± 误差传递计算器（任意公式、和/差/积/商/幂）Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power)
输入公式（如 a*b/c、a^2*sin(b)）和各变量的测量值与不确定度，按一阶泰勒展开（偏导数平方和开方）计算结果的不确定度、相对不确定度和各变量的贡献；也可以用和、差、积、商、幂的快速公式。假设各变量误差独立。本地处理。Enter a formula (such as a*b/c or a^2*sin(b)) with each variable value and uncertainty to get the uncertainty of the result by first-order Taylor expansion (root sum of squared partial terms), the relative uncertainty and each variable contribution, or use the quick rules for sums, differences, products, quotients and powers. Errors are assumed independent. Processed locally.
使用说明与常见问题How-to & FAQ
「误差传递计算器（任意公式、和/差/积/商/幂）」是什么？What is the Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power) tool?
输入公式（如 a*b/c、a^2*sin(b)）和各变量的测量值与不确定度，按一阶泰勒展开（偏导数平方和开方）计算结果的不确定度、相对不确定度和各变量的贡献；也可以用和、差、积、商、幂的快速公式。假设各变量误差独立。本地处理。Enter a formula (such as a*b/c or a^2*sin(b)) with each variable value and uncertainty to get the uncertainty of the result by first-order Taylor expansion (root sum of squared partial terms), the relative uncertainty and each variable contribution, or use the quick rules for sums, differences, products, quotients and powers. Errors are assumed independent. Processed locally.
如何使用「误差传递计算器（任意公式、和/差/积/商/幂）」？How do I use the Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power) tool?
在「误差传递计算器（任意公式、和/差/积/商/幂）」中，填写「运算」、「a」、「δa」、「b」、「δb」等参数，结果会随输入实时计算并显示在下方。In the Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power): Fill in "Operation", "a", "δa", "b", "δb"; results are recalculated instantly as you type.
常用规则有哪些？What are the basic rules?
和与差：δ = √(δa² + δb²)；积与商：δ/|y| = √((δa/a)² + (δb/b)²)；幂 aⁿ：δ/|y| = |n| × δa/|a|。前提是各误差相互独立、比较小。Sums and differences: δ = √(δa² + δb²). Products and quotients: δ/|y| = √((δa/a)² + (δb/b)²). Powers aⁿ: δ/|y| = |n| × δa/|a|. These assume small, independent errors.
任意公式怎么算？How does the any-formula mode work?
δf = √Σ(∂f/∂xᵢ × δxᵢ)²，偏导数用数值差分求出。表格列出每个变量的贡献，可以看出哪个测量对结果误差影响最大。δf = √Σ(∂f/∂xᵢ × δxᵢ)², with partial derivatives found numerically. The table shows each variable contribution, so you can see which measurement matters most.
变量相关时怎么办？What if the variables are correlated?
本页假设误差独立。如果变量之间有相关性，需要加上协方差项，结果可能更大或更小。This page assumes independent errors. Correlated variables need covariance terms, which can make the result larger or smaller.
能举个计算实例吗？Can you show a worked example?
例如在「误差传递计算器（任意公式、和/差/积/商/幂）」中填 运算 = 和 a + b，a = 10，δa = 0.1，b = 5，δb = 0.2，结果区显示：结果：15 ± 0.2236；不确定度 δ：0.223607；相对不确定度：1.4907 %For example, entering Operation = Sum, a = 10, δa = 0.1, b = 5, δb = 0.2 in the Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power) gives: Result: 15 ± 0.2236; Uncertainty δ: 0.223607; Relative uncertainty: 1.4907 %
哪些输入会被检查？Which inputs are validated?
「误差传递计算器（任意公式、和/差/积/商/幂）」会检查这些输入——「a」、「指数 n」：所选方式用到时必须填写；「δa」：必须填写；「b」、「δb」、「c」、「δc」、「d」、「δd」：可留空；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power) checks its inputs — "a", "Exponent n": required when the chosen option uses it; "δa": required; "b", "δb", "c", "δc", "d", "δd": optional; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.
有哪些选项可以调整？What options can I adjust?
「误差传递计算器（任意公式、和/差/积/商/幂）」的设置：「运算」可选 和 a + b、差 a − b、积 a·b、商 a、幂 aⁿ、任意公式（默认 和 a + b）；「a」默认 10；「δa」默认 0.1；「b」默认 5；「δb」默认 0.2；「指数 n」默认 2；「公式（变量 a、b、c、d）」默认 a*b/c；「c」默认 2；「δc」默认 0.05；「d」默认 0；「δd」默认 0。每次生成结果都按当前设置处理。Settings in the Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power) — "Operation": Sum, Difference, Product, b / Quotient, Power, Any formula (default Sum); "a" default 10; "δa" default 0.1; "b" default 5; "δb" default 0.2; "Exponent n" default 2; "Formula (variables a, b, c, d)" default a*b/c; "c" default 2; "δc" default 0.05; "d" default 0; "δd" default 0. The result always uses the current settings.
计算精度如何？How precise are the results?
「误差传递计算器（任意公式、和/差/积/商/幂）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字）；极大或极小的数值可能存在精度误差。The Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power) uses double-precision floating point (about 15–16 significant digits); very large or very small values may lose precision.
「误差传递计算器（任意公式、和/差/积/商/幂）」免费吗？需要注册吗？Is the Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power) free? Do I need an account?
「误差传递计算器（任意公式、和/差/积/商/幂）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Error Propagation Calculator (Any Formula, Sum, Difference, Product, Quotient, Power) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.