# Questions tagged [floating-point]

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

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### The floating point function of Chopping. Absolute error and Relative error.

Consider a number $(x)_\beta$ : $$x = \pm 0.d_1d_2 \ldots d_pd_{p+1}d_{p+2} \ldots \times \beta^E$$ The function $chop(x)$ considers only the first $p$ digits ignoring digits from $(p+1)$th to ...
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### How to deal with underflow issues in high-dimensional entropy calculation?

I was not sure if the question makes sense here or should better be placed in a computational/CS forum, but I hope you can give me some insights. I am working in image processing and use the ...
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### Error Propagation in Floating-Point Multiplication

Wikipedia (Machine epsilon) tells me that the result of a multiplication between 2 floating-point numbers, with a rounding induced relative error ϵ, still only has the relative error ϵ. Why do the ...
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### Numerical operations when numbers are very large?

Explain the best way to evaluate $f(x,y) = \sqrt{(x^2 + y^2)}$ numerically when $x$ or $y$ are very large. Does anyone have any insight to this? I'm lost. I usually know how to deal with these types ...
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### Random sum of Random variable with floating numbers

Let $L$ and $S$ integer valued random variables $(0,1,2, \ldots).$ $R$ is another random variable which can be a floating numbers. The RV $S$ is the least required number of $R$ to be equal or ...
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### Why is 1.4 - 1.3 == 0.9999+ but 0.4 - 0.3 == 1.000000003

I'm not sure if this is a maths question or a programming question or a how-does-your-computer-work question. Sorry about that. I remember from university that 0.999999 ... == 1 since 1 - 0.999999 ......
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### Square roots by Newton’s method

The following Python program implements Newton’s method for computing the square root of a number: ...
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### Solution for matrix equation with gaussian + column pivoting

\begin{array}{ccc|c} 7.000E+0 & 1.000E+0 & 1.000E+0 & 1.000E+1\\ 1.000E+1 & 1.000E+0 & 1.000E+0 & 1.300E+1\\ 1.000E+3 & 0.000E+0 & 1.000E+0 & 1.001E+3\\\...
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### Max Mantissa $2^{bits}-1$

if we look at a $5$ bit mantissa, the max value will be $11111$ which is $2^5-1$, Why is it in the form of $2^{bits}-1$ is it a combinatorial explanation?
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### Floating Point Arithmetic dealing with a Taylor expansion for e^-x

Suppose we want to compute $e^{-a}$ for $a>>1$. Which of the following techniques should I use? (a) Taylor expansion for $e^{-x}$ about $x=0$ or (b) Taylor expansion for $e^x$ about $x=0$, ...
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### Floating point operation/arithmetic

I read my lecture script and I have a question regarding this statement. "The approximation using floating point numbers with fixed mantissa length is useful, because the cost of each floating point ...
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### Rounding in floating point operations

I'm given this set of floating point numbers: And I'm given the function: I'm then asked to find the value of f for: So I've done this: Now my guess is that I need to convert the exact value (1....
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### Floating point and LU decomposition

So in a numerical linear algebra book I'm studying (Matrix Computations 4th edition) there is this example on floating points and LU decomposition: Suppose 3-digit floating point arithmetic is used ...
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### Why is $0.09 + 0.01$ not exactly representable as $0.1$ in floating-point systems but $0.085 + 0.015$ is?

I understand that $0.09 + 0.01$ in floating-point systems cannot be exactly represented as $0.1$ because the binary equivalent of $0.1$ does not exist—it has infinitely repeating bits. But why can a ...
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### Absolute error bound for floating-point number?

For a input real number $x$ with some interval $[-a, b)$, which is not performed by any computation for now, how can we bound the absolute error of it for IEEE-754 binary64? Most of the articles ...
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### How many discrete points can be expressed in $[-1/2, 1/2)$with IEEE 754 double floats and what is the meaning of precision?

I know that IEEE754 double floats (64-bit floating number) is known to provide 52 bits of precision (or 53 bits including implicit 1). But I do not know the exact meaning of the precision. Suppose we ...