Questions tagged [floating-point]

Approximate representation of numbers as a fixed number of digits multiplied by a logarithmic scale.

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2 answers
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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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1 vote
1 answer
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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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25 views

How many computer numbers can there be, with a basis $\beta$, and $n$ digits for the mantissa?

Cheers, I have to prove that, given a basis $\beta$ (e.g. $\beta = 2$ for binary), $n$ digits for the mantissa and $e$ for exponent ($ m \leq e \leq M, m \lt 0 \lt M$) for the floating point ...
1 vote
1 answer
20 views

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-...
0 votes
1 answer
38 views

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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2 votes
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46 views

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 ...
2 votes
0 answers
113 views

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 ...
4 votes
3 answers
1k views

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 ...
0 votes
1 answer
35 views

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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1 vote
1 answer
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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 ...
1 vote
0 answers
27 views

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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2 answers
59 views

(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 ...
3 votes
1 answer
76 views

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 ...
1 vote
0 answers
136 views

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 ...
0 votes
1 answer
38 views

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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4 votes
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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 ...
1 vote
0 answers
32 views

(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 ...
1 vote
1 answer
51 views

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 ...
1 vote
2 answers
364 views

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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1 vote
1 answer
246 views

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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1 vote
1 answer
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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 ...
0 votes
0 answers
53 views

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 ...
2 votes
1 answer
138 views

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 ...
1 vote
2 answers
502 views

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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1 vote
1 answer
276 views

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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3 votes
2 answers
134 views

Python versus Matlab on the quantity 1/0

Python and Matlab seem to disagree on the division by 0. Python: ...
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3 answers
2k views

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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1 vote
2 answers
193 views

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?
1 vote
2 answers
148 views

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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2 votes
2 answers
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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 $...
0 votes
0 answers
76 views

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 ...
1 vote
1 answer
346 views

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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1 vote
0 answers
42 views

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 ...
1 vote
3 answers
129 views

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 ...
3 votes
1 answer
323 views

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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0 answers
30 views

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 ...
2 votes
1 answer
5k views

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 $\beta$ --------------------...
0 votes
0 answers
39 views

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 ...
-1 votes
3 answers
458 views

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: ...
1 vote
1 answer
791 views

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 ...
1 vote
2 answers
45 views

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)}{...
1 vote
2 answers
69 views

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 ...
0 votes
0 answers
43 views

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 ...
0 votes
2 answers
172 views

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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1 vote
1 answer
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Help comparing relative error for different parenthesizations of addition

I am given two functions: $ fl(fl(x+y)+z) $ and $ fl(x+fl(y+z)) $ and asked to derive their relative error. Then, given a set of conditions: a) $ x < y < x $ b) $ x > 0, y < 0, z > 0 $...
0 votes
0 answers
140 views

Floating point binary number to a 7 segment decimal display

I have covered floating point (32 bit) conversion from float to decimal and decimal to float. I am happy with the theory and I have created a conversion tool in Excel VBA which works just fine ...
1 vote
1 answer
168 views

What is the 2's complement answer of 16.5?

According to this post it is saying Two's complement is only for integers, but in Wolframalpha is is saying the Two's complement of 16.5 is 0010000.1, how?

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