Questions tagged [floating-point]
Approximate representation of numbers as a fixed number of digits multiplied by a logarithmic scale.
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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 ...
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1answer
15 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 ...
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1answer
23 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:
...
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1answer
34 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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2answers
15 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
23 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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1answer
27 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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0answers
41 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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0answers
38 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
20 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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0answers
17 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 ...
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2answers
56 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 ...
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1answer
20 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 ...
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0answers
33 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 ...
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1answer
22 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 ...
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1answer
30 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) -...
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2answers
47 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 ...
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1answer
25 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 : $$\frac{1}{1-x}-\...
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0answers
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Special operands in floating point standards
An interview with William Kahan, the primary force behind the adoption of the IEEE 754 floating point standard, said the following in an interview.
"From that time on the standard ground slowly ...
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1answer
35 views
Using a 16-bit 2,s Complement normalised floating-point representation; 10-bit fractional mantissa and a 6-bit integer exponent: express 2.171875
So far I know how to normalize when you are given, say:
1111010010 Mantissa, and 000100 exponent
and are told that it's a positive number:
...
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1answer
19 views
Number of IEEE 754 doubles between two adjacent single-precision floats
Between an adjacent pair of nonzero IEEE single precision real numbers, how many IEEE double precision numbers are there?
I was also wondering if this question has something to do with the hidden ...
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1answer
51 views
Why are floats converted from different integers sometimes equal?
I'm trying to understand the following algorithm:
...
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2answers
29 views
Can area-partitioning lose included points due to floating point precision?
I'm currently partitioning a big area $A$ into $n$ areas $B_i$ such that
$$\bigcup_{i=1}^n B_i = A$$
I have geo-coordinates which I know are in $A$ (also with the finite precision of floats).
...
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2answers
39 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 ...
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1answer
56 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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3answers
58 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 ...
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0answers
30 views
Using half float to represent scaled short (int16), do I lose precision comparing to using double?
A device is generating 14-bit integer which is stored as int16 (short), a scaling process will then scale the data to value of order 10E-3.
Does it then matter if I store these number with half float ...
2
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1answer
34 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
147 views
How to calculate machine epsilon
In a binary system we know that the next floating point number after 4 is 4+1/32. What is the machine epsilon? Is it 1/32 and if yes, why?
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1answer
47 views
What does floating point biased notation have to do with 2's complement?
During lecture the professor talked about Integer Compare for floats and kept mentioning two's complement.
What's the relation between them?
All I understand is that two's complement can take us ...
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0answers
10 views
Why does economising the power series give me more error?
Suppose I want to economise $\sin x$ with the following taylor series:
$$P_{2n-1}(x) = \sum^n_{i=1}(-1)^{i-1}\frac{x^{2i-1}}{(2i-1)!}$$
for the interval $[-1, 1]$ for $P_5(x)$.
For $P_5(x)$, I have ...
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1answer
58 views
Computing the error bound of floating-point expression
How should I compute the maximum absolute and relative error of the following IEEE-754 floating-point expression?
a.y + (x - a.x) * ((b.y - a.y) / (b.x - a.x))
...
3
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1answer
47 views
Understanding truncation and rounding error in IEEE floating point system?
I'm trying to understand the theory behind finding the optimal $h$ value for differentiation in this definition:
$$ \frac{f(x+h) - f(x)}{h}$$
as $h$ tends to 0.
Here is my understanding:
...
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1answer
141 views
How to Add IEEE 754 Floating Point Numbers
I'm trying to add two 16 bit numbers that use a similar format as IEEE 754. The format is in the image below:
I can't figure out how to add together two numbers for the life of me.
Specifically, ...
3
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1answer
47 views
Why do I get really different results with my benchmarking code I made?
I'm doing research work for my last year in high school. My work is about processors and for the experimental part i've coded an app that can mesure how many Floating Point Operation can a processor ...
4
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1answer
80 views
Imaginary numbers and negative zero
I've been studying the low-level hardware implementations of floating point numbers and doing an exercise to design a custom floating point implementation.
I know that being able to represent ...
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2answers
142 views
Is IEEE 754 float arithmetic associative, commutative, distributive, etc? Why?
Does the associative/commutative/distributive/etc property hold for arithmetic performed with IEEE 754 floats?
Obviously the answer is no to most of those questions, but do any of the properties of ...
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2answers
46 views
Union of fixed and floating point types
Say I have two real number types. They may be floating or fixed point. How can I construct a new type whose values are at least the union of the two with the minimal number of bits?
There are 3 cases ...
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4answers
92 views
Float multiplication
While doing the multiplication 1.4*0.8 in a python program, I got the result as 1.1199999999999999.
Why didn't I get ...
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2answers
39 views
Float type variable uncertainty
I developed an image processing software and I need to do a numerical analysis of it, considering the error propagation associated to its operations and the uncertainty of float type variables caused ...
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0answers
44 views
Inverting an IEEE 754 double
While preparing for an exam I've come across this problem:
Let $x$ be an IEEE 754 double precision number. Show that
$$
\texttt{fl}(x * \texttt{fl}(1/x))
$$
has only 2 possible results ...
0
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1answer
325 views
What is exponent of denormalized real in IEEE 754 floating point format?
I was reading this link.
Here I noticed that, in table titles "Float Values (b = bias)", in exponent column, entry for denormalized reals is $00..00$, but in value column, the entry is $-b+1$. Why is ...
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1answer
104 views
Genetic Algorithm Getting Stuck on Certain Values
I'm attempting to teach myself something about genetic algorithms. I found a simple tutorial on it here, and it made enough sense that I was able implement the suggested example of generating a string ...
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1answer
41 views
How do proof assistants nowadays avoid floatpoint error?
I heard that Coq is using "Dedekind cut", but how's that related to floatpoint errors?
Can we just use something like
...
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1answer
23 views
Exact Value of Denormalised number related to floating point numbers
I faced with the following issue:
if the value of a floating point format number is in the range of underflow,
it must be 0 or a denormal number.If the value is not 0 than what is the exact value of ...
3
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1answer
165 views
Why do floating point values have infinity?
Why do floating point values have infinity instead of just a higher exponent?
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1answer
38 views
Converting a 32-bit bit pattern into floating point number
In The Scientist and Engineer's Guide to
Digital Signal Processing
By Steven W. Smith, Ph.D. i have read something like that,
Single precision or 32-bit floating point forms the floating point number,...
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1answer
411 views
How does conversion from fixed-point to floating-point happen?
I came across to the code that convert 32-bit signed fixed-point number (16.16) to a float and it looks like (pseudocode)
floating = fixed / 65536.0
Could you ...
0
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1answer
31 views
Shortest decimal expansion within binary interval
Consider an interval $[x-2^n,x+2^n]$ defined by a binary float $x$ and a power of two $2^n$ typically much smaller than $x$. I would like to know an efficient algorithm exists to determine the ...
2
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0answers
23 views
Heuristic Consistency with float multiplications
I'm running A* search with a theoretically consistent heuristic. I'm doing a lot of float multiplications, and I'm not sure how best to ensure that my heuristic remains consistent down to the last ...
