Why Doesn't Floating-Point Addition Always Follow the Associative Law?
Dec 20, 2024 am 02:02 AMFloating-Point Arithmetic: Exploring Associativity in Floating-Point Operations
In the realm of programming, floating-point numbers are essential for working with continuous values. However, floating-point arithmetic can sometimes behave differently from the mathematics we learned in school, especially when it comes to associativity.
Consider the example of adding three floating-point values and comparing them to 1:
Curiously, the two expressions produce different results. This discrepancy stems from the fact that floating-point addition is not always associative. In other words, changing the order in which we add values can alter the final result.
The phenomenon of non-associativity arises due to the limitations of floating-point representation. Floating-point numbers use a binary format with a finite number of bits, which limits their accuracy. As a result, computations with floating-point numbers can introduce rounding errors that accumulate over multiple operations.
In the example above, the addition of 0.7, 0.2, and 0.1 results in slightly different intermediate values depending on the order of operations. These slight differences propagate through the subsequent addition, ultimately leading to different final results.
To understand this concept further, let's consider an example from David Goldberg's seminal paper "What Every Computer Scientist Should Know about Floating Point Arithmetic":
In this case, the parentheses deeply affect the result. Adding 1e30 and -1e30 in parentheses results in a near-zero value due to rounding, which then gets added to 1. On the other hand, adding 1e30 and the result of -1e30 1 produces a non-zero value.
Therefore, when working with floating-point arithmetic, it is crucial to be aware of potential non-associativity and to be cautious when manipulating expressions involving repeated additions or other associative operations.
The above is the detailed content of Why Doesn't Floating-Point Addition Always Follow the Associative Law?. For more information, please follow other related articles on the PHP Chinese website!

Hot AI Tools

Undress AI Tool
Undress images for free

Undresser.AI Undress
AI-powered app for creating realistic nude photos

AI Clothes Remover
Online AI tool for removing clothes from photos.

Clothoff.io
AI clothes remover

Video Face Swap
Swap faces in any video effortlessly with our completely free AI face swap tool!

Hot Article

Hot Tools

Notepad++7.3.1
Easy-to-use and free code editor

SublimeText3 Chinese version
Chinese version, very easy to use

Zend Studio 13.0.1
Powerful PHP integrated development environment

Dreamweaver CS6
Visual web development tools

SublimeText3 Mac version
God-level code editing software (SublimeText3)

Hot Topics

People who study Python transfer to C The most direct confusion is: Why can't you write like Python? Because C, although the syntax is more complex, provides underlying control capabilities and performance advantages. 1. In terms of syntax structure, C uses curly braces {} instead of indentation to organize code blocks, and variable types must be explicitly declared; 2. In terms of type system and memory management, C does not have an automatic garbage collection mechanism, and needs to manually manage memory and pay attention to releasing resources. RAII technology can assist resource management; 3. In functions and class definitions, C needs to explicitly access modifiers, constructors and destructors, and supports advanced functions such as operator overloading; 4. In terms of standard libraries, STL provides powerful containers and algorithms, but needs to adapt to generic programming ideas; 5

std::chrono is used in C to process time, including obtaining the current time, measuring execution time, operation time point and duration, and formatting analysis time. 1. Use std::chrono::system_clock::now() to obtain the current time, which can be converted into a readable string, but the system clock may not be monotonous; 2. Use std::chrono::steady_clock to measure the execution time to ensure monotony, and convert it into milliseconds, seconds and other units through duration_cast; 3. Time point (time_point) and duration (duration) can be interoperable, but attention should be paid to unit compatibility and clock epoch (epoch)

C STL is a set of general template classes and functions, including core components such as containers, algorithms, and iterators. Containers such as vector, list, map, and set are used to store data. Vector supports random access, which is suitable for frequent reading; list insertion and deletion are efficient but accessed slowly; map and set are based on red and black trees, and automatic sorting is suitable for fast searches. Algorithms such as sort, find, copy, transform, and accumulate are commonly used to encapsulate them, and they act on the iterator range of the container. The iterator acts as a bridge connecting containers to algorithms, supporting traversal and accessing elements. Other components include function objects, adapters, allocators, which are used to customize logic, change behavior, and memory management. STL simplifies C

Learn C You should start from the following points when playing games: 1. Proficient in basic grammar but do not need to go deep into it, master the basic contents of variable definition, looping, condition judgment, functions, etc.; 2. Focus on mastering the use of STL containers such as vector, map, set, queue, and stack; 3. Learn fast input and output techniques, such as closing synchronous streams or using scanf and printf; 4. Use templates and macros to simplify code writing and improve efficiency; 5. Familiar with common details such as boundary conditions and initialization errors.

In C, cin and cout are used for console input and output. 1. Use cout to read the input, pay attention to type matching problems, and stop encountering spaces; 3. Use getline(cin, str) when reading strings containing spaces; 4. When using cin and getline, you need to clean the remaining characters in the buffer; 5. When entering incorrectly, you need to call cin.clear() and cin.ignore() to deal with exception status. Master these key points and write stable console programs.

STL (Standard Template Library) is an important part of the C standard library, including three core components: container, iterator and algorithm. 1. Containers such as vector, map, and set are used to store data; 2. Iterators are used to access container elements; 3. Algorithms such as sort and find are used to operate data. When selecting a container, vector is suitable for dynamic arrays, list is suitable for frequent insertion and deletion, deque supports double-ended quick operation, map/unordered_map is used for key-value pair search, and set/unordered_set is used for deduplication. When using the algorithm, the header file should be included, and iterators and lambda expressions should be combined. Be careful to avoid failure iterators, update iterators when deleting, and not modify m

As a beginner graphical programming for C programmers, OpenGL is a good choice. First, you need to build a development environment, use GLFW or SDL to create a window, load the function pointer with GLEW or glad, and correctly set the context version such as 3.3. Secondly, understand OpenGL's state machine model and master the core drawing process: create and compile shaders, link programs, upload vertex data (VBO), configure attribute pointers (VAO) and call drawing functions. In addition, you must be familiar with debugging techniques, check the shader compilation and program link status, enable the vertex attribute array, set the screen clear color, etc. Recommended learning resources include LearnOpenGL, OpenGLRedBook and YouTube tutorial series. Master the above

volatile tells the compiler that the value of the variable may change at any time, preventing the compiler from optimizing access. 1. Used for hardware registers, signal handlers, or shared variables between threads (but modern C recommends std::atomic). 2. Each access is directly read and write memory instead of cached to registers. 3. It does not provide atomicity or thread safety, and only ensures that the compiler does not optimize read and write. 4. Constantly, the two are sometimes used in combination to represent read-only but externally modifyable variables. 5. It cannot replace mutexes or atomic operations, and excessive use will affect performance.
