Questions tagged [floating-point]

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

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2answers
33 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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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 ...
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1answer
35 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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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 ...
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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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41 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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1answer
110 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
101 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)) ...
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1answer
155 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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1answer
36 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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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 ...
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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 ...
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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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27 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 ...
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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. ...
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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, ...
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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
48 views

numerically stable log1pexp calculation

What are good approximations for computing log1pexp for single precision and double precision floating point numbers? Note: ...
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1answer
67 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 ...
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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)...
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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 ...
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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 ...
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3answers
597 views

What is the 1's and 2's complement of 0.01101?

What is the 1's and 2's complement of 0.01101? I'm unable to find any details on this from google. Basically how do we represent the floating points in 1's and 2's complement forms? Even wikipedia ...
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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 ...
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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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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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2answers
155 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 \...
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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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1answer
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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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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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5answers
843 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 $n-1$. ...
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1answer
429 views

Simple algorithm for IEEE-754 division on 8-bit CPU?

IEEE Std 754-2008 is the modern definition of Floating-Point Arithmetic. It requires that division (among other operations) performs as if it first produced an intermediate result correct to ...
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2answers
66 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 ...
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1answer
29 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?
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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
124 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
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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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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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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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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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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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 ...
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2answers
104 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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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
73 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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2answers
60 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
47 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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1answer
994 views

half precision floating point multiplication

A = 0 10011 0011110111 B = 1 00011 0010011000 exponent is 15, mantissa is 10 bits and first bit is implicit. Can somebody please tell me the final answer cause I am having trouble figuring ...
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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 ...