# Tagged Questions

Mathematical questions concerning floating point numbers, a finite approximation of the real numbers used in computing.

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Suppose a machine with the floating-point system $\beta = 10$, $p = 8$, $L = -50$, and $U = 50$ is used to calculate the roots of a quadratic equation $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are ...
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### Checking if the Hessian is the derivative of the gradient

Suppose $f: \Bbb R^n \to \Bbb R$. I have a code that computes the gradient of $f$. I have another code that computes the Hessian of $f$ times a vector. Now I want to check if they are correct. ...
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### The arctangent is a strange floating point number

I have 2 players in a game (Cod 4). I read X, Y, Z and store them in: ...
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### error bound in function approximation algorithm

Suppose we have the set of floating point number with "m" bit mantissa and "e" bits for exponent. Suppose more over we want to approximate a function "f". From the theory we know that usually a "...
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### Why is this number the smallest positive normalised binary value?

In the AQA A2 Computing textbook (Bond and Langfield, 2009), they say that this number is the smallest positive normalised value, given a 10 bit mantissa and a 6 bit exponent: ...
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### Minimum error in floating point approximation of an elementary function.

I need a confirmation of a thing that probably is silly. Let $x$ a floating point number representable using $e$ bits for exponent and $m$ bits for mantissa, let $f$ a be an elementary function, you ...
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### Floating point rounding

Having trouble proving these two statements are true if we assume no overflow occurs and all rounding modes(round down, round up, round to zero, round to nearest) are valid. 1) If x is non zero ...
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### Stability of (floating point) computed variance

Homework Question from Accuracy and Stability of Numerical Algorithms, 2nd Edition, by Nicholas J. Higham, page 33: So every time we store an number and do a operation, we introduce an error bounded ...
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### standard notation to handle representation of a real number on a computer

Is there a standard notation to handle the effective representation of some real number $x$ on a finite machine ? I have in mind some kind of braces, but I am not sure it is appropriate. Let me try to ...
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### Floating point number,Mantissa,Exponent

In this computer, numbers are stored in 12-bits. We will also assume that for a floating point (real) number, 6 bits of these bits are reserved for the mantissa (or significand) with 2^(k-1)-1 as the ...
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### How would you convert the following 32-bit IEEE floating-point to decimal form?

I have got -1.101 1100 1011 1010 1001 1000 * 2^(9) How can I convert this to decimal form?
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### Convergence and limit of Muller sequence

The Muller sequence is given by the recursive definition: $U(n+1)=111-\frac{1130}{U(n)}+\frac{3000}{U(n)U(n-1)}$ with $U(0)=5.5$ and $U(1)=\frac{61}{11}$. This sequence is interesting in floating-...
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### Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
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### Floating Point Number System

I really have no idea of how to do these questions - in fact I have no idea of how to do any question in the paper - but I have tried to figure out what's going on in the course called Computational ...
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### How do we tell arithemetic addition or subtraction from floating point numbers?

Source: C++ for Engineers and Scientists, Gary J. Bronson Source: Programmable Logic Controllers: The Complete Guide to the Technology by Clarence T. Jones In the table $12345.67_{10}$ = 1.234567+...
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### If $1/2 ≤ p/q ≤ 2$ , then $p-q$ is representable exactly on the computer

I've found the following affirmation in an article. I've been thinking about it but I don't know the way to prove it: It is not hard to prove that if $p$ and $q$ are two of a computer’s floating–...
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### How to bound the input parameters to a chaotic function to obtain exact result in a finite precision setting?

While I was reading the paper entitled (http://dx.doi.org/10.1109/ISCAS.2003.1204947) Kocarev, Ljupco, and Zarko Tasev. "Public-key encryption based on Chebyshev maps." Circuits and Systems, ...
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### Rounding up a binary number

I just converted the fraction $\frac{3}{5}$ to the following floating point binary number: $(1.001100110011001100110011\cdots)_{2}\times 2^{-1}$. Now, I am trying to find the two nearest machine ...
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### smallest poisitive integer does not belong in floating point system F

A floating-point number system F is a subset of real numbers whose elements have the form: $x=\pm(m/\beta^t)\beta^e$ base (beta), precision (t), and exponent range (emin and emax). The mantissa m ...
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### Machine Floating Point Theorem

Completely stuck on this floating point question. Let $x \in \mathbb{R}$ have the following floating point representation: $$x = (-1)^s[0.a_1a_2\dots a_ta_{t+1}\dots]\cdot \beta^e$$ [Where $\beta$ ...
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### Displaying $2 \cdot 10^5$ as $200000$

How can I in Maple write out the number $2 \cdot 10^5$, so it displays the full number with all digits: $200000$? By typing increasing zeros I can test Maples treshold, and I get: $20$, $200$, $2000$,...
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### Convert double precision number to rational fraction plus exponent

I have a double precision quantity (either pixels per cm or pixels per inch) that gets converted into pixels per meter. I then need to convert this number into a rational fraction, with numerator and ...
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### How do I round this binary number to the nearest even

I have this binary representation of 0.1: 0.00011001100110011001100110011001100110011001100110011001100110 I need to round it ...
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### Automatic curve fitting to find order of an algorithm?

I'm a newbies in mathematics. I'm looking for an automatic best curve fitting function to find the order of an algorithm. I would like to know if it does exists a math library function that would ...
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### Whether assigning of single precision IEEE754 float to double is reversible?

Within scope of IEEE754 standard let's assign single precision variable s to double precision variable d and then assign d to single precision variable s'. Whether this operation is reversible(...
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### What should be the precision to uniquely recover the binary double float number from the decimal one?

Assume we store binary float number in IEEE 754 format to decimal format. Then we recover decimal to binary back. With regard to single floats - according to this article When a binary IEEE single ...
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### Is there any difference between fixed point and decimal point?

Source: Introduction to Computers' 1999 Ed.1999 Edition Fixed point number 774.3675 is just a decimal number with a decimal point to show a fractional part 3675/10000. I see no difference in the fixed ...
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### Given a base, a mantissa size and a range for the exponent, what's the largest number?

I was introduced to floating-point numbers week ago or so, and although I think I understood the basics, I'm still not sure how to apply them. A floating-point number is usually represented with the ...
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### Simple computations on decimal floating point arithmetic

Can anyone please check my working on the computations of $2*0.3-0.6$ and $3*0.3-0.9$ on decimal floating point? My working: $2*0.3-0.6=(2.0*10^0*3.0*10^{-1})-6.0*10^{-1}=6.0*10^{-1}-6.0*10^{-1}=0$ ...
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### What are 2*0.3-0.6 and 3*0.3-0.9 in decimal floating point numbers?

Investigate the values $a=2*0.3-0.6$ and $b=3*0.3-0.9$. What do we expect to get on a computer using decimal floating point numbers? My reason and doubt: It doesn't really mention how many degree ...
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### Floating point binary to half precision floating point

I am trying to convert to $16$ bit half precision floating point however I ran into a possible error and am unsure if a negative exponent is ok. I am trying to convert $0010011100010000$ I separate ...
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### What line-polygon clipping algorithm can I use to ensure that the resultant endpoints are always within the polygon?

I have a 2D plane, partitioned into n-sided, convex polygons. I'm using WRF's PNPOLY algorithm for polygon inclusion to ensure that a point belongs inside one and only one polygon. Is there an ...
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### floating point subtraction for binary numbers

Consider that I want to do a binary operation on the following floating point numbers: 0.35-0.62 I can reach the end but I can not figure out how the sign bit is ...
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### Fixed point exponent faithfull rounding? does it make any sense?

i've a stupid question that just come up in my mind. Let's say we have a floating point number $x = (-1)^{s_x} 2^{e_x} 1.m_x$, let's say for some reason we need to divide the exponent by $2$, maybe ...
### Longest string whose $\log$ probability can be represented
The problem starts out with a binary string of length $n$, so the probability of any random string in the set is $\cfrac{1}{2^n}$. The smallest positive floating point value that the system can ...