All Questions
Tagged with floating-point computer-arithmetic
17
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Proof that $\epsilon_{mach} \leq \frac{1}{2} b^{1-n}$
I have a question about the proof of the following statement:
For each set of machine numbers $F(b, n, E_{min}, E_{max})$ with $E_{min} < E_{max}$ the following inequality holds: $\epsilon_{mach} \...
4
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1
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151
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1
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1
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168
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On the axioms of floating-point arithmetic
As I understand there are two "axioms" that should be satisfied in floating-point arithmetic:
$$\forall x\in \mathbb R,\ \exists |\varepsilon|\leq\varepsilon_{\text{machine}},\ \mbox{fl} (x) ...
- 1,347
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2
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164
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Evaluating $a(b + c)$ more accurately with FMA
I'm using machine-precision floating-point arithmetic, and every so often it happens that I need to evaluate an expression of the form $a(b + c)$. I found that the accuracy can be improved using FMA (...
- 249
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82
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Numerically stable evaluation of factored univariate real polynomial
Suppose we have a real univariate factored polynomial, meaning we have its factors: an arbitrary number of polynomials of degree less than or equal to two. To simplify things, if necessary, let's ...
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126
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The gap size of floating point
In Trefethen Bau Numerical Algebra, floating point set F is defined by
$\textbf{F} = \left\{\pm(m/\beta^{t})\beta ^{e}| 1 \leq m \leq \beta^{t}, e \in \mathbb{Z} \right\}$
Equivalently, by making $m$ ...
- 23
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82
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Accuracy in rounding
I want to see if the following two rounding statements are true or false. If it is true, I want to prove it, and if it is false, I want to give a counterexample. I assume no overflow occurs in the ...
- 11
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40
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A different type of IEEE single format
To practice with different kinds of IEEE single format types, I am trying a format where the width of the exponent field is $4$ instead of $8$ and the width of the fraction field is $4$ instead of $23$...
- 11
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97
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Floating Point Rounding of 2 multiplied single precision numbers
I have built a floating point multiplier in Logisim (digital design tool) for only single precision normal inputs. I have realised 2 different round to even rounding algorithms and I am not sure which ...
- 23
3
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293
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Error bound for floating-point interval dot product
In Handbook of Floating-Point Arithmetic (Birkhäuser, 2010, Chapter 6) Muller et al. presented the following absolute forward error bound for the floating-point recursive dot product:
$$
\left|...
1
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2
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98
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Relativistiv kinetic energy and floating point
My function is $E(v)=mc^2(\frac{1}{\sqrt{1-v^2/c^2}} - 1)$, (c=3e8, m=1) and I have to calculate it for values of v between 1e-6 and 2.99e8.
The point of this problem is floating point precision.
For ...
- 23
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163
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What is the maximum number of significant bits lost when the computer evaluates x − y using IEEE 64 bits?
Consider two positive numbers $x = p2^m$ and $y = q2^n$ such that $m > n$, $1 < p < 2$, and $1 < q < 2$.
Both of these numbers can be stored using the IEEE 64 bit standard.
What is ...
- 15
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2
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1k
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How to represent the decimal value $- 7.5$ as a floating point number?
Consider the following $32$-bit floating-point representation scheme as shown in the format below :
A $1$ bit sign field
A $24$ bit fraction field
and a $7$ bit exponent field (in excess-$64$ ...
- 2,664
2
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2
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310
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How can I rearrange this logarithmic formula to be computer friendly?
I've had a look through the logarithmic identities on Wikipedia, but nothing fits the bill.
Basically, I have a formula which shows how much more 'risky' one number is compared to another, where 0 = ...
- 645
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Find original inputs $x$ and $^y$ for a given product, possible or not? [closed]
$387,381,625,547,900,583,936$ is the product of this calculation $21\cdot2^{64}$.
If I only have the product and the multiplier $2$ (without the exponent) would it be possible to find the other inputs ...
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