Questions tagged [floating-point]

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

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213 views

Are IEEE floating point numbers intervals or point values?

The context is IEEE 754-2008 floating point number systems. The systems defined by the standard comprise, as far as I understand it, a bit-level representation and a set of guarantees on the precision ...
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1answer
1k views

How to work out if an IEEE 754 floating point number is normalized?

How do I tell whether a particular IEEE 754 floating point number is a normalized floating point number? Is there some way to recognize an IEEE 754 normalized floating point number?
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2answers
277 views

Why is there more frequent overflow in normalised Floating Point

I read that overflow is more frequent when we work with normalised mantissas. Why is this? Is it because when we adopt a normalised representation, our range is smaller than in a unnormalised ...
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1answer
752 views

What is 0.1 converted to 8bit IEEE754?

0.1 via $32\text{ bit}$ is rather easy: Sign: $0_2 = 0_{10}$ Exponent: $123_{10} = 01111011_2$ Mantissa: $5033165_{10} = 100110011001100110011001101_2$ Now, how do you calculate this, if you've ...
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775 views

Implicit Leading 1 in Binary Floating Point

Is the convention of dropping the leading 1 when storing the significand a given in all binary floating point representations or not necessarily?
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1answer
36 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^...
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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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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 ...
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2answers
105 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
46 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
689 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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3answers
548 views

Can a unnormalized floating point number be recognized also when exponent is not zero?

From Tanebaum's Structed Computer Organization. Exercise 4 of Appendix B The following binary floating-point number consist of a sign bit, an excess $64$, radix $2$ exponent, and a $16$-bit ...
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74 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 ...
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28 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 ...
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32 views

numeric stability of map reduce operations

I am building a small library for computing information retrieval metrics for classifiers (precision, recall, f1, accuracy, whatever). Typically each metric is built by calculating a single value for ...
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0answers
212 views

Why mantissa and exponent are stored differently in a float?

As we know, in IEEE 754 standard, float number's exponent and mantissa are stored differently. While the exponent is stored as an unsigned number, taking advantage of the bias, the mantissa is in sign-...
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59 views

Normalised Floating Point System

I have a floating point number system and I have a number for which I need to calculate the exact relative error after rounding. The number is clearly an overflow. Does anyone know what I should do? ...
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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 ...
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3answers
818 views

Comparing floating-point numbers as integers

In the article http://floating-point-gui.de/errors/comparison/, a method for comparing floating-point numbers is suggested: There is an alternative to heaping conceptual complexity onto such an ...
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1answer
334 views

Why doesn't the binary fraction representation match the decimal fraction representation?

The Problem: What value does the hexadecimal number x55544552 represent in data type IEEE floating point? My Work:     I first wrote out that hexadecimal number in binary and got ...
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2answers
318 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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1answer
103 views

What factors influence machine epsilon?

I was wondering what factors effect a machine's epsilon value. I was thinking about how modern computers can calculate expressions to higher accuracy then their predecessors, but what hardware and ...
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4answers
64 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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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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1answer
95 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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1answer
44 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
114 views

What are some of the practical applications of functions that extract the exponent and mantissa of a floating point number?

I'm talking about functions such as Python's math.frexp() : math.frexp(x) Return the mantissa and exponent of x as the pair (m, e). m is a float and e is an integer such that x == m * 2**e ...
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1answer
45 views

Why do floating point additions sometimes produce equal results? [duplicate]

I have tried two sentences on my computer : 2 + 10^(-18) == 2 2^(-55) + 2^(-57) == 2^(-55) My computer gives TRUE and FALSE respectively. Why does the computer ...
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1answer
4k views

8-bit floating-point representation

I'm studying about representing fractional numbers as floating-point values. It is going to be an 8-bit representation. Somewhere in the text, it is said that: "We use the first bit to represent ...
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1answer
5k views

Interpretation of '1/3' in IEEE floating point representation

For a rational number 1/3 below is the floating point representation(64 bit) of decimal expansion 0.3333333.... As per the above bit structure, I would like to ...
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1answer
649 views

Machine epsilon vs least positive number

What is the difference between machine epsilon and least positive number in floating point representation? If I try to show the floating point number on a number line .Is the gap between exact 0 and ...
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1answer
78 views

Floating point operations; Exception, Flags, and Trap Handlers

I am reading over the article found here specifically §D.4.4 Exceptions, Flags and Trap Handlers and am confused by the table D-4 in that section. Specifically the arguments sent to the trap handler ...
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1answer
127 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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2answers
53 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
75 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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1answer
556 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, ...
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4answers
112 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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1answer
40 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
61 views

Compress 32 bit integer into a normalized float

and sorry if this is a duplicate of some sort, my poor technical vocabulary doesn't get me very far with google. I am trying to implement some well-known PRNG algorithms. For the time, the algorithm ...
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2answers
1k views

What are the reasons for and against using normalized floating point?

Floating point number formats can be normalized or not, meaning that ‘normal’ floating point numbers have an implicit (hidden) leading bit 1 in the significand. For example, the binary IEEE 754 ...
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1answer
369 views

Convert 24(decimal) to modified IEEE 754 floating point format?

Here's what I have as the tweaked format I'm to use: S EEE MMMM (excess 8 format) s = sign bit, E = exponent bits, M = mantissa/fraction bits Otherwise, it follows the IEEE 754 in principle. The ...
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2answers
58 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
48 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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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 ...
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2answers
52 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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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 ...
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1answer
1k 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 ...
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1answer
56 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 whether an efficient algorithm exists to determine the ...
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1answer
163 views

how to convert any fraction to decimal

So for a project I am working on. I need to take a string representation of a floating point number, lets say "1.5" and do arithmetic on it with another floating point number. The catch is I cannot ...
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1answer
411 views

Neural Networks: Simulate the working of Dynamic Fixed Point representation of the weights on hardware

I am looking to implement a neural network on hardware using Verilog. I have completed and tested with floating point representation and a 20 bit fixed point representation. I want to further reduce ...