# 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 meant by “uneven spacing” on number line when dealing with floating point numbers in computers?

I read some examples online on this subject, but none of them really gave an explanation. When I visualize the number line all I can see is evenly spaced numbers, I'm not grasping the notation of "...
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### Comparing floating-point numbers as integers

In the article http://floating-point-gui.de/errors/comparison/, a method for comparing floating-point numbers is suggested: There is an alternative to heaping conceptual complexity onto such an ...
204 views

### NaN simple precision IEEE 754 [closed]

How many words exist in the format of the simple precision IEEE 754 standard to represent the NaN value?
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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### Inequality caused by float inaccuracy

At least in Java, if I write this code: float a = 1000.0F; float b = 0.00004F; float c = a + b + b; float d = b + b + a; boolean e = c == d; the value of $e$ ...
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### MSB and LSB in floating point number

What is MSB and LSB in significant of a floating point number? If I have a significant for example 0011010000 (half-precision) but 1. bit is implicit what is MSB: 0 or 1? In this case LSB is 0, ...
91 views

### 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 ...
891 views

### Floating Point Systems - Rounding Error in Taylor series

I have a question about rounding error. I have created a function to approximate exp(x) by summing the terms of its taylor series until the sum stops changing. (That is, 1 + x + x^2/2! ...). This ...
896 views

### How to add two numbers in IEE754 half precision format?

i'm trying to understand how to add two numbers in IEE754 half precision format , i haven't found a good reference/tutorial on this , so i've tried to follow the 32 bit floating point addition ...
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### 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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### Formal notations of equivalencence between floating point expressions

In an informal context, most programmers might call these two expressions "equal" and no one would be mislead: (a + b) + c a + (b + c) For floating point numbers,...
434 views

### Find the duplicates in a list of floating point numbers

I receive a list of real numbers ( float ) between $0$ and $1$. The list has length $N+1$ and I need to find two numbers on the list which are $\le \frac{1}{N}$ ...
130 views

### What factors influence machine epsilon?

I was wondering what factors effect a machine's epsilon value. I was thinking about how modern computers can calculate expressions to higher accuracy then their predecessors, but what hardware and ...
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 ...
606 views

### Whether assigning of single precision IEEE754 float to double is reversible?

Using the IEEE754 standard, let's assign single precision variable s to double precision variable d and then assign d to single precision variable s'. Is this operation is reversible (lossless) for ...
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### How to work out if an IEEE 754 floating point number is normalized?

How do I tell whether a particular IEEE 754 floating point number is a normalized floating point number? Is there some way to recognize an IEEE 754 normalized floating point number?
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### 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 ...
1k views

### half precision floating point multiplication

A = 0 10011 0011110111 B = 1 00011 0010011000 exponent is 15, mantissa is 10 bits and first bit is implicit. Can somebody please tell me the final answer cause I am having trouble figuring ...
140 views

### Difference between ways to compare floating-point numbers

There seems to be many approaches to judge whether two floating-point numbers are identical. Here are some examples I've found: ...
156 views

### Approximate a float using a minimal fraction

This sounds like it's probably a well-known problem, but I haven't been able to find references to it by searching. Given a floating point value $x$ and an error range $\varepsilon$, how can I ...
586 views

### Implementation of Naive Bayes

I am implementing a Naive Bayes algorithm for text categorization with Laplacian smoothing. The problem I am having is that the probability approaches zero because I am multiplying many small ...
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### Why do floating point additions sometimes produce equal results? [duplicate]

I have tried two sentences on my computer : 2 + 10^(-18) == 2 2^(-55) + 2^(-57) == 2^(-55) My computer gives TRUE and FALSE respectively. Why does the computer ...
382 views

### What happens if we add denormalized number and normalized number?

Denormalized numbers can represent numbers smaller than 2^(-1022) whereas normalized number cannot. So I'm curious what happens if we add denormalized number and normalized number. Actually, I have ...
5k views

### 8-bit floating-point representation

I'm studying about representing fractional numbers as floating-point values. It is going to be an 8-bit representation. Somewhere in the text, it is said that: "We use the first bit to represent ...
1k views

### Smallest number close to 0 in IEEE754 (64bits)?

I thought the smallest number close to 0 would be : 0 00000000001 (exponent) 0000000000000000000000000000000000000000000000000000 (significand) But this site (http://binaryconvert.com/result_double....
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### 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 ...
583 views

### Distribution of IEEE 754 single precision floating point over number line

"What is the maximum and minimum difference between two successive real numbers representable in IEEE 754 Single Precision and Double Precision Floating Point Representations respectively?" In ...
966 views

### What is the 1's and 2's complement of 0.01101?

What is the 1's and 2's complement of 0.01101? I'm unable to find any details on this from google. Basically how do we represent the floating points in 1's and 2's complement forms? Even wikipedia ...
258 views

### Is < binary relation a strict partial order on IEEE doubles?

To me it looks that it is: irreflexivity: NaN < NaN == false transitivity: if a < b and b < c then a < c (the antecedent is never true for NaNs) asymmetry: if a < b then not b < a (...
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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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### Interpretation of '1/3' in IEEE floating point representation

For a rational number 1/3 below is the floating point representation(64 bit) of decimal expansion 0.3333333.... As per the above bit structure, I would like to ...
0.1 via $32\text{ bit}$ is rather easy: Sign: $0_2 = 0_{10}$ Exponent: $123_{10} = 01111011_2$ Mantissa: $5033165_{10} = 100110011001100110011001101_2$ Now, how do you calculate this, if you've ...