Questions tagged [floating-point]
Approximate representation of numbers as a fixed number of digits multiplied by a logarithmic scale.
183
questions
1
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1answer
9 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
votes
0answers
20 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
1answer
19 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?
3
votes
4answers
139 views
fast and stable x * tanh(log1pexp(x)) computation
$$f(x) = x \tanh(\log(1 + e^x))$$
The function can be easily implemented using a stable log1pexp without any significant loss of precision. Unfortunately, this is computationally heavy.
Is it ...
0
votes
1answer
44 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
44 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/...
-1
votes
2answers
135 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 two’s complement.
(a) Write the smallest positive number that can be represented by ...
1
vote
1answer
37 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
39 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
29 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
$$
\...
0
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2answers
147 views
Why is there a precision loss for floating-point numbers?
It could be a silly question, yet I'm not able to understand.
In modern day, computers, we have integers with million digits length. Even in my ordinary 2GB laptop, I can calculate values of numbers ...
2
votes
1answer
44 views
Max flow algorithm for floating-point weights and E~=10*V
Could you, please, suggest a maximum flow algorithm for a graph with floating-point weights and the number of edges approximately equal to the number of vertices? I.e. ...
0
votes
1answer
58 views
Normalization in IBM hexadecimal floating point
According to the Wikipedia link of IBM hexadecimal floating point:
Consider encoding the value −118.625 as an IBM single-precision
floating-point value.
The value is negative, so the sign ...
1
vote
1answer
42 views
Convert scientific notation decimal number to binary
I'm given the number:
$$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
67 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
169 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
29 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
24 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
87 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
35 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)/...
0
votes
0answers
12 views
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 ...
2
votes
1answer
45 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
42 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
71 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
52 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
27 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
71 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
61 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
104 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, ...
2
votes
1answer
115 views
numerically stable log1pexp calculation
What are good approximations for computing log1pexp for single precision and double precision floating point numbers?
Note: ...
2
votes
1answer
97 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 ...
0
votes
2answers
191 views
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 ...
3
votes
3answers
66 views
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)...
0
votes
0answers
69 views
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 ...
1
vote
0answers
16 views
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 ...
0
votes
2answers
138 views
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 ...
2
votes
0answers
231 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 ...
4
votes
2answers
413 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 \...
2
votes
1answer
68 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 ...
0
votes
1answer
95 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 ...
-2
votes
1answer
51 views
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.
7
votes
5answers
1k 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$. ...
0
votes
2answers
375 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}
$$
This can be implemented as
...
0
votes
1answer
48 views
How does IEEE 754 decimal encoding work? [closed]
This may be a silly question, but if a computer works in binary how can you encript numbers using decimal?
5
votes
0answers
156 views
What is the state of the algorithmic art for floating point arithmetic on complex numbers?
Most modern compilers and processors implement the IEEE 754 binary formats for floating point numbers. IEEE 754 guarantees that the addition, subtraction, multiplication, division, and square root ...
1
vote
1answer
360 views
Binary Floating Point Range/Precision for 4-bit Mantissa and 4-bit Exponent
I'm trying to understand binary floating point and using just a 4-bit mantissa and a 4-bit exponent (both 2s compliment) to keep things simple.
As far as I can tell, the largest denary number I can ...
0
votes
2answers
52 views
Floating point precision from rearranging equation
When I run $x^2 - y^2$ with x=8.8888888888 and y=9.9999999999 in python, I get the following result:
...
1
vote
1answer
44 views
Why do I get different results from two calculation methods?
I am wondering what the reason for the following is
We know that ,
exponential has a taylor representation :
$$exp(x)=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...$$
Using the first n terms , in R , ...
1
vote
2answers
63 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 ...
