Questions tagged [floating-point]
Approximate representation of numbers as a fixed number of digits multiplied by a logarithmic scale.
203
questions
0
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0answers
6 views
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 ...
-1
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0answers
31 views
Designing a format for floating point numbers (IEEE 754)
I have tried to figure out how to solve the following question for a while now, but cannot seem to find anything on the internet or anywhere else. Everything i can find only seems to be about ...
0
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0answers
23 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 ...
-2
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0answers
35 views
Biasing in floating point numbers
Here is my question Q- Assume a 4 bit exponent in a floating point number has a bias of 8 .how will you represent the following exponent values (binary) (also indicate if a value cannot be represented)...
1
vote
1answer
31 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 ...
1
vote
0answers
29 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 ...
0
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2answers
63 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 ...
2
votes
1answer
37 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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0answers
10 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 ...
1
vote
1answer
64 views
How to represent zero as floating point number?
For the floating-point number, we have the form
$\pm d_0.d_1d_2Ā·Ā·Ā·d_{P-1}\times\beta^E$
$\pm$ --------------------------- sign
$d_0.d_1d_2Ā·Ā·Ā·d_{p-1}$ --------- significant
$\beta$ --------------------...
0
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0answers
37 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 ...
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2answers
66 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
1answer
73 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
2answers
42 views
Floating-point oblivious way to compute multiset numbers
I have to compute $R = \left(\!\!{n + 1\choose k}\!\!\right)$, which happens to be:
$$
R = \left(\!\!{n+1\choose k }\!\!\right) = \binom{n+k}{k} = \frac{(n + k)!}{n!k!} = \frac{(n+1)(n+2)\cdots(n+k)}{...
1
vote
2answers
40 views
Validity of Algorithm for Testing Two Floating Point Numbers
This question is related to the epsilon- (or delta- if you prefer) test for floating point equality. But my question is not how to do it. Instead I have a related algorithm for testing equality, and I ...
0
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0answers
35 views
New way of representing floating point
I want to create a new way of representing floating point. In standard IEEE floating point, we have 1 bit to represent sign, 8 bits to represent exponential and 23 bits to represent significand. In ...
0
votes
1answer
54 views
IEEE 754 addition wrong result floating point numbers
I want to add two IEEE 754 numbers.
I followed the steps to add two 754 numbers. However the result it not correct.
Number 1:
S:0
E:01111111
M:11111111111111111111111
Number 2:
S:0
E:01111111
M:...
0
votes
0answers
42 views
Is $x$ in the working range [3,2]?
Consider $x$ = $(0.1001)_b$ and $F$[3,2]. ($F$[3,2] is the set of all floating point numbers with 3 digits in the mantissa and 2 digits in the exponent.)
The question is $x$ in the working range [3,2]?...
0
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2answers
51 views
Cancellation of inequalities in floating point arithmetic
In finite precision floating point arithmetic the associative property of addition is not satisfied. This is, it is not always the case that $$(a+b)+c=a+(b+c)$$
Even $a=(a+b)-b$ is not always true.
To ...
0
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0answers
15 views
Why do many floating point numbers appear compact in base16 and base32 representation? (IEEE 754)
Consider 3 randomly chosen floating point values and their base conversions:
...
0
votes
1answer
26 views
How does value decompression work for Facebook's Gorilla in the case where count of leading zeroes is not stored
I am referring to this paper: http://www.vldb.org/pvldb/vol8/p1816-teller.pdf
My question is regarding section 4.1.2 where it says:
When XOR is non-zero, calculate the number of leading and ...
1
vote
1answer
16 views
Help comparing relative error for different parenthesizations of addition
I am given two functions:
$ fl(fl(x+y)+z) $ and $ fl(x+fl(y+z)) $ and asked to derive their relative error. Then, given a set of conditions:
a) $ x < y < x $
b) $ x > 0, y < 0, z > 0 $...
0
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0answers
53 views
Floating point binary number to a 7 segment decimal display
I have covered floating point (32 bit) conversion from float to decimal and decimal to float. I am happy with the theory and I have created a conversion tool in Excel VBA which works just fine ...
1
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1answer
74 views
What is the 2's complement answer of 16.5?
According to this post it is saying Two's complement is only for integers, but in Wolframalpha is is saying the Two's complement of 16.5 is 0010000.1, how?
5
votes
5answers
309 views
fast and stable x * tanh(log1pexp(x)) computation
$$f(x) = x \tanh(\log(1 + e^x))$$
The function (mish activation) can be easily implemented using a stable log1pexp without any significant loss of precision. Unfortunately, this is computationally ...
0
votes
1answer
120 views
Converting a number to 16-bit Floating Point Format
I want to convert the number -29.375 to IEEE 745 16-bit floating point format. Here is my solution:
The format of the floating point number is:
1 sign bit
unbiased exponent in 4 bits plus a ...
2
votes
1answer
54 views
Numerical Accuracy & Sorting Algorithms?
Sedgewick and Wayne talk about how sorting algorithms and, specifically, priority queues are used in devising ways to improve accuracy in floating-point calculations: https://algs4.cs.princeton.edu/...
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2answers
703 views
Write the smallest positive number that can be represented by the floating point system
Using a normalised floating point representation box with an 8-bit mantissa and a 4-bit exponent, both stored using twoās complement.
(a) Write the smallest positive number that can be represented by ...
1
vote
1answer
46 views
Is the exponent bias $2^{n-1}-1$ or $2^{n-1}$
I'm a bit confused with the exponent bias. The sources I found online claim that it is either $2^{n-1}-1$ or $2^{n-1}$, $n$ is the number of bits used for the exponent. In my book when given examples ...
0
votes
1answer
51 views
Understanding How Double Precision Numbers are Stored in a Computer
I am reading Numerical Analysis by Walter Gautschi. I am somewhat confused by the following quote from page $5$:
To increase the precision, one can use two machine registers to represent a machine ...
2
votes
1answer
42 views
Question About Floating Point System
I have begun reading Numerical Analysis by Walter Gautschi. On page $3$, the author introduces the floating point number system as follows: a floating point number is a number representible as
$$
\...
0
votes
2answers
585 views
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 ...
2
votes
1answer
63 views
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. ...
0
votes
1answer
160 views
Normalization in IBM hexadecimal floating point
According to the Wikipedia link of IBM hexadecimal floating point:
Consider encoding the value ā118.625 as an IBM single-precision
floating-point value.
The value is negative, so the sign ...
1
vote
1answer
71 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 ...
2
votes
1answer
187 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 ...
1
vote
0answers
431 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 ...
1
vote
0answers
38 views
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 ...
1
vote
1answer
30 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 ...
0
votes
2answers
201 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 $\epsilon_m$ ...
1
vote
1answer
37 views
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)/...
2
votes
1answer
51 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^...
2
votes
2answers
83 views
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 ...
1
vote
4answers
76 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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0answers
54 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 ...
0
votes
0answers
29 views
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 ...
2
votes
1answer
76 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 ...
1
vote
0answers
29 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 ...
0
votes
1answer
62 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. ...
3
votes
2answers
114 views
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, ...
