Questions tagged [rounding]

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Rounding of $2-10^{20}$ in IEEE double precision

How do we get the rounding of $2-10^{20}$ in IEEE double precision? The textbook says it is $-10^{20}$, but I do not know why. I think my textbook only explains the rule for rounding mantissa.
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0answers
37 views

An LP with two covering constraints - how to round

I came across an LP with two covering problems, and I wonder how to find a good approximation. For the relevant part of the LP: We have a set $E$ , for each $e\in E$ we have a corresponding set $Y_{e}\...
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2answers
59 views

Increased rounding relative error when subtracting

I'm reading the book "Lessons in Scientific Computing" by Schoerghofer and it says: If x and y are real numbers of the same sign, their sum x + y has an absolute error that adds the two ...
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24 views

IEEE-754 and machine numbers

I've been trying to wrap my head around machine numbers like the unit roundoff (u) and epsilon (e) in combination with the IEEE 754 standard. My textbook states some things that don't really make ...
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0answers
45 views

Compile-time error control vs. interval arithmetic?

I use interval arithmetic for reliable computing. Now, a procedure coded in a good implementation of interval arithmetic takes perhaps about eight times as much as the same procedure carried out ...
1
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1answer
56 views

Shortest decimal expansion within binary interval

Consider an interval $[x-2^n,x+2^n]$ defined by a binary float $x$ and a power of two $2^n$ typically much smaller than $x$. I would like to know whether an efficient algorithm exists to determine the ...
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1answer
220 views

Are IEEE floating point numbers intervals or point values?

The context is IEEE 754-2008 floating point number systems. The systems defined by the standard comprise, as far as I understand it, a bit-level representation and a set of guarantees on the precision ...
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1answer
155 views

Rounding logarithm to next integer — Potential function

The problem is IV-3 of this pdf: potential function. Defining a potential function as $\Phi(i) = 2i - 2^{\lfloor{\log_2i}\rfloor+1} + 1$ The solution states that if $i$ is not an exact power of $2$,...
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2answers
4k views

Why can't we round results of linear programming to get integer programming?

If linear programming suggests that we need $2.5$ trucks to deliver goods, why can't we round up and say $3$ trucks are needed? If linear programming suggests we can afford only $3.7$ workers, then ...
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2answers
321 views

Why are transcendental functions of large numbers inaccurate on computers?

For instance, why is it hard to accurately compute sin(1e99)? I suspect it has something to do with rounding error.
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0answers
60 views

Normalised Floating Point System

I have a floating point number system and I have a number for which I need to calculate the exact relative error after rounding. The number is clearly an overflow. Does anyone know what I should do? ...
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1answer
141 views

floating point rounding (1/x)*x

I'm trying to figure out what the smallest positive integer x such that the floating point expression round(round(1/x)*x) is not equal to 1 in single precision. I have that the answer is 41, but when ...
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1answer
17k views

Normalizing the mantissa in floating point representation

How to represent $0.148 * 2^{14}$ in normalized floating point arithmetic with the format 1 - Sign bit 7 - Exponent in Excess-64 form 8 - Mantissa $(0.148)_{10} =...
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1answer
608 views

In a 32-bit floating number with normalized mantissa and excess-64 exponent base 16, the number $16^{-65}$ denotes

In a 32-bit floating number with normalized mantissa and excess-64 exponent base 16, the number $16^{-65}$ denotes Floating point overflow. Negative floating point overflow. All 0's in the exponent ...
13
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1answer
408 views

Floating point rounding

Can an IEEE-754 floating point number < 1 (i.e. generated with a random number generator which generates a number >= 0.0 and < 1.0) ever be multiplied by some integer (in floating point form) to ...