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1answer
21 views

Machine epsilon vs least positive number

What is the difference between machine epsilon and least positive number in floating point representation? If I try to show the floating point number on a number line .Is the gap between exact 0 and ...
1
vote
1answer
30 views

Why is the precision of floating point numbers worse for smaller numbers?

Why is the machine error/epsilon higher between a pair of two lower numbers than a pair of two high numbers? For example, between the two smallest numbers possible in 5 bit mantissa and the two ...
1
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2answers
52 views

Gap between numbers in fixed-point vs. floating point arithmetic

If $r$ is a machine-representable number and $f(r)$ is the next larger machine representable number, are the following true or false? In fixed-point arithmetic, the distance between $r$ and $f(r)$ ...
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9answers
2k views

Represent a real number without loss of precision

Current floating point (ANSI C float, double) allow to represent an approximation of a real number. Is there any way to represent real numbers without errors? Here's an idea I had, which is anything ...
1
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1answer
37 views

Floating point operations; Exception, Flags, and Trap Handlers

I am reading over the article found here specifically §D.4.4 Exceptions, Flags and Trap Handlers and am confused by the table D-4 in that section. Specifically the arguments sent to the trap handler ...
2
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1answer
225 views

Numerical stability of C++: will a C++ program using a float or double library return the same arithmetical results on two different computers?

Say I am using boost or the built-in float or double mathematical libraries of my C++ compiler. I distribute the program. Will the execution of my C++ program on different machines given different ...
0
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1answer
24 views

floating point normalised value of -1

I have, lets say, 8 bits mantissa and 4 bit exponent. Then, -1=1111 1111 there are no 0s so how can I normalise -1 in 2's complement form?
6
votes
0answers
89 views

Why does floating point modulus exactness matters?

Most Smalltalk dialects currently implement a naive inexact floating modulus (fmod/remainder). I just changed this to improve Squeak/Pharo and eventually other Smalltalk adherence to standards (IEEE ...
3
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1answer
60 views

Can we improve the precision of IEEE floats by dropping leading zeros in the mantissa?

It seems like it would be possible to add more precision to the IEEE 32-bit mantissa system if the leading zeroes were also dropped, just like the leading 1 is dropped due to it being implicitly ...
2
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1answer
28 views

Floating point format: why must `1−emax ≤ q+p−1 ≤ emax`?

From the Wikipedia page on the IEEE Standard for Floating-Point Arithmetic, The possible finite values that can be represented in a format are determined by the base (b), the number of digits in ...
0
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2answers
86 views

Problem in finding the floating point representation?

So, i was trying: $(-10.75)_{10}$ and to convert it into 32 bit binary floating point representation. i did this: According to IEEE standard: $(-1)^{-s} * 1.M * 2^{E-bias} $ ...
1
vote
1answer
50 views

When does the IEEE-754 64-bit float break as a counter

As a matter of curiosity I've been trying to determine at what point a 64-bit float no longer reflects the addition of 1 as expected; that is, at what point the digits as printed do not correspond to ...
1
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1answer
84 views

Implicit Leading 1 in Binary Floating Point

Is the convention of dropping the leading 1 when storing the significand a given in all binary floating point representations or not necessarily?
2
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0answers
80 views

Can a unnormalized floating point number be recognized also when exponent is not zero?

From Tanebaum's Structed Computer Organization. Exercise 4 of Appendix B The following binary floating-point number consist of a sign bit, an excess $64$, radix $2$ exponent, and a $16$-bit ...
5
votes
1answer
275 views

Confused by Floating Point Spacing

I'm currently taking a numerical analysis class in college and we're covering floating point systems. For the most part, I have a good grasp on it. However, something I can't seem to visualize, and ...
6
votes
2answers
246 views

What is the reason of inaccuracy of operations on float numbers?

I wonder why in JavaScript 0.1 + 0.2 // return 0.30000000000000004 4%0.1 // return 0.09999999999999978 http://jsbin.com/oHISAfU/1/edit (Example) In C the ...
0
votes
2answers
159 views

Implications of truncation of numbers when converted into binary

I have been posed with a question whereby a computer truncates numbers to x number of digits. Due to this, if this computer is trying to store a decimal number which has a binary equivalent greater ...
3
votes
1answer
4k 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} ...
0
votes
1answer
351 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 ...
1
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1answer
318 views

Calculating Binet's formula for Fibonacci numbers with arbitrary precision

Binet's formula for the nth Fibonacci numbers is remarkable because the equation "converts" via a few arithmetic operations an irrational number $\phi$ into an integer sequence. However, using finite ...
5
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3answers
902 views

Why floating point representation uses a sign bit instead of 2's complement to indicate negative numbers

Consider a fixed point representation which can be regarded as a degenerate case of a floating number. It is entirely possible to use 2's complement for negative numbers. But why is a sign bit ...
11
votes
1answer
227 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 ...