Questions tagged [floating-point]
Approximate representation of numbers as a fixed number of digits multiplied by a logarithmic scale.
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Absolute difference between largest IEEE754 number and its predecesor
In simple precision format, the largest possible positive number is
$A = 0 ~~~ 11111110 ~~~ 111\ldots 111$
Its predecessor is
$B = 0 ~~~ 11111110 ~~~ 111 \ldots 110$
But what is the absolute ...
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Comparison of different algorithms for summing floating point numbers
I am exploring several approaches to summing floating point values, such as:
Naive summation, for comparison
Summing sorted values
summing with numpy, again for comparison
Kahan's algorithm
Pairwise ...
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Convert a rational number to a floating-point number exactly
We have two integers, $n$ and $d$. They are coprime (the only positive integer that is a divisor of both of them is $1$). They may be implemented as something that fits in a machine register, or they ...
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Multiplication of subnormal and normal numbers under IEEE 754
As far as I understand, when doing this operation, we first need to identify the subnormal value, normalise it, adjust the exponent, and then multiply the significands. Wouldn't it be advantageous to ...
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Floating-point modular multiplication algorithm
Is there a well-known algorithm for modular multiplication of floating-point numbers?
I would like to multiply some large angle in single precision (6-7 significant digits) and wrap it back to 360 ...
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FFT of logarithmic input data
Is there a reasonably accurate method of computing an FFT of logarithmically-represented input data (with a sign bit, that is $±2^{\text{double-precision value}}$)?
The naive method (convert to linear ...
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Floating-point rounding - bit patterns of values that are halfway between two possible results
I am working through the book Computer Systems: A Programmer's Perspective.
The authors explain that round-to-even rounding can be applied for values that are halfway between two possible results. For ...
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Why XMM register width is 128-bit while double precision floating point is 64-bit?
As far as I know, XMM is used to store floating point. But Highest width floating point according IEEE754 standard is double precision (64-bit). But why XMM register width is 128-bit. Is another 64-...
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How much larger is the next representable value if 2^59 is stored in a double?
This is an exam question I couldn't solve
If I store 2^59 as double, that would give me
1 * 2^58. Is the answer just 2? I.e. next value is 2^60??
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Best way to constrain a complex number to being within the unit circle?
What is the best way of implementing the following function,
$$f(x) = \frac{x}{\max(1, |x|)},$$
where $x$ is complex, using a Cartesian representation of $x$ with IEEE 754 floating point ...
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What are the use cases for the IEEE 754 inexact flag?
The IEEE 754 standard for floating point numbers defines a flag that is set when a result from floating point calculation isn't exact, i.e. has to be rounded. What algorithms are there that utilize ...
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Is significand same as mantissa in IEEE754?
I'm trying to understand IEEE 754 floating point. when I try convert 0.3 from decimal to binary with online calculator, it said the significand value was ...
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How does CPU determine Reserved Exponent cases?
Using IEEE 754 algorithm i assume, that it can be implemented in a branchless way.
But how does CPU determine special cases (Reserved Exponent values):
Exponent
Significand
is
11111111
000000000...
...
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What does it mean unambiguously that a number is value 0 up to numerical precision?
I was reading that a quantity $x$ is $0$ upt to numerical precision. What does this statement formally mean -- especially in the context of numerical methods or real computers.
I looked up in google ...
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Standard for representing a float scaled to a particular range?
TL;DR Is there a "standard" way to represent a float scaled to a particular range, such that we get maximum precision for the given bit depth, within that range?
I'll start with my general ...
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(Numerical Analysis) What is the largest double float represented for the gamma function and $n!$
Consider that
\begin{align}
\Gamma(n+1) = n!
\end{align}
for any integers. I then got the following two questions:
What is the largest value of $n$ for which $Γ(n+1)$ and $n!$ can be exactly ...
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Bisecting Intervals of floating point numbers containing 0 and infinity fairly
It is seldom considered that floating points are not evenly distributed in the real number line. I've been working with interval arithmetic and noticed when bisecting $[a,b]$ on the real number line ...
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Can Radix Sort be modified for signed ints and/or floats?
A few months ago I learned about the magic that allows radix sort to run in O(n) time and space. Most tutorials on radix sort say it is useful for very large ...
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IEEE 754 conversion
I'm trying to convert 3.2 into IEEE 754 format. We find that $(3)_2=11$ and we also find that
$0.2*2=0.4 -0$
$0.4*2=0.8 -0$
$0.8*2=1.6 -1$
$0.6*2=1.2 -1$
and this cycle repeats so $.2=00110011...$
...
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Uniformly random decimal numbers
Due to finite precision of number representations, we face situations like:
In: 0.1+0.1+0.1==0.3
Out: False
(on my ...
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(Branchless) Bitonic Sorting Network for a Set of Floating Point Numbers
In the past I've implemented a branchless Bitonic Sorting Network on a gpu using CUDA, for integers.
I am facing a related problem:
In my Order Independent Transparency implementation, I would like to ...
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How can vector angle comparison between lattice points be done without using floating-points? (Convex Hull)
Let's say I have a point $(x_0, y_0)$, and some other points $(x_1, y_1), (x_2, y_2) ... (x_n, y_n)$, such that all of them are lattice points; all have integer coordinates. Let's further assume that ...
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How many Integers can be represent in Double-Precision floating-point form
How to calculate the number of Integers that can be represent in Double-Precision floating-point form?
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Prove every number in double precision 32-bit floating-point format can be represented in 64-bit format
Theorem: Prove every number in double precision 32-bit floating-point format can be represented in double precision 64-bit floating point-format.
64-bit format:
Attempt: Let $ b = b_0 ,...,b_{31} $ ...
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Why does floating point become less accurate as the powers of 2 increase?
https://fabiensanglard.net/floating_point_visually_explained/
I was reading this article where the exponent and the mantissa are explained as the window and offset respectively. As the gap between ...
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Is there a way to convert FLOPS to bit operation per second
My problem is the following: I have $N$ inner products to compute in parallel every second.
Each of the vectors in those inner product is composed of $7$ bits.
I want to know for which $N$ it starts ...
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1
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Unit conversion - Better to divide by an integer or multiply by a double?
I currently have a long timestamp measured in units of 100ns elapsed since January 1st, 1900. I need to convert it to milliseconds.
I have the choice of either ...
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Half precision floating point question -- smallest non-zero number
There's a floating point question that popped up and I'm confused about the solution. It states that
IEEE 754-2008 introduces half precision, which is a binary
floating-point representation that uses ...
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Floating Point Arithmetic with 3 bits mantissa
Find all values of $ x ∈ R $ such that x + 1 = 1 in floating point arithmetic with 3 bits mantissa.
How do we represent number 1 in floating point arithmetic with 3 bits mantissa I wonder? After that, ...
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Python versus Matlab on the quantity 1/0
Python and Matlab seem to disagree on the division by 0.
Python:
...
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Negative Numbers in 32 bit Floating Point IEEE Numbers
So I understand the logic behind converting positive decimal numbers to IEEE 32 bit floating numbers but I'm not completely sure behind the negative one's. If for example we have a decimal number say -...
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Adding two numbers in base 2(floating point) vs Multiplying two numbers in base 2(floating point)
Is it true that adding two numbers in base 2 is more complex than multiplying them? If so can someone please explain why this is the case?
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Prove that $1^\text{nan} = 1.00$
I know that for most computation involve nan (not a number) the result is a nan itself except for some cases.
For example, $1^{\text{nan}} = 1.00$ which proven by mathematicians to be true.
I tried to ...
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Floating point bitwise comparator. If f1 and f2 are floating point numbers with the following properties can we always say f1 > f2?
Recall floating-point representation:
Suppose $f$ is a floating-point number then we can express f as,
If $f$ is normal:
$$(-1)^{s}\cdot2^{e-127}(1 + \sum\limits_{k=1}^{23} b_{23-k}\cdot 2^{-k})$$
If $...
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Convert $8.75×10^{6}$ to IEEE-32 format?
There is a similar question already asked on this site but does not have an answer as to how the 10x was converted into 2y. I know how to convert 8.75 or 875 into IEEE representation. But what about ...
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What is the machine epsilon and number of mantissa bits for TI-83?
I am trying to determine how many bits the TI-83 Plus uses to store floating point numbers. I am using the algorithm for approximating the machine epsilon given in "Numerical Mathematics and ...
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How close are current computer technologies in terms of energy efficiency to the Landauer Limit?
I'm trying to figure out how close (in orders of magnitude) current computer technologies are in terms of energy efficiency to the Landauer Limit. However, I'm finding (seemingly) conflicting ...
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How to determine the set of real numbers corresponding to a given floating point number?
Let's say we consider IEEE 754 double precision floating-point numbers, and we use RNTE - Round To Nearest, Ties to Even - rounding.
I know that the RNTE rounding works this way:
given two consecutive ...
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Logic behind choosing the exponent bias as $2^7 -1$ instead of $2^7$ in $32$ bits IEEE-754 floating point representation
The $\text{IEEE-754}$ uses $32$ bits to represent single precision floating point numbers. The partitions of the register are as follows:
...
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What is the currently know fastest algorithms to convert each individual Java primitive and Number types to a Char Array/String
What is the currently know fastest algorithms to convert each individual Java primitive and Number types to a Char Array/String number representation while ...
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How to represent zero as floating point number?
For the floating-point number, we have the form
$\pm d_0.d_1d_2···d_{P-1}\times\beta^E$
$\pm$ --------------------------- sign
$d_0.d_1d_2···d_{p-1}$ --------- significant
$\beta$ --------------------...
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Are floating-point numbers normalised by computers or humans?
I keep seeing posts and articles about how to normalise a floating-point number, why that's done and how such a number is represented in binary. But no one seems to mention who/what the normalisation ...
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How are bitwise operators used in normalisation of floating-point numbers?
After having spent a significant amount of time googling this topic, I am still struggling to confidently answer the following question:
...
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Converting Decimal Numbers between 0 and 1 to Binary
I've been playing around with a program I wrote that converts decimal numbers to binary numbers and i've noticed that eventually, after applying the algorithm (multiply by 2, subtract 1 if greater ...
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2
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Floating-point oblivious way to compute multiset numbers
I have to compute $R = \left(\!\!{n + 1\choose k}\!\!\right)$, which happens to be:
$$
R = \left(\!\!{n+1\choose k }\!\!\right) = \binom{n+k}{k} = \frac{(n + k)!}{n!k!} = \frac{(n+1)(n+2)\cdots(n+k)}{...
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Validity of Algorithm for Testing Two Floating Point Numbers
This question is related to the epsilon- (or delta- if you prefer) test for floating point equality. But my question is not how to do it. Instead I have a related algorithm for testing equality, and I ...
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New way of representing floating point
I want to create a new way of representing floating point. In standard IEEE floating point, we have 1 bit to represent sign, 8 bits to represent exponential and 23 bits to represent significand. In ...
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IEEE 754 addition wrong result floating point numbers
I want to add two IEEE 754 numbers.
I followed the steps to add two 754 numbers. However the result it not correct.
Number 1:
S:0
E:01111111
M:11111111111111111111111
Number 2:
S:0
E:01111111
M:...
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Cancellation of inequalities in floating point arithmetic
In finite precision floating point arithmetic the associative property of addition is not satisfied. This is, it is not always the case that $$(a+b)+c=a+(b+c)$$
Even $a=(a+b)-b$ is not always true.
To ...
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How does value decompression work for Facebook's Gorilla in the case where count of leading zeroes is not stored
I am referring to this paper: http://www.vldb.org/pvldb/vol8/p1816-teller.pdf
My question is regarding section 4.1.2 where it says:
When XOR is non-zero, calculate the number of leading and ...
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