Questions tagged [floating-point]

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

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Adding two numbers in half-precision IEEE754 standart

I have to add two numbers using IEEE754: -14.1875 and -7.4375. I managed to convert them to half-precision numbers in IEEE754: 1 10010 1100011000 (-14.1875) ...
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83 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 ...
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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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32 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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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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138 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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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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Finding fraction from a double [duplicate]

You are given the decimal expansion (double) of a rational number $p/q$, where $\gcd(p,q)=1$. How can you efficiently determine what $p$ and $q$ are? I don't know how to solve this, but I think we ...
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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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62 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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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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539 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 $\beta$ --------------------...
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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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1answer
224 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 ...
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44 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)}{...
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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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84 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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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 $...
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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 ...
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103 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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fast and stable x * tanh(log1pexp(x)) computation

$$f(x) = x \tanh(\log(1 + e^x))$$ The function (mish activation) can be easily implemented using a stable log1pexp without any significant loss of precision. Unfortunately, this is computationally ...
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323 views

Converting a number to 16-bit Floating Point Format

I want to convert the number -29.375 to IEEE 745 16-bit floating point format. Here is my solution: The format of the floating point number is: 1 sign bit unbiased exponent in 4 bits plus a ...
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Numerical Accuracy & Sorting Algorithms?

Sedgewick and Wayne talk about how sorting algorithms and, specifically, priority queues are used in devising ways to improve accuracy in floating-point calculations: https://algs4.cs.princeton.edu/...
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Write the smallest positive number that can be represented by the floating point system

Using a normalised floating point representation box with an 8-bit mantissa and a 4-bit exponent, both stored using two’s complement. (a) Write the smallest positive number that can be represented by ...
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61 views

Is the exponent bias $2^{n-1}-1$ or $2^{n-1}$

I'm a bit confused with the exponent bias. The sources I found online claim that it is either $2^{n-1}-1$ or $2^{n-1}$, $n$ is the number of bits used for the exponent. In my book when given examples ...
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Understanding How Double Precision Numbers are Stored in a Computer

I am reading Numerical Analysis by Walter Gautschi. I am somewhat confused by the following quote from page $5$: To increase the precision, one can use two machine registers to represent a machine ...
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Question About Floating Point System

I have begun reading Numerical Analysis by Walter Gautschi. On page $3$, the author introduces the floating point number system as follows: a floating point number is a number representible as $$ \...
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Why is there a precision loss for floating-point numbers?

It could be a silly question, yet I'm not able to understand. In modern day, computers, we have integers with million digits length. Even in my ordinary 2GB laptop, I can calculate values of numbers ...
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Max flow algorithm for floating-point weights and E~=10*V

Could you, please, suggest a maximum flow algorithm for a graph with floating-point weights and the number of edges approximately equal to the number of vertices? I.e. ...
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222 views

Normalization in IBM hexadecimal floating point

According to the Wikipedia link of IBM hexadecimal floating point: Consider encoding the value −118.625 as an IBM single-precision floating-point value. The value is negative, so the sign ...
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122 views

Convert scientific notation decimal number to binary

I'm given the number: $$8.881784197001252 \cdot 10^{-16}$$ I know it is $2^{-50}$ but suppose I didn't know. I need to convert it to binary. One way is to apply the school taught algorithm on the ...
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293 views

Do all programming languages adhere to the IEEE 754-2008 standard?

I am well aware that the float data type's maximum value is approximately 1.8E308, which means that across the various ...
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669 views

What are the smallest and biggest negative floating point numbers in IEEE 754 32 bit?

I am stuck with a question that asks for smallest and biggest negative floating point numbers in IEEE 754 32-bit (their representation and decimal numerical value from which one can approximate the ...
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How can I calculate the exponential integral?

(I'm not sure this is the right forum.) I'm writing a program that uses the prime-counting function. Right now, I'm using x/log(x), but I want to switch to ...
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1answer
31 views

What's the least signifcant bit of a mantissa system?

If Mantissa is a 1-dot-M fixed-point number whose most significant bit is always 1 then, how is the least significant bit calculated? I know the least and most significant bit of the mantissa ...
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262 views

Question about machine epsilon

I am studying over my notes, and there is something I don't understand about $e_m$. We represent the floating point numbers as $1.d_1d_2...d_t \times \beta^e$. Now, my professor defines $\epsilon_m$ ...
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Floating point arithmetic on division

I am trying to figure out how $(x/y)$ in floating point arithmetic $fl(fl(x) / fl(y))$ where $fl(x) = x(1-\delta_1)$, $fl(y) = y(1-\delta_2)$, $fl = (1-\delta_3)$ I have: $= x/y \cdot ((1-\delta_1)/...
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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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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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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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