For questions concerning finite precision arithmetic in computers and other related concepts.
Questions tagged [computer-arithmetic]
116 questions
12
votes
3 answers
Why do so many computer programming language implementations have trouble with the remainders of negative integers?
As most of us know, or should know, $-7 \equiv 1 \pmod 4$.
But if you use Java's modulus operator %, you get -3 for the answer, not 1. That's technically correct, but it can cause problems if you're not aware of this as you write a program. C# and…
David R.
- 1,316
8
votes
1 answer
Prerequisite reading for Concrete Mathematics?
I'm a freshman computer science major who has just started reading Concrete Mathematics, mathematics for computer science. Is there any prerequisite reading or learning I should do before embarking on reading this book?
Ben Moore
- 123
5
votes
1 answer
Is there still a fast invsqrt magic number for float128?
template
T union_cast(U data){
union{U a;T b;}t{data};
return t.b;
}
float128_t quick_invsqrt_with_magic_num(int128_t mnum,float128_t X){
auto x= union_cast(mnum - (union_cast(X) >> 1));
return…
5
votes
1 answer
Mathematics of two's complement
I am trying to understand the underlying mathematics of two's complement. Googling the topic gives me a lot of articles on how to invert the digits and add one, and why computers use this system rather than more straight forward binary addition.…
Avatrin
- 1,615
4
votes
3 answers
Differences between signed and unsigned decimal values
What are examples of signed and unsigned decimal values? What are the differences between them?
Bilis
- 125
3
votes
2 answers
Why does Eulers Theorem not work in this case?
In my number theory textbook I am tasked with finding the value of $5^{30}\mod 62$. As the last section had been about Eulers Theorem which states that for any $a,n\in\mathbb{Z}$ where the $\gcd(a,n)=1$
$$a^{\phi(n)}\equiv1\mod n$$
The first thing I…
Ilikemath
- 358
3
votes
1 answer
UPD: Structure of subgroups of $S_{2^n}$ generated by $\langle x \mapsto ax \mod 2^n \rangle$ and linear groups
It's a very well known result by Gauss that $(\mathbb{Z}/2^n \mathbb{Z})^\times = \langle -1 \rangle \times \langle 3 \rangle \cong C_2 \times C_{2^{n-2}}$.
Consider a faithful action $\mathrm{mul}: (\mathbb{Z}/2^n \mathbb{Z})^\times \to…
Aleksei Averchenko
- 7,627
3
votes
1 answer
How to represent subset using 5-bit binary code?
For the set $V=\{a, e, i, o, u\}$, give the $5$-bit binary string that codes each of the following subsets:
$\{a, i,o\}; \{e\}; V; \emptyset$;
Which subset is represented by the $5$-bit string $10001$?
Can I know how do you get the 5-bit binary…
J_fruitty
- 31
3
votes
0 answers
Error bound for floating-point interval dot product
In Handbook of Floating-Point Arithmetic (Birkhäuser, 2010, Chapter 6) Muller et al. presented the following absolute forward error bound for the floating-point recursive dot product:
$$
\left|RecursiveDot (a, b) - \sum_{i=1}^n a_ib_i\right| \le…
3
votes
3 answers
Floating point approximation to logarithms by powers and bit counting?
Earlier I both asked questions and found some interesting sources on the internet of how to do approximate division by combining and counting number of $0$s following most significant bit in denominator.
Example:
We represent a fraction $7/3$ as two…
mathreadler
- 26,534
3
votes
1 answer
SICP: Why does this recursion-based sine approximation work?
Here is the question and solution to Structure and Interpretation of Computer Programs' exercise 1.15 (see here). My problem is, I don't know how the combination of these formulae actually work:
$$sin(x) = 3sin(x/3) - 4sin^3(x/3)$$
and
$$sin(x) =…
147pm
- 1,162
3
votes
2 answers
Converting $\frac{2}{7}$ to a binary number in a $32$ bit computer
I want to convert $\frac{2}{7}$ to a binary number in a $32$ bit computer. That is, $1$ bit is assigned to the sign of the number, $8$ bits are assigned to the exponent, and $23$ bits are assigned to the mantissa.
So $x = \pm q \times 2^{m}$ where…
user463756
- 117
3
votes
1 answer
Determine sign of sum of square roots
Problem
Given positive square-free integers $r_i$ and non-zero integers $a_i$, is there an algorithm for determining the sign of $\sum_{i=1}^n a_i\sqrt{r_i}$ without calculating approximations for the square roots?
If $n=2$ it is easy and I hope it…
user70612
2
votes
1 answer
IEEE754 32-bit single precision format
I have a question like this:
Show how the number $-12.75$D is stored in the computer's storage using IEEE754 32-bit single precision format. You are required to show your conversion steps clearly.
My answer…
Bilis
- 125
2
votes
0 answers
On the complexity of big integer multiplication
There seems to be something I am deeply missing about the assumptions while calculating the complexity of multiplication
Let us say we have two number m,n, that we want to multiply and we have n>m, and Log(n) = bwe have a set of primes $p_1, p_2,…
