Questions tagged [floating-point]
Mathematical questions concerning floating point numbers, a finite approximation of the real numbers used in computing.
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pow and its relative error
Investigating the floating-point implementation of the $\operatorname{pow}(x,b)=x^b$ with $x,b\in\Bbb R$ in some library implementations, I found that some pow ...
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Algebraic Structures involving 𝙽𝚊𝙽 (absorbing element).
IEEE 754 floating point numbers contain the concept of 𝙽𝚊𝙽 (not a number), which "dominates" arithmetical operations ($+,-,⋅,÷$ will return ...
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Solve $10^{10^z} = 10^{10^x}+10^{10^y}$ for $z$ with floating point accuracy
In the following equation
$$10^{10^z} = 10^{10^x}+10^{10^y}$$
I want to find an algorithm that computes $z$ in a floating point accurate manner given any values of $x$ and $y$ (e.g. $x=y=2000$). The ...
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How to transform this expression to a numerically stable form?
I have this function
$$f(x, t)=\frac{\left(1+x\right)^{1-t}-1}{1-t}$$
Where $x \ge 0$ and $t \ge 0$.
I want to use it in neural network, and thus need it to be differentiable.
While it has a ...
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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} \...
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Why does TI-84 show scientific notation for zeros sometimes but not others?
When graphing a function and then going through the process to calculate the zeroes (left bound, right bound, guess), is there a reason that sometimes it shows y = 0, but there are other times when it ...
- 141
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Numerically stable way to compute ugly double fraction
I am looking for a numerically stable version of this (ugly) equation
$$
s^2=\frac{1}{\frac{1}{\beta_1}+\frac{1}{\beta_2}W}
$$
where
$$
\beta_1 = c_1-c_2m+(m-c_2)b\\
\beta_2 = \frac{1}{2}\left((a-m)^2-...
- 272
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Fundamental Axiom of Floating Point Arithmetic for Complex Numbers Multiplication
I am trying to prove the fundamental axiom of floating point arithmetic also applies to complex number multiplication.
First, let $fl$ be a function that maps a number to its closest floating point ...
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How do calculators represent floating points (somewhat) perfectly?
If you ask a programming language to calculate 0.6 + 0.7, you’ll get something like 1.2999998, and that’s because of how floating point numbers are represented in computers. But if you ask a ...
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Calculating coordinates of vertices, given dimensions in an architectural floorplan
So, one of my friend is trying to learn autocad. They were given a floorplan. The floorplan had the dimensions. And they were asked to find the coordinates of the all the vertices of the plan. So we ...
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Proof that $\frac 1{10}$ has no finite binary float representation
I am supposed to prove that $\frac 1{10}$ is not representable as a finite binary float. I tried proving this via induction but that did not seem to work, now I am out of ideas. Thank you
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Absolute difference between largest IEEE754 number and its predecesor
In simple precision format, the largest possible positive number is
$A = 0 ~~~ 11111110 ~~~ 111\ldots 111$
Its predecessor is
$B = 0 ~~~ 11111110 ~~~ 111 \ldots 110$
But what is the absolute ...
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Proof of `TWOSUM` implementation in "double-double" arithmetic
"double-double" / "compensated" arithmetic uses unevaluated sums of floating point numbers to obtain higher precision.
One of the basic algorithms is ...
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Specify the conditions Exponent and Mantissa sizes must meet, so that the minimal distance between representable numbers is no more than 1.
Using the following floating-point representation:
s - one sign bit
m - mantissa - real number in range [1, 2), in which 1 and the comma are skipped, size of M bits
c - Exponent - natural number, ...
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What is the computational complexity of calculating determinants for matrices of finite precision floating-point numbers?
Following up from this older question, I understand that calculation of determinants for integer-valued matrices is possible with polynomial scaling. However, I have been unable to locate any ...
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