# Questions tagged [floating-point]

A method of representing numbers by a fixed number of significant digits, and the exponent of some base number. They are characterized in the form ${(significant digits)}*base^{exponent}$. Typically, numbers are represented with respect to base = 2 (binary).

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### Numerical algorithms made stable by unums which are unstable on IEEE floats

For unums, there is good evidence (see figure 5) that accuracy is better than IEEE floats. (Note: I use the term "unum" broadly to refer to any of the various iterations and revisions to the ...
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### float in C: Size of the exponent

This question is very related to the question Number of decimal of float and double in C. In the second table we may see that the exponent of the float are from -38 to +38. But the IEEE754 standard ...
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### Accuracy loss in single-precision Euclidean norm computation

I do hydrodynamics simulations with Fortran and recently I met with this issue: I have a single-precision array b of length ...
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### GPU vs CPU FLOP counts

I apologise if this is somewhat of a rookie question. So, from my understanding, on a GPU board, far more of the space is allocated to ALUs compared to CPUs which have far more cache available. This ...
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### Polynomial approximation for floating-point arithmetic

I cannot remember where I picked this up, but during my time reading about polynomial approximation for floating-point arithmetic of sin(x), I vaguely remember that ...
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### What are some good strategies to test a floating point arithmetic implementation for double numbers?

For IEEE, the single representation is 1-bit sign, 8-bit exponent and 23-bit mantissa. This means that at each exponent value, you can test all 2^23-1 (roughly 9mil cases) possible combination of ...
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### How to interpret if $\displaystyle \sum_{j = 0}^{n} \frac {1}{j!}$ is a stable algorithm for computing $e$?

I am trying to solve problem $15.1$ from Numerical Linear Algebra by Trefethen and Bau, which reads Determine whether the algorithm is backward stable, stable but not backward stable, or unstable. ...
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### Is Highams' computation of mean worth the price?

In Accuracy and Stability of Numerical Algorithms, equation 1.6a, Higham gives the following update formula for the mean: $$M_{1} := x_1, \quad M_{k+1} := M_{k} + \frac{x_k - M_k}{k}$$ Ok, one ...
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### SLATEC Routine Computes Givens Rotation in Unexpected Way

Some Background I am working on a C++ translation of a SLATEC routine, R1UPDT, which performs a Givens rotation: $$r = \frac{1.0}{\sqrt{a^2 + b^2}}$$ Usually, ...
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### Why is the method of im2col with GEMM is more efficient than the method of direction implementation with SIMD in CNN

The convolutional layers are most computationally intense parts of Convolutional neural networks (CNNs).Currently the common approach to impement convolutional layers is to expand the image into a ...
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### Fast and Numerically Stable Pairwise Distance Algorithms

I'm looking for resources on fast, numerically stable pairwise euclidean distance algorithms. In particular, suppose $A \in \mathbb{R}^{M \times D}$ and $B \in \mathbb{R}^{N \times D}$ are two sets of ...