# Questions tagged [floating-point]

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

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### How to determine the set of real numbers corresponding to a given floating point number?

Let's say we consider IEEE 754 double precision floating-point numbers, and we use RNTE - Round To Nearest, Ties to Even - rounding. I know that the RNTE rounding works this way: given two consecutive ...
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### Logic behind choosing the exponent bias as $2^7 -1$ instead of $2^7$ in $32$ bits IEEE-754 floating point representation

The $\text{IEEE-754}$ uses $32$ bits to represent single precision floating point numbers. The partitions of the register are as follows: ...
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### What is the currently know fastest algorithms to convert each individual Java primitive and Number types to a Char Array/String

What is the currently know fastest algorithms to convert each individual Java primitive and Number types to a Char Array/String number representation while ...
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### How to represent zero as floating point number?

For the floating-point number, we have the form $\pm d_0.d_1d_2···d_{P-1}\times\beta^E$ $\pm$ --------------------------- sign $d_0.d_1d_2···d_{p-1}$ --------- significant $\beta$ --------------------...
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### Are floating-point numbers normalised by computers or humans?

I keep seeing posts and articles about how to normalise a floating-point number, why that's done and how such a number is represented in binary. But no one seems to mention who/what the normalisation ...
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### How are bitwise operators used in normalisation of floating-point numbers?

After having spent a significant amount of time googling this topic, I am still struggling to confidently answer the following question: ...
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### Converting Decimal Numbers between 0 and 1 to Binary

I've been playing around with a program I wrote that converts decimal numbers to binary numbers and i've noticed that eventually, after applying the algorithm (multiply by 2, subtract 1 if greater ...
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### Why is there a precision loss for floating-point numbers?

It could be a silly question, yet I'm not able to understand. In modern day, computers, we have integers with million digits length. Even in my ordinary 2GB laptop, I can calculate values of numbers ...
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### 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. ...
119 views

### Normalization in IBM hexadecimal floating point

According to the Wikipedia link of IBM hexadecimal floating point: Consider encoding the value −118.625 as an IBM single-precision floating-point value. The value is negative, so the sign ...
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### 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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### Do all programming languages adhere to the IEEE 754-2008 standard?

I am well aware that the float data type's maximum value is approximately 1.8E308, which means that across the various ...
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### 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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### 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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### What's the least signifcant bit of a mantissa system?

If Mantissa is a 1-dot-M fixed-point number whose most significant bit is always 1 then, how is the least significant bit calculated? I know the least and most significant bit of the mantissa ...
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I am studying over my notes, and there is something I don't understand about $e_m$. We represent the floating point numbers as $1.d_1d_2...d_t \times \beta^e$. Now, my professor defines $\epsilon_m$ ...
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### Why multiplying float number by multiple of 10 seems to preserve better precision?

It is famous that for float numbers: .1 + .2 != .3 but 1+2=3 It seems that multiplying floats by 10 allows you to preserve ...
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### How does a computer compute negative(-) and positive(+) Infinity?

If we divide (1.0/0.0) we will get +Infinity and if we divide (-1.0/0.0) we will get -Infinity. How does a computer calculate this value internally?
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### Can we represent $\sqrt{2}$ exactly even with infinite bits in mantissa [closed]

Can we represent $\sqrt{2}$ exactly even with infinite bits in mantissa in floating point notation or otherwise. We actually have to prove this is not possible. But why can't we if we have infinite ...
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### 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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### Polynomials - using Newton's method, or not?

I have to find a root of polynomial of degree $n\ge2$. I need to write code to calculate the root for different values of $n$. Only 1 real positive solution is needed. I can use general Newton's ...
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### How many floating point ops were performed worldwide over a time interval [closed]

I am looking for information regarding the evolution of computing capability. Specifically I would like to know how many floating point operations were performed worldwide from, say, the deployment of ...
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### Computing an Expression

I am writing code to evaluate the following expression: $$\frac{(a+b+c)!}{a! b! c!}$$ where $a$, $b$ and $c$ are on the range of $10$ to $500$. The result is going to be a floating point number. ...
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### How much can we trust mathematical software when working with large numbers, and how much memory it needs to work with these numbers?

For example, I want to evaluate the expression: $3^{3^{{3}^{3}}}$ so I used wolframalpha.com (it's free, and I don't own any software), which returned the scientific notation of the number above, ...
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### numerically stable log1pexp calculation

What are good approximations for computing log1pexp for single precision and double precision floating point numbers? Note: ...
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### Floating point arithmetic

I need to change x1 = 0.3 and x2 = -0.29 to a FP(floating point) number with one sign bit, a 4 bit mantissa, a 3 bit exponent. The results I got are: x1: 0 001 0011 x2: 1 001 0010 I am also trying ...
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### Convert $1.75\times10^{15}$ to IEEE-32 format?

$1.75\times10^{15}$ I know how to convert decimal to binary $(1.75)_{10}$ is equal to $(1.11)_2$ But to represent $10^{15}$ is the main problem for me. I can solve the question but this is the ...
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### Guarantees on computing $a+x(b-a)$ in floating point

I want to implement the function $f(x,a,b) = a + x(b-a)$ where all the inputs are floating point (doubles, say), such that (a) $f(0,a,b)=a$ exactly; (b) $f(1,a,b)=b$ exactly; (c) \$f(x,a,b) \le f(y,a,b)...