n-1$. ...
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35 views

arithmetic coding for generating random number with desired distribution

Hi i want to convert random number with uniform distribution to desired distribution using arithmetic coding. It has been done in the following research paper called arithmetic distribution coding ...
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1answer
213 views

Is arithmetic turing complete?

Maybe my question doesn't make sense, because I lack some more thorough understanding, but I was curious if arithmetic was Turing complete? As I understand it, a "model of computation" is a mechanism ...
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1answer
148 views

Why integer division is of equal complexity as multiplication

I am trying to understand the fact that integer division is no more difficult than integer multiplication. I found some references - here and this lecture note. Wikipedia says if there is a way to ...
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1answer
42 views

Find value of $a$ or $b$ of two XOR equations

Is it possible to find $a$ or $b$ given that $a \oplus b = c$ and $c \oplus b = a$ when I only have the value of $b$?
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2answers
62 views

Increased rounding relative error when subtracting

I'm reading the book "Lessons in Scientific Computing" by Schoerghofer and it says: If x and y are real numbers of the same sign, their sum x + y has an absolute error that adds the two ...
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1answer
105 views

Addition errors in IEEE754 floating point representation

So in class, we were talking about the idea of floating point precision in IEEE754 format, and how, when some numbers are added, precision is lost. My professor then gave the following example of a ...
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1answer
2k views

Overflow rule in two's complement arithmetic

In the book by William Stallings the overflow rule overflow rule for 2's complement addition is stated as follows: Overflow rule: If two numbers are added, and they are both positive or both ...
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1answer
186 views

Algorithms for elementary operations using other elementary operators

The question asks to provide an algorithm to compute $(i)$ The product of $n$-bit numbers using reciprocation operation and addition operation but not using multiplication and squaring. $(ii)$ The ...
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0answers
97 views

What is the simplest automaton that can compute the sum of two integers of arbitrary length?

It should be obvious that a Turing machine is capable of computing the sum of two integers. However, what is the simplest automaton that can compute the sum of two integers of arbitrary length? I ...
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1answer
31 views

Why Does The Division Alorithim Need [Register Size] + 1 Iterations

Following this flow diagram for division hardware I made a program to "simulate" division on $N^+$. ...
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2answers
81 views

Given $0 <(x, y) < z < 2^{64}$, How can I compute $\lfloor \frac{xy}{z} \rfloor$ using only 64-bit arithmetic operations?

I can compute this easily in the case that $xy < 2^{64}$. But I'm not sure how to do this if $xy \geq 2^{64}$. I know that $\lfloor \frac{xy}{z} \rfloor = \frac{xy - (xy\ \text{mod} \ z)}{z}$, but ...
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1answer
56 views

Efficient way to compute mod(w +1) or mod(w - 1) where w= 2^p

Knuth in his book provides a method of how to efficiently calculate mod(w +1) or mod(w-1) where w is a power of 2. I am not sure I could understand his assembly language completely. Could you explain ...
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1answer
220 views

IP Address CIDR Bitmask Conversion

In looking for a programming function to validate an IP address within a subnet, I had to calculate the subnet mask from a given CIDR bitmask value ie. 192.168.0.0/24, the value 24 in this example. ...
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1answer
365 views

Algorithm for implementing the modulus “%” operator?

How can an efficient modulus operator be implemented? Here's a naive way of defining A % B: given $(a,b) \in \mathbb{Z}$ (represented as ...
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0answers
798 views

Booth bit-pair recoding of multipliers

1) In Booth's bit-pair recording technique how to multiply a multiplicand with 2? 2) In booth's algorithm for multiplication/Booth's bit-pair recording of multipliers, the sign bit extension of the ...
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1answer
22 views

Emulating equal operator using multiplication

I have two values $A$ and $B$, I want to know if I can implement the equals $=$ operation as the product of the two values. I can apply any function to $A$ and any function to $B$, but I need to use ...
2
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1answer
62 views

Implementing Gauss–Legendre algorithm using arbitrary-length rationals

I am trying to re-implement SuperPI myself in Rust, but the results I get are not very accurate. SuperPI computes pi using the Gauss-Legendre algorithm. The Gauss–Legendre algorithm is quite simple, ...
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1answer
32 views

How can the binary OR function be computed by a MOD3 gate of constant fan-in?

I've been working on a problem and in order to prove the bigger picture, I need to understand how a binary OR function can be computed by a constant fan-in MOD3 gate. I would seem that the output ...
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1answer
48 views

Complexity of numerical operations

I have written a program which contains a while loop: while k < sqrt(n), so clearly my program evaluates $\sqrt{n}$ at each iteration of the while loop. (Note ...

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