Lo que todo programador debería saber sobre aritmética de punto flotante
more work
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README.txt
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README.txt
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floating-point.gui.de aims to provide both short and simple
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answers to the common recurring questions of novice programmers
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about floating point numbers not "adding up" correctly, and
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more in-depth information about how IEEE 754 floats work,
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when and how to use them correctly, and what to use instead
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when they are not appropriate.
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The site is build using the nanoc static site generator:
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http://nanoc.stoneship.org/
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floating-point.gui.de aims to provide both short and simple
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answers to the common recurring questions of novice programmers
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about floating point numbers not "adding up" correctly, and
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more in-depth information about how IEEE 754 floats work,
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when and how to use them correctly, and what to use instead
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when they are not appropriate.
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The site is build using the nanoc static site generator:
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http://nanoc.stoneship.org/
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Rules
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Rules
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# don’t filter or layout
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end
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compile '/favicon/' do
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# don’t filter or layout
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end
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compile '/logo/' do
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# don’t filter or layout
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end
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compile '*' do
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filter :erb
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filter :kramdown
···
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route '/stylesheet/' do
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'/style.css'
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end
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route '/favicon/' do
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'/favicon.ico'
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end
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route '/logo/' do
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'/logo.png'
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end
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route '*' do
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content/basic.html
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content/basic.html
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---
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title: Basic Answers
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---
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<%= @item[:title] %>
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====================
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### Why don't my numbers, like 0.1 + 0.2 add up to a nice round 0.3, and instead I get a weird result like 0.30000000000000004?
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···
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It's not stupid, just different. Decimal numbers cannot accurately
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represent a number like 1/3, so you have to round to 0.33 - and you
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don't expect 0.33 + 0.33 + 0.33 to result in 1, either.
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don't expect 0.33 + 0.33 + 0.33 to add up to 1, either - do you?
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Computers use binary numbers because they're faster at dealing with
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those, and because for most calculations, a tiny error in the 17th
···
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### What can I do to avoid this problem?
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That depends on what kind of calculations you're doing:
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That depends on what kind of calculations you're doing.
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* If you are doing technical calculations with short numbers and just don't want to end up with all those extra decimal places: simply [round your result](/rounding) to a fixed number of decimal places.
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* If you really need your results to add up exactly, especially when you work with money: use a special [decimal datatype](/decimal). Look at the "languages" section for details.
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* If you just don't want to end up with all those extra decimal places: simply [round your result](/rounding) to a fixed number of decimal places.
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* If you have no decimal datatype available, an alternative is to work with integers, e.g. do money calculations entirely in cents. But this is more work and has some drawbacks.
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content/favicon.ico
content/favicon.ico
This is a binary file and will not be displayed.
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content/fp.html
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content/fp.html
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---
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title: A New Item
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title: Floating Point Numbers
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---
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How to mess with people who've come to *expect* rounding errors in floating point math.
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Floating Point Numbers
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======================
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Binary Fractions
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----------------
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As a programmer, you are familiar with the concept of binary integers, i.e.
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the representation of integer numbers as a series of bits:
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<table>
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<tr><th colspan="9">Decimal (<span class="num_base">base 10</span>)</th><th></th><th colspan="17">Binary (<span class="num_base">base 2</span>)</th></tr>
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<tr class="base_example">
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<td class="digit">1</td><td>⋅</td><td class="num_base">10<sup>1</sup></td><td>+</td>
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<td class="digit">3</td><td>⋅</td><td class="num_base">10<sup>0</sup></td><td>=</td>
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<td class="digit">13<sub class="num_base">10</sub></td><td class="separator">=</td>
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<td class="digit">1101<sub class="num_base">2</sub></td><td>=</td>
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<td class="digit">1</td><td>⋅</td><td class="num_base">2<sup>3</sup></td><td>+</td>
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<td class="digit">1</td><td>⋅</td><td class="num_base">2<sup>2</sup></td><td>+</td>
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<td class="digit">0</td><td>⋅</td><td class="num_base">2<sup>1</sup></td><td>+</td>
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<td class="digit">1</td><td>⋅</td><td class="num_base">2<sup>0</sup></td>
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</tr><tr class="base_example">
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<td class="digit">1</td><td>⋅</td><td>10</td><td>+</td>
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<td class="digit">3</td><td>⋅</td><td>1 </td><td>=</td>
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<td class="digit">13<sub class="num_base">10</sub></td><td class="separator">=</td>
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<td class="digit">1101<sub class="num_base">2</sub></td><td>=</td>
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<td class="digit">1</td><td>⋅</td><td>8</td><td>+</td>
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<td class="digit">1</td><td>⋅</td><td>4</td><td>+</td>
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<td class="digit">0</td><td>⋅</td><td>2</td><td>+</td>
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<td class="digit">1</td><td>⋅</td><td>0</td>
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</tr></table>
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This is how computers store integer numbers internally. And for fractional numbers, they do the same thing:
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content/languages/java.html
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content/languages/java.html
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Decimal Types
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-------------
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Java has an arbitrary-precision decimal type named <code>BigDecimal</code>, which
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also allows to choose the rounding mode.
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Java has an arbitrary-precision decimal type named <code>java.math.BigDecimal</code>, which
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also allows to choose the rounding mode:
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new BigDecimal("0.1").add(new BigDecimal("0.2")) // returns a BigDecimal representing exactly 0.3
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How to Round
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------------
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To get a String:
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String.format("%.2f", 1.2399) // returns "1.24"
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String.format("%.3f", 1.2399) // returns "1.240"
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String.format("%.2f", 1.2) // returns "1.20"
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To print to standard output (or any <code>PrintStream</code>):
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System.out.printf("%.2f", 1.2399) // same syntax as String.format()
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If you don't want trailing zeroes:
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new DecimalFormat("0.00").format(1.2)// returns "1.20"
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new DecimalFormat("0.##").format(1.2)// returns "1.2"
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content/logo.png
content/logo.png
This is a binary file and will not be displayed.
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content/stylesheet.css
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content/stylesheet.css
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background: #ffffee;
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}
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table {
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border: 1px solid black;
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border-collapse: collapse;
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}
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td{
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border: 1px solid black;
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padding: 5px;
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}
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tr.base_example td{
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border-left:none;
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border-right:none;
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border-top: 1px solid black;
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border-bottom: 1px solid black;
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padding: 2px;
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vertical-align:bottom;
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}
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tr.base_example td.separator{
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border: 1px solid black;
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padding: 5px;
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}
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tr.base_example td.digit{
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font-weight:bold;
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color: #ff0000
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}
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th {
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border: 1px solid black;
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padding: 5px;
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background: #fff0f0;
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text-align: left;
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}
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.num_base{
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color: #0000ff
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}
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.num_base sup{
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color: #000000
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}
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#main {
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position: absolute;
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···
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#main h1 {
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font-size: 30px;
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font-weight: normal;
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line-height: 30px;
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letter-spacing: -1px;
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}
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#main p {
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content/xkcd.html
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content/xkcd.html
···
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---
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title: xkcd
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---
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e<sup>π</sup> - π
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====================
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[xkcd](http://www.xkcd.com/217/): How to mess with people who've come to *expect* rounding errors in floating point math.
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layouts/default.html
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layouts/default.html
···
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<meta charset="utf-8">
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<title>The Floating-Point Guide - <%= @item[:title] %></title>
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<link rel="stylesheet" type="text/css" href="/style.css" media="screen">
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<link rel="shortcut icon" href="/favicon.ico" type="image/vnd.microsoft.icon">
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<meta name="generator" content="nanoc 3.1.2">
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</head>
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<body>
···
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<%= yield %>
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</div>
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<div id="sidebar">
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<img src="/logo.png">
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<h2>The Floating-Point Guide</h2>
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<ul>
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<li><a href="/">Home</a></li>
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<li><a href="/basic">Basic Answers</a></li>
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<li><a href="/fp">Floating Point</a></li>
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<li><a href="/decimal">Decimal types</a></li>
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<li><a href="/rounding">Rounding</a></li>
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<li><a href="/basic/">Basic Answers</a></li>
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<li><a href="/fp/">Floating Point</a></li>
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<li><a href="/decimal/">Decimal types</a></li>
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<li><a href="/rounding/">Rounding</a></li>
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<li><a href="/xkcd/">xkcd</a></li>
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</ul>
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<h2>Languages</h2>
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<ul>
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<li><a href="/languages/csharp">C#</a></li>
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<li><a href="/languages/java">Java</a></li>
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<li><a href="/languages/javascript">JavaScript</a></li>
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<li><a href="/languages/php">PHP</a></li>
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<li><a href="/languages/csharp/">C#</a></li>
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<li><a href="/languages/java/">Java</a></li>
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<li><a href="/languages/javascript/">JavaScript</a></li>
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<li><a href="/languages/php/">PHP</a></li>
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</ul>
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</div>
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</body>