Questions tagged [floating-point]
Approximate representation of numbers as a fixed number of digits multiplied by a logarithmic scale.
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questions
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9 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 ...
2
votes
1answer
38 views
Smallest integer i stored as a float such that i+1=i
So I had an assignment which asked me to find the smallest integer $i$ which when represented as a float is such that $i+1=i$
My approach- By making a simple C++ program , we get $i=16777216$ or $i=2^...
2
votes
2answers
30 views
Why multiplying float number by multiple of 10 seems to preserve better precision?
It is famous that for float numbers:
.1 + .2 != .3
but
1+2=3
It seems that multiplying floats by 10 allows you to preserve ...
1
vote
4answers
66 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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0answers
44 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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0answers
24 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 ...
2
votes
1answer
68 views
Polynomials - using Newton's method, or not?
I have to find a root of polynomial of degree $n\ge2$. I need to write code to calculate the root for different values of $n$. Only 1 real positive solution is needed.
I can use general Newton's ...
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vote
0answers
28 views
How many floating point ops were performed worldwide over a time interval [closed]
I am looking for information regarding the evolution of computing capability. Specifically I would like to know how many floating point operations were performed worldwide from, say, the deployment of ...
0
votes
1answer
60 views
Computing an Expression
I am writing code to evaluate the following expression:
$$ \frac{(a+b+c)!}{a! b! c!} $$
where $a$, $b$ and $c$ are on the range of $10$ to $500$. The result is going
to be a floating point number. ...
3
votes
2answers
97 views
How much can we trust mathematical software when working with large numbers, and how much memory it needs to work with these numbers?
For example, I want to evaluate the expression:
$3^{3^{{3}^{3}}}$
so I used wolframalpha.com (it's free, and I don't own any software), which returned the scientific notation of the number above, ...
2
votes
1answer
52 views
numerically stable log1pexp calculation
What are good approximations for computing log1pexp for single precision and double precision floating point numbers?
Note: ...
2
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1answer
73 views
Floating point arithmetic
I need to change x1 = 0.3 and x2 = -0.29 to a FP(floating point) number with one sign bit, a 4 bit mantissa, a 3 bit exponent. The results I got are:
x1: 0 001 0011
x2: 1 001 0010
I am also trying ...
0
votes
2answers
100 views
Convert $1.75\times10^{15}$ to IEEE-32 format?
$1.75\times10^{15}$
I know how to convert decimal to binary
$(1.75)_{10}$ is equal to $(1.11)_2$
But to represent $10^{15}$ is the main problem for me. I can solve the question but this is the ...
3
votes
3answers
63 views
Guarantees on computing $a+x(b-a)$ in floating point
I want to implement the function $f(x,a,b) = a + x(b-a)$ where all the inputs are floating point (doubles, say), such that (a) $f(0,a,b)=a$ exactly; (b) $f(1,a,b)=b$ exactly; (c) $f(x,a,b) \le f(y,a,b)...
0
votes
0answers
33 views
How does normalised floating point binary work with two's complement?
I'm doing AQA a-level computer science, and the specification for which states that:
Exam questions on floating point numbers will
use a format in which both the normalised mantissa and
exponent are ...
1
vote
0answers
12 views
Where can I find some free benchmarks to evaluate a MCU? [closed]
At present, I'm designing a soft processor with single-precision floating point unit (FPU). I am going to put my soft core into an FPGA and do some performance evaluation. The benchmarks are supposed ...
0
votes
2answers
63 views
Float number to binary
I would like to convert 0,347 and 0,9828 to binary, how can I do that?
I know that sucessive multiplication by 2 can do this, but this method seems very painful and even ineffective since the size of ...
2
votes
0answers
91 views
Stable and fast computation of the squared euclidean distance matrix
Let's say I want to compute the matrix $M$ of the squared euclidean distances between each pair of vectors $(x, y)$ belonging to two sets $X$ and $Y$ respectively. The sets of vectors $X$ and $Y$ have ...
4
votes
2answers
171 views
Proof that (x-y)(x+y) is more accurate than x²-y²
I was carrying on my reading of What Every Computer Scientist Should Know About Floating-Point Arithmetic but got stuck on the proof of Theorem 2 (page 34).
At some point it says:
\begin{align}
(x \...
2
votes
1answer
49 views
Proof that a guard digit bound the error of subtraction
I was reading What Every Computer Scientist Should Know About Floating-Point Arithmetic, which is extremely interesting. But I have some troubles understanding the proof of Theorem 9 (page 33).
First ...
0
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1answer
54 views
Why isn't it necessary to store an integer part of significant in IEEE754 floating point notation?
We see that there is a sign, exponent, and mantissa part for the notation. But, there is no location for the significant bit.
Why isn't it necessary to store an integer part of significant in IEEE754 ...
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votes
1answer
40 views
Doubt in definition of Float
Can anyone tell the meaning of the bold portion?
Float: It is used to store decimal numbers (numbers with floating point value) with single precision.
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votes
5answers
953 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 $n-1$. ...
0
votes
2answers
91 views
How to test for overflow when multiplying floats
I am trying to implement a 3-term recurrence relation:
$$
p_{n+1} = ap_n + bp_{n-1}
$$
This can be implemented as
...
0
votes
1answer
32 views
How does IEEE 754 decimal encoding work? [closed]
This may be a silly question, but if a computer works in binary how can you encript numbers using decimal?
5
votes
0answers
97 views
What is the state of the algorithmic art for floating point arithmetic on complex numbers?
Most modern compilers and processors implement the IEEE 754 binary formats for floating point numbers. IEEE 754 guarantees that the addition, subtraction, multiplication, division, and square root ...
1
vote
1answer
164 views
Binary Floating Point Range/Precision for 4-bit Mantissa and 4-bit Exponent
I'm trying to understand binary floating point and using just a 4-bit mantissa and a 4-bit exponent (both 2s compliment) to keep things simple.
As far as I can tell, the largest denary number I can ...
0
votes
2answers
41 views
Floating point precision from rearranging equation
When I run $x^2 - y^2$ with x=8.8888888888 and y=9.9999999999 in python, I get the following result:
...
1
vote
1answer
37 views
Why do I get different results from two calculation methods?
I am wondering what the reason for the following is
We know that ,
exponential has a taylor representation :
$$exp(x)=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$$
Using the first n terms , in R , ...
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vote
2answers
59 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 ...
0
votes
3answers
151 views
IEEE754 representation in hexadecimal?
In class, I've heard hexadecimal representation for IEEE754 mentioned and described in 32bit length as a format that consists of one bit for sign, normalized 6-digit fraction (with an implied leading ...
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vote
1answer
55 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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vote
0answers
2k views
Understanding denormalized numbers in floating point representation
I am confused about how denormalized numbers work in floating point representation.
I was referring to Stallings book and this article. The book initially explains floating point number format in ...
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vote
0answers
44 views
Accuracy and performance between a division and subtraction for a ratio in decibels
For comparing two images, one can use the Peak Signal-to-Noise Ratio (PSNR) metric, defined as follows:
$\mathrm{PSNR} = 10 \cdot \log_{10}\left(\frac{\mathrm{MAX}^2}{\mathrm{MSE}}\right) = 20 \cdot \...
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0answers
27 views
Numbers lying between single and double precision
The IEEE floating point number format is defined as
$$s\underbrace{c_1\dots c_m}_\text{exponent}\underbrace{f_1\dots f_n}_\text{fraction}\text{ (*)}$$
with $s, c_i, f_j$ being either $1$ or $0$. The ...
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votes
0answers
24 views
IEEE-754 and machine numbers
I've been trying to wrap my head around machine numbers like the unit roundoff (u) and epsilon (e) in combination with the IEEE 754 standard. My textbook states some things that don't really make ...
2
votes
2answers
117 views
Algorithm challenge: build a pile of 'n' cubes whose total volume adds up to 'm'
I'm working on solving an algorithm problem defined as follows (important parts in bold):
Your task is to construct a building which will be a pile of n cubes. The cube at the bottom will have a ...
0
votes
1answer
37 views
algorithm for correctly rounded floating point radix conversion
Is there any generic algorithm which implements a floating point radix conversion?
Lets say we have a $p$-digit FP number
$A = \sum_{i=0}^{p-1} A_i \beta^{e-i}$
in radix $\beta$ and with $0 \leq ...
1
vote
0answers
186 views
Why is the method of im2col with GEMM is more efficient than the method of direction implementation with SIMD in CNN
The convolutional layers are most computationally intense parts of Convolutional neural networks (CNNs).Currently the common approach to impement
convolutional layers is to expand the image into a ...
0
votes
1answer
83 views
Floating point Euclidean norm optimization
Past few days I've been struggling with floating point number exercises, trying to learn how they work and what are their limitations. The exercise I've encountered today is as follows:
The ...
0
votes
1answer
207 views
How to compute relative error for the rounding of floating point numbers when the rounded number is 0?
I have asked this question on Stack Overflow, I am asking it here in the hope to get more traction.
The relative rounding error for a floating point number x is defined as
$e_r = |\frac{(round(x) -...
1
vote
2answers
61 views
Double floating precision exercise
today I had to deal with this exercise:
If $x \approx y$, we might expect some cancellation in computing $log(x)- log(y)$. On the other hand, $log(x) - log(y) = log(\frac{x}{y})$, and the latter ...
0
votes
1answer
54 views
Floating point number fractions excercise
During my preparation for an exam, I've come across an exercise that was focused on precision calculation with floating point numbers. It goes like this:
Consider the expression : $$\frac{1}{1-x}-\...
0
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1answer
121 views
Using a 16-bit 2,s Complement normalised floating-point representation; 10-bit fractional mantissa and a 6-bit integer exponent: express 2.171875
So far I know how to normalize when you are given, say:
1111010010 Mantissa, and 000100 exponent
and are told that it's a positive number:
...
1
vote
1answer
23 views
Number of IEEE 754 doubles between two adjacent single-precision floats
Between an adjacent pair of nonzero IEEE single precision real numbers, how many IEEE double precision numbers are there?
I was also wondering if this question has something to do with the hidden ...
0
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1answer
56 views
Why are floats converted from different integers sometimes equal?
I'm trying to understand the following algorithm:
...
0
votes
2answers
35 views
Can area-partitioning lose included points due to floating point precision?
I'm currently partitioning a big area $A$ into $n$ areas $B_i$ such that
$$\bigcup_{i=1}^n B_i = A$$
I have geo-coordinates which I know are in $A$ (also with the finite precision of floats).
...
1
vote
2answers
55 views
Hypothetical question on floating point normalization
I've seen a few questions posted regarding the upsides/downsides of floating point formats that have explicit integrals vs. formats that have implicit integrals, but have not seen an answer that ...
1
vote
1answer
113 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 ...
1
vote
3answers
59 views
How does computer store non-repeating decimal very accurately?
Maybe it is not accurate at all, but it looks very accurate for me.
I'm making a scientific calculator for my portfolio, and I found an interesting phenomenon.
Because computers store decimals not ...
