Floating-point Numbers Aren't Real

本文探讨了浮点数运算中的误差来源及其可能导致的问题,如精度损失和数值稳定性问题,并提供了一些避免这些陷阱的方法。

Floating-point numbers are not "real numbers" in the mathematical sense, even though they are called real in some programming languages, such as Pascal and Fortran. Real numbers have infinite precision and are therefore continuous and non-lossy; floating-point numbers have limited precision, so they are finite, and they resemble "badly-behaved" integers, because they're not evenly spaced throughout their range.

To illustrate, assign 2147483647 (the largest signed 32-bit integer) to a 32-bit float variable (x, say), and print it. You'll see 2147483648. Now print x - 64. Still 2147483648. Now print x-65 and you'll get 2147483520! Why? Because the spacing between adjacent floats in that range is 128, and floating-point operations round to the nearest floating-point number.

IEEE floating-point numbers are fixed-precision numbers based on base-two scientific notation: 1.d1d2...dp-1 × 2e, where p is the precision (24 for float, 53 for double). The spacing between two consecutive numbers is 21-p+e, which can be safely approximated by ε|x|, where ε is the machine epsilon (21-p).

Knowing the spacing in the neighborhood of a floating-point number can help you avoid classic numerical blunders. For example, if you're performing an iterative calculation, such as searching for the root of an equation, there's no sense in asking for greater precision than the number system can give in the neighborhood of the answer. Make sure that the tolerance you request is no smaller than the spacing there; otherwise you'll loop forever.

Since floating-point numbers are approximations of real numbers, there is inevitably a little error present. This error, called roundoff, can lead to surprising results. When you subtract nearly equal numbers, for example, the most significant digits cancel each other out, so what was the least significant digit (where the roundoff error resides) gets promoted to the most significant position in the floating-point result, essentially contaminating any further related computations (a phenomenon known as smearing). You need to look closely at your algorithms to prevent such catastrophic cancellation. To illustrate, consider solving the equation x2 - 100000x + 1 = 0 with the quadratic formula. Since the operands in the expression -b + sqrt(b2 - 4) are nearly equal in magnitude, you can instead compute the root r1 = -b - sqrt(b2 - 4), and then obtain r2 = 1/r1, since for any quadratic equation, ax2 + bx + c = 0, the roots satisfy r1r2 = c/a.

Smearing can occur in even more subtle ways. Suppose a library naively computes ex by the formula 1 + x + x2/2 + x3/3! + .... This works fine for positive x, but consider what happens when x is a large negative number. The even-powered terms result in large positive numbers, and subtracting the odd-powered magnitudes will not even affect the result. The problem here is that the roundoff in the large, positive terms is in a digit position of much greater significance than the true answer. The answer diverges toward positive infinity! The solution here is also simple: for negative x, compute ex = 1/e|x|.

It should go without saying that you shouldn't use floating-point numbers for financial applications — that's what decimal classes in languages like Python and C# are for. Floating-point numbers are intended for efficient scientific computation. But efficiency is worthless without accuracy, so remember the source of rounding errors and code accordingly!

By Chuck Allison

This work is licensed under a Creative Commons Attribution 3

请查看以下的C++代码的编写要求,请根据代码要求开始编写代码 PURPOSE: This file is a proforma for the EEET2246 Laboratory Code Submission/Test 1. This file defines the assessment task which is worth 10% of course in total - there is no other documentation. At the BASIC FUNCTIONAL REQUIREMENTS level, your goal is to write a program that takes two numbers from the command line and perform and arithmetic operations with them. Additionally your program must be able to take three command line arguments where if the last argument is 'a' an addition is performed, and if 's' then subtraction is performed with the first two arguments. At the FUNCTIONAL REQUIREMENTS level you will be required to extend on the functionality so that the third argument can also be 'm' for multiplication,'d' for division and 'p' for exponential operations, using the first two arguments as the operands. Additionally, at this level basic error detection and handling will be required. The functionality of this lab is relatively simple: + - / * and "raised to the power of" The emphasis in this lab is to achieve the BASIC FUNCTIONALITY REQUIREMENTS first. Once you a basic program functioning then you should attempt the FUNCTIONALITY REQUIREMENTS and develop your code so that it can handle a full range of error detection and handling. ___________________________________________________________________________________________ ___ GENERAL SPECIFICATIONS (mostly common to all three EEET2246 Laboratory Code Submissions): G1. You must rename your file to lab1_1234567.cpp, where 1234567 is your student number. Your filename MUST NEVER EVER contain any spaces. _under_score_is_Fine. You do not need to include the 's' in front of your student number. Canvas will rename your submission by adding a -1, -2 etc. if you resubmit your solution file - This is acceptable. G2. Edit the name/email address string in the main() function to your student number, student email and student name. The format of the student ID line is CSV (Comma Separated Variables) with NO SPACES- student_id,student_email,student_name When the program is run without any operands i.e. simply the name of the executable such as: lab1_1234567.exe the program MUST print student ID string in Comma Separated Values (CSV) format with no spaces. For example the following text should be outputted to the console updated with your student details: "1234567,s1234567@student.rmit.edu.au,FirstName_LastName" G3. All outputs are a single error character or a numerical number, as specified by the FUNCTIONAL REQURMENTS, followed by a linefeed ( endl or \n). G4. DO NOT add more than what is specified to the expected console output. Do NOT add additional information, text or comments to the output console that are not defined within the SPECIFICATIONS/FUNCTIONAL REQURMENTS. G5. DO NOT use 'cin', system("pause"), getchar(), gets(), etc. type functions. Do NOT ask for user input from the keyboard. All input MUST be specified on the command line separated by blank spaces (i.e. use the argv and argc input parameters). G6. DO NOT use the characters: * / \ : ^ ? in your command line arguments as your user input. These are special character and may not be processed as expected, potentially resulting in undefined behaviour of your program. G7. All input MUST be specified on the command line separated by blank spaces (i.e. use the argc and argv[] input parameters). All input and output is case sensitive unless specified. G8. You should use the Integrated Debugging Environment (IDE) to change input arguments during the development process. G9. When your code exits the 'main()' function using the 'return' command, you MUST use zero as the return value. This requirement is for exiting the 'main()' function ONLY. A return value other than zero will indicate that something went wrong to the Autotester and no marks will be awarded. G10. User-defined functions and/or class declarations must be written before the 'main()' function. This is a requirement of the Autotester and failure to do so will result in your code scoring 0% as it will not be compiled correctly by the Autotester. Do NOT put any functions/class definitions after the 'main()' function or modify the comments and blank lines at the end of this file. G11. You MUST run this file as part of a Project - No other *.cpp or *.h files should be added to your solution. G12. You are not permitted to add any other #includes statements to your solution. The only libraries permitted to be used are the ones predefined in this file. G13. Under no circumstances is your code solution to contain any go_to labels - Please note that the '_' has been added to this description so that this file does not flag the Autotester. Code that contains go_to label like syntax will score 0% and will be treated as code that does not compile. G14. Under no circumstances is your code solution to contain any exit_(0) type functions. Please note that the '_' has been added to this description so that this file does not flag the Autotester. Your solution must always exit with a return 0; in main(). Code that contains exit_(0); label like syntax will score 0% and will be treated as code that does not compile. G15. Under no circumstances is your code solution to contain an infinite loop constructs within it. For example usage of while(1), for(int i; ; i++) or anything similar is not permitted. Code that contains an infinite loop will result in a score of 0% for your assessment submission and will be treated as code that does not compile. G16. Under no circumstances is your code solution to contain any S_l_e_e_p() or D_e_l_a_y() like statements - Please note that the '_' has been added to this description so that this file does not flag the Autotester. You can use such statements during your development, however you must remove delays or sleeps from your code prior to submission. This is important, as the Autotester will only give your solution a limited number of seconds to complete (i.e. return 0 in main()). Failure for your code to complete the required operation/s within the allotted execution window will result in the Autotester scoring your code 0 marks for that test. To test if your code will execute in the allotted execution window, check that it completes within a similar time frame as the provided sample binary. G17. Under no circumstances is your code solution to contain any characters from the extended ASCII character set or International typeset characters. Although such characters may compile under a normal system, they will result in your code potentially not compiling under the Autotester environment. Therefore, please ensure that you only use characters: a ... z, A ... Z, 0 ... 9 as your variable and function names or within any literal strings defined within your code. Literal strings can contain '.', '_', '-', and other basic symbols. G18. All output to console should be directed to the standard console (stdout) via cout. Do not use cerr or clog to print to the console. G19. The file you submit must compile without issues as a self contained *.cpp file. Code that does not compile will be graded as a non-negotiable zero mark. G20. All binary numbers within this document have the prefix 0b. This notation is not C++ compliant (depending on the C++ version), however is used to avoid confusion between decimal, hexadecimal and binary number formats within the description and specification provided in this document. For example the number 10 in decimal could be written as 0xA in hexadecimal or 0b1010 in binary. It can equally be written with leading zeroes such as: 0x0A or 0b00001010. For output to the console screen you should only ever display the numerical characters only and omit the 0x or 0b prefixes (unless it is specifically requested). ___________________________________________________________________________________________ ___ BASIC FUNCTIONAL REQUIREMENTS (doing these alone will only get you to approximately 40%): M1. For situation where NO command line arguments are passed to your program: M1.1 Your program must display your correct student details in the format: "3939723,s3939723@student.rmit.edu.au,Yang_Yang" M2. For situation where TWO command line arguments are passed to your program: M2.1 Your program must perform an addition operation, taking the first two arguments as the operands and display only the result to the console with a new line character. Example1: lab1_1234567.exe 10 2 which should calculate 10 + 2 = 12, i.e. the last (and only) line on the console will be: 12 M3. For situations where THREE command line arguments are passed to your program: M3.1 If the third argument is 'a', your program must perform an addition operation, taking the first two arguments as the operands and display only the result to the console with a new line character. M3.2 If the third argument is 's', your program must perform a subtraction operation, taking the first two arguments as the operands and display only the result to the console with a new line character. The second input argument should be subtracted from the first input argument. M4. For situations where less than TWO or more than THREE command line arguments are passed to your program, your program must display the character 'P' to the console with a new line character. M5. For specifications M1 to M4 inclusive: M5.1 Program must return 0 under all situations at exit. M5.2 Program must be able to handle integer arguments. M5.3 Program must be able to handle floating point arguments. M5.4 Program must be able to handle one integer and one floating point argument in any order. Example2: lab1_1234567.exe 10 2 s which should calculate 10 - 2 = 8, i.e. the last (and only) line on the console will be: 8 Example3: lab1_1234567.exe 10 2 which should calculate 10 + 2 = 12, i.e. the last (and only) line on the console will be: 12 Example4: lab1_1234567.exe 10 4 a which should calculate 10 + 4 = 14, i.e. the last (and only) line on the console will be: 14 ___________________________________________________________________________________________ ___ FUNCTIONAL REQUIREMENTS (to get over approximately 50%): E1. For situations where THREE command line arguments (other than 'a' or 's') are passed to your program: E1.1 If the third argument is 'm', your program must perform a multiplication operation, taking the first two arguments as the operands and display only the result to the console with a new line character. E1.2 If the third argument is 'd', your program must perform a division operation, taking the first two arguments as the operands and display only the result to the console with a new line character. E1.3 If the third argument is 'p', your program must perform an exponential operation, taking the first argument as the base operand and the second as the exponent operand. The result must be display to the console with a new line character. Hint: Consider using the pow() function, which has the definition: double pow(double base, double exponent); Example5: lab1_1234567.exe 10 2 d which should calculate 10 / 2 = 5, i.e. the last (and only) line on the console will be: 5 Example6: lab1_1234567.exe 10 2 p which should calculate 10 to power of 2 = 100, i.e. the last (and only) line on the console will be: 100 NOTE1: DO NOT use the character ^ in your command line arguments as your user input. Question: Why don't we use characters such as + - * / ^ ? to determine the operation? Answer: Arguments passed via the command line are processed by the operating system before being passed to your program. During this process, special characters such as + - * / ^ ? are stripped from the input argument stream. Therefore, the input characters: + - * / ^ ? will not be tested for by the autotester. See sections G6 and E7. NOTE2: the pow() and powl() function/s only work correctly for given arguments. Hence, your code should output and error if there is a domain error or undefined subset of values. For example, if the result does not produce a real number you code should handle this as an error. This means that if the base is negative you can't accept and exponent between (but not including) -1 and 1. If you get this then, output a MURPHY's LAW error: "Y" and return 0; NOTE3: zero to the power of zero is also undefined, and should also be treated MURPHY's LAW error. So return "Y" and return 0; In Visual Studio, the 0 to the power of 0 will return 1, so you will need to catch this situation manually, else your code will likely calculate the value as 1. ___ REQUIRED ERROR HANDLING (to get over approximately 70%): The following text lists errors you must detect and a priority of testing. NB: order of testing is important as each test is slight more difficult than the previous test. All outputs should either be numerical or upper-case single characters (followed by a new line). Note that case is important: In C, 'V' is not the same as 'v'. (No quotes are required on the output). E2. Valid operator input: If the third input argument is not a valid operation selection, the output shall be 'V'. Valid operators are ONLY (case sensitive): a addition s subtraction m multiplication d division p exponentiation i.e. to the power of: 2 to the power of 3 = 8 (base exponent p) E3. Basic invalid number detection (Required): Valid numbers are all numbers that the "average Engineering graduate" in Australia would consider valid. Therefore if the first two arguments are not valid decimal numbers, the output shall be 'X'. For example: -130 is valid +100 is valid 1.3 is valid 3 is valid 0.3 is valid .3 is valid ABC123 is not valid 1.3.4 is not valid 123abc is not valid ___ ERROR HANDLING (not marked by the autotester): E4. Intermediate invalid number detection (NOT TESTED BY AUTOTESTER - for your consideration only): If the first two arguments are not valid decimal numbers, the output shall be 'X'. Using comma punctuated numbers and scientific formatted numbers are considered valid. For example: 0000.111 is valid 3,000 is valid - NB: atof() will read this as '3' not as 3000 1,000.9 is valid - NB: atof() will read this as '1' not as 1000.9 1.23e2 is valid 2E2 is valid -3e-0.5 is not valid (an integer must follow after the e or E for floating point number to be valid) 2E2.1 is not valid e-1 is not valid .e3 is not valid E5. Advanced invalid number detection (NOT TESTED BY AUTOTESTER - for your consideration only): If the first two arguments are not valid decimal numbers, the output shall be 'X'. 1.3e-1 is valid 1,00.0 is valid - NB: if the comma is not removed atof() will read this as '1' not as 100 +212+21-2 is not valid - NB: mathematical operation on a number of numbers, not ONE number 5/2 is not valid - NB: mathematical operation on a number of numbers, not ONE number HINT: consider the function atof(), which has the definition: double atof (const char* str); Checking the user input for multiple operators (i.e. + or -) is quite a difficult task. One method may involve writing a 'for' loop which steps through the input argv[] counting the number of operators. This process could also be used to count for decimal points and the like. The multiple operator check should be considered an advanced task and developed once the rest of the code is operational. E6. Input number range checking: All input numbers must be between (and including) +2^16 (65536) or -2^16 (-65536). If the operand is out of range i.e. too small or too big, the output shall be 'R'. LARGE NUMBERS: is 1.2e+999 acceptable input ? what happens if you enter such a number ? try and see. Hint: #INF error - where and when does it come up ? SMALL NUMBERS: is 1.2e-999 acceptable input ? what happens if you enter such a number ? try and see. Test it by writing your own test program. E7. ERROR checks which will NOT be performed are: E7.1 Input characters such as: *.* or / or \ or : or any of these characters: * / ^ ? will not be tested for. E7.2 Range check: some computer systems accept numbers of size 9999e999999 while others flag and infinity error. An infinity error becomes an invalid input Therefore: input for valid numbers will only be tested to the maximum 9.9e99 (Note: 9.9e99 is out of range and your program should output 'R') E8. Division by zero should produce output 'M' E9. Error precedence: If multiple errors occur during a program execution event, your program should only display one error code followed by a newline character and then exit (using a return 0; statement). In general, the precedence of the error reported to the console should be displayed in the order that they appear within this proforma. However to clarify the exact order or precedence for the error characters, the precedence of the displayed error code should occur in this order: 'P' - Incorrect number of input command line arguments (see M4) 'X' - Invalid numerical command line argument 'V' - Invalid third input argument 'R' - operand (command line argument) value out of range 'M' - Division by zero 'Y' - MURPHY'S LAW (undefined error) Therefore if an invalid numerical command line argument and an invalid operation argument are passed to the program, the first error code should be displayed to the console, which in this case would be 'X'. Displaying 'V' or 'Y' would be result in a loss of marks. E10. ANYTHING ELSE THAT CAN GO WRONG (MURPHY'S LAW TEST): If there are any other kinds of errors not covered here, the output shall be 'Y'. Rhetorical question: What for example are the error codes that the Power function returns ? If this happens then the output shall be 'Y'. See section E1.3, NOTE2. ___________________________________________________________________________________________ ___ HINTS: - Use debug mode and a breakpoint at the return statement prior to program finish in main. - What string conversion routines, do you know how to convert strings to number? Look carefully as they will be needed to convert a command line parameter to a number and also check for errors. - ERROR CHECKING: The basic programming rules are simple (as covered in lectures): 1) check that the input is valid. 2) check that the output is valid. 3) if any library function returns an error code USE IT !!! CHECK FOR IT !!! - Most conversion routines do have inbuilt error checking - USE IT !!! That means: test for the error condition and take some action if the error is true. If that means more than 50% of your code is error checking, then that's the way it has to be. ____________________________________________________________________________________________ */ // These are the libraries you are allowed to use to write your solution. Do not add any // additional libraries as the auto-tester will be locked down to the following: #include <iostream> #include <cstdlib> #include <time.h> #include <math.h> #include <errno.h> // leave this one in please, it is required by the Autotester! // Do NOT Add or remove any #include statements to this project!! // All library functions required should be covered by the above // include list. Do not add a *.h file for this project as all your // code should be included in this file. using namespace std; const double MAXRANGE = pow(2.0, 16.0); // 65536 const double MINRANGE = -pow(2.0, 16.0); // All functions to be defined below and above main() - NO exceptions !!! Do NOT // define function below main() as your code will fail to compile in the auto-tester. // WRITE ANY USER DEFINED FUNCTIONS HERE (optional) // all function definitions and prototypes to be defined above this line - NO exceptions !!! int main(int argc, char *argv[]) { // ALL CODE (excluding variable declarations) MUST come after the following 'if' statement if (argc == 1) { // When run with just the program name (no parameters) your code MUST print // student ID string in CSV format. i.e. // "studentNumber,student_email,student_name" // eg: "3939723,s3939723@student.rmit.edu.au,Yang_Yang" // No parameters on command line just the program name // Edit string below: eg: "studentNumber,student_email,student_name" cout << "3939723,s3939723@student.rmit.edu.au,Yang_Yang" << endl; // Failure of your program to do this cout statement correctly will result in a // flat 10% marks penalty! Check this outputs correctly when no arguments are // passed to your program before you submit your file! Do it as your last test! // The convention is to return Zero to signal NO ERRORS (please do not change it). return 0; } //--- START YOUR CODE HERE. // The convention is to return Zero to signal NO ERRORS (please do not change it). // If you change it the AutoTester will assume you have made some major error. return 0; } // No code to be placed below this line - all functions to be defined above main() function. // End of file.
08-16
【语音分离】基于平均谐波结构建模的无监督单声道音乐声源分离(Matlab代码实现)内容概要:本文介绍了基于平均谐波结构建模的无监督单声道音乐声源分离方法,并提供了相应的Matlab代码实现。该方法通过对音乐信号中的谐波结构进行建模,利用音源间的频率特征差异,实现对混合音频中不同乐器或人声成分的有效分离。整个过程无需标注数据,属于无监督学习范畴,适用于单通道录音场景下的语音与音乐分离任务。文中强调了算法的可复现性,并附带完整的仿真资源链接,便于读者学习与验证。; 适合人群:具备一定信号处理基础和Matlab编程能力的高校学生、科研人员及从事音频处理、语音识别等相关领域的工程师;尤其适合希望深入理解声源分离原理并进行算法仿真实践的研究者。; 使用场景及目标:①用于音乐音频中人声与伴奏的分离,或不同乐器之间的分离;②支持无监督条件下的语音处理研究,推动盲源分离技术的发展;③作为学术论文复现、课程项目开发或科研原型验证的技术参考。; 阅读建议:建议读者结合提供的Matlab代码与网盘资料同步运行调试,重点关注谐波建模与频谱分解的实现细节,同时可扩展学习盲源分离中的其他方法如独立成分分析(ICA)或非负矩阵分解(NMF),以加深对音频信号分离机制的理解。
内容概要:本文系统介绍了新能源汽车领域智能底盘技术的发展背景、演进历程、核心技术架构及创新形态。文章指出智能底盘作为智能汽车的核心执行层,通过线控化(X-By-Wire)和域控化实现驱动、制动、转向、悬架的精准主动控制,支撑高阶智能驾驶落地。技术发展历经机械、机电混合到智能三个阶段,当前以线控转向、线控制动、域控制器等为核心,并辅以传感器、车规级芯片、功能安全等配套技术。文中还重点探讨了“智能滑板底盘”这一创新形态,强调其高度集成化、模块化优势及其在成本、灵活性、空间利用等方面的潜力。最后通过“2025智能底盘先锋计划”的实车测试案例,展示了智能底盘在真实场景中的安全与性能表现,推动技术从研发走向市场验证。; 适合人群:汽车电子工程师、智能汽车研发人员、新能源汽车领域技术人员及对智能底盘技术感兴趣的从业者;具备一定汽车工程或控制系统基础知识的专业人士。; 使用场景及目标:①深入了解智能底盘的技术演进路径与系统架构;②掌握线控技术、域控制器、滑板底盘等关键技术原理与应用场景;③为智能汽车底盘研发、系统集成与技术创新提供理论支持与实践参考。; 阅读建议:建议结合实际车型和技术标准进行延伸学习,关注政策导向与行业测试动态,注重理论与实车验证相结合,全面理解智能底盘从技术构想到商业化落地的全过程。
【顶级EI复现】计及连锁故障传播路径的电力系统 N-k 多阶段双层优化及故障场景筛选模型(Matlab代码实现)内容概要:本文介绍了名为《【顶级EI复现】计及连锁故障传播路径的电力系统 N-k 多阶段双层优化及故障场景筛选模型(Matlab代码实现)》的技术资源,重点围绕电力系统中连锁故障的传播路径展开研究,提出了一种N-k多阶段双层优化模型,并结合故障场景筛选方法,用于提升电力系统在复杂故障条件下的安全性与鲁棒性。该模型通过Matlab代码实现,具备较强的工程应用价值和学术参考意义,适用于电力系统风险评估、脆弱性分析及预防控制策略设计等场景。文中还列举了大量相关的科研技术支持方向,涵盖智能优化算法、机器学习、路径规划、信号处理、电力系统管理等多个领域,展示了广泛的仿真与复现能力。; 适合人群:具备电力系统、自动化、电气工程等相关背景,熟悉Matlab编程,有一定科研基础的研究生、高校教师及工程技术人员。; 使用场景及目标:①用于电力系统连锁故障建模与风险评估研究;②支撑高水平论文(如EI/SCI)的模型复现与算法验证;③为电网安全分析、故障传播防控提供优化决策工具;④结合YALMIP等工具进行数学规划求解,提升科研效率。; 阅读建议:建议读者结合提供的网盘资源,下载完整代码与案例进行实践操作,重点关注双层优化结构与场景筛选逻辑的设计思路,同时可参考文档中提及的其他复现案例拓展研究视野。
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