Questions tagged [floating-point]

Approximate representation of numbers as a fixed number of digits multiplied by a logarithmic scale.

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What is the state of the algorithmic art for floating point arithmetic on complex numbers?

Most modern compilers and processors implement the IEEE 754 binary formats for floating point numbers. IEEE 754 guarantees that the addition, subtraction, multiplication, division, and square root ...
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Representing Computations on Transcendental Numbers

Consider the set of transcendental numbers that are not compressible to a finite base-2 representation. How can I compute multiples (more generally, any algebraic computation) of one of these numbers,...
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Question About Floating Point System

I have begun reading Numerical Analysis by Walter Gautschi. On page $3$, the author introduces the floating point number system as follows: a floating point number is a number representible as $$ \...
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1answer
38 views

Max flow algorithm for floating-point weights and E~=10*V

Could you, please, suggest a maximum flow algorithm for a graph with floating-point weights and the number of edges approximately equal to the number of vertices? I.e. ...
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199 views

Stable and fast computation of the squared euclidean distance matrix

Let's say I want to compute the matrix $M$ of the squared euclidean distances between each pair of vectors $(x, y)$ belonging to two sets $X$ and $Y$ respectively. The sets of vectors $X$ and $Y$ have ...
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31 views

Heuristic Consistency with float multiplications

I'm running A* search with a theoretically consistent heuristic. I'm doing a lot of float multiplications, and I'm not sure how best to ensure that my heuristic remains consistent down to the last ...
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32 views

numeric stability of map reduce operations

I am building a small library for computing information retrieval metrics for classifiers (precision, recall, f1, accuracy, whatever). Typically each metric is built by calculating a single value for ...
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240 views

Why mantissa and exponent are stored differently in a float?

As we know, in IEEE 754 standard, float number's exponent and mantissa are stored differently. While the exponent is stored as an unsigned number, taking advantage of the bias, the mantissa is in sign-...
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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
30 views

Convert scientific notation decimal number to binary

I'm given the number: $$8.881784197001252 \cdot 10^{-16}$$ I know it is $2^{-50}$ but suppose I didn't know. I need to convert it to binary. One way is to apply the school taught algorithm on the ...
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57 views

What are the smallest and biggest negative floating point numbers in IEEE 754 32 bit?

I am stuck with a question that asks for smallest and biggest negative floating point numbers in IEEE 754 32-bit (their representation and decimal numerical value from which one can approximate the ...
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28 views

How can I calculate the exponential integral?

(I'm not sure this is the right forum.) I'm writing a program that uses the prime-counting function. Right now, I'm using x/log(x), but I want to switch to ...
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45 views

Accuracy and performance between a division and subtraction for a ratio in decibels

For comparing two images, one can use the Peak Signal-to-Noise Ratio (PSNR) metric, defined as follows: $\mathrm{PSNR} = 10 \cdot \log_{10}\left(\frac{\mathrm{MAX}^2}{\mathrm{MSE}}\right) = 20 \cdot \...
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273 views

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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58 views

Dual Signed Kahan Summation

NOTE: This is for a project I'm working on for fun, NOT production code. So I'm working on a pet project that involves reading data in from a sensor and summing it up. The values are mostly ...
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50 views

Understanding exponential computation by digit recurrence

I've met in a book the following algorithm that computes the exponential: Input: $t, n$ ($n$ is the number of steps) Output: $E_n$ $\begin{array}{l} \mbox{define $t_0 = 0$ ; $E_0 = 1$} \\ \mbox{...
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Why have only 15 bits been apportioned for the exponent in the 128-bit quad-precision datatype?

I look forward to the day we can start using quad-precision numbers, but was disappointed to see that in the specification, only 15 bits out of a whopping 128 were assigned to the exponent as shown by ...
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192 views

Addition in IEEE 754

Hi I have this simple question. When adding a positive number with a negative number, both of same exponent. Imagine that after complementing the negative we have something like this: 1.111 + 0.001 .....
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Is Decimal (correctly-rounded arbitrary precision decimal floating point arithmetic) fixed-point, floating-point or something else?

The Decimal data type I am referring to is GNU MPFR(https://en.wikipedia.org/wiki/GNU_MPFR), or libmpdec (http://www.bytereef.org/mpdecimal/doc/libmpdec/index.html). I have been searching for ...
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27 views

Floating point substraction

if $x=1.0e38=1.0 * 10^{38}$ and $y=3.0$ i want to find $ (x-x)+y $ and $(x+y)-x$ i think the value of (x-x)+y will be just substract $x-x=0 + y=3.0 = 3.0$ but how can i perfom addition of different ...
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56 views

How does normalised floating point binary work with two's complement?

I'm doing AQA a-level computer science, and the specification for which states that: Exam questions on floating point numbers will use a format in which both the normalised mantissa and exponent are ...
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1answer
44 views

algorithm for correctly rounded floating point radix conversion

Is there any generic algorithm which implements a floating point radix conversion? Lets say we have a $p$-digit FP number $A = \sum_{i=0}^{p-1} A_i \beta^{e-i}$ in radix $\beta$ and with $0 \leq ...
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2answers
415 views

How to compute relative error for the rounding of floating point numbers when the rounded number is 0?

I have asked this question on Stack Overflow, I am asking it here in the hope to get more traction. The relative rounding error for a floating point number x is defined as $e_r = |\frac{(round(x) -...
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1answer
173 views

Using a 16-bit 2,s Complement normalised floating-point representation; 10-bit fractional mantissa and a 6-bit integer exponent: express 2.171875

So far I know how to normalize when you are given, say: 1111010010 Mantissa, and 000100 exponent and are told that it's a positive number: ...
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Using half float to represent scaled short (int16), do I lose precision comparing to using double?

A device is generating 14-bit integer which is stored as int16 (short), a scaling process will then scale the data to value of order 10E-3. Does it then matter if I store these number with half float ...
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1answer
140 views

Computing the error bound of floating-point expression

How should I compute the maximum absolute and relative error of the following IEEE-754 floating-point expression? a.y + (x - a.x) * ((b.y - a.y) / (b.x - a.x)) ...
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Inverting an IEEE 754 double

While preparing for an exam I've come across this problem: Let $x$ be an IEEE 754 double precision number. Show that $$ \texttt{fl}(x * \texttt{fl}(1/x)) $$ has only 2 possible results ...
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When do you know whether an exponent represents a movement of the decimal point to the left?

I'm learning about floating point numbers and I don't quite understand when one should interpret an exponent as moving the decimal point to the left. By book shows an example of converting -10.5 ...
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114 views

How can a Number Represented without Normalization Technique?

Consider the following floating point Representation Mantissa is pure fraction in signed magnitude form . What is the representation of the binary number 1111.1111 $\times 2^{2} $ in hexadecimal ...
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3k views

How do the Guard, sticky, and round bits work and affect floating point Representation?

I've got a test on this tomorrow, my instructor's lectures aren't up, and the notes are ambiguous... Anyhow, we've been covering floating point operations, specifically addition, and he mentioned ...
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How to determine the range of a number represented in IEEE-754 format?

Given a format: say 12-bit floating point with sign bit, 5 - bit biased exponent and 6- bit significant with implied bit, how do i go about determining the range of ...
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42 views

Logarithms using digit recurrence, how many bits to store in a LUT?

In a known method for approximating logarithms the following decomposition is adopted (assuming $x \in [1,2)$): $$ log_2(x) = 1 + \sum_{j=1}^{+\infty} w_i log_2(1+2^{-j}) $$ Where all the ...
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Are floats abstract data types?

Source:https://en.wikipedia.org/wiki/Abstract_data_type Can we say that float data types (or all primitive data types or only integer) are abstract data types?
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810 views

Calculating sums of floating point numbers by hand

how do you calculate the sum of 2.6125 * 10^1 and 4.150390625 * 10-1 by hand, assuming A and B are stored in the 16-bit half precision. i cant figure out how to do this by hand with out losing most of ...
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Write the smallest positive number that can be represented by the floating point system

Using a normalised floating point representation box with an 8-bit mantissa and a 4-bit exponent, both stored using two’s complement. (a) Write the smallest positive number that can be represented by ...
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556 views

Convert between IEEE 754-2008 decimal64 and IEEE double precision floating point number

I need to know the algorithm for converting between IEEE 754-2008 decimal64 and IEEE 754-1985 double precision floating point number. I have been working on this for the past 2 days and I match the ...