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'''Hypot''' is a mathematical function defined to calculate the length of the [[hypotenuse]] of a right-angle triangle. It was designed to avoid errors arising due to limited-precision calculations performed on computers.
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Calculation of the length of the hypotenuse of a triangle is possible to do using the square root function but hypot(''x'',&nbsp;''y'') avoids possible problems with very large or very small numbers.
 
The magnitude of the hypotenuse from (0,&nbsp;0) to (''x'',&nbsp;''y'') can be calculated using:
 
: <math>r = \sqrt { x^2 + y^2 } \, </math>
 
However the squares of very large or small values of ''x'' and ''y'' may exceed the range of machine precision when calculated on a computer, leading to an inaccurate result (see [[Arithmetic underflow|underflow]], [[Arithmetic overflow|overflow]]). The hypot function was designed to calculate the result without causing this problem.
 
The hypot function may typically be used together with the [[atan2]] function to convert from [[Cartesian coordinate system|Cartesian]] to [[polar coordinates]]:
 
:&nbsp;''r''&nbsp;=&nbsp;hypot(''x'',&nbsp;''y'')&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;''θ''&nbsp;=&nbsp;atan2(''y'',&nbsp;''x'')
 
This operation is also known as [[Pythagorean addition]].
 
== Implementation ==
The difficulty with the naive implementation is that ''x''<sup>2</sup> or ''y''<sup>2</sup> may over- or underflow, unless the intermediate result is computed with [[extended precision]].  A common implementation technique is to exchange the values, if necessary, so that |''x''|&nbsp;>&nbsp;|''y''|, and then use the equivalent form:
 
: <math>\begin{align}
r & = \sqrt { x^2 + y^2 } \\
  & = \sqrt { x^2 ( 1 + (y/x)^2) } \\
  & = |x|  \sqrt {1 + (y/x)^2 }
\end{align}</math>
 
The computation of ''y''/''x'' cannot overflow, and underflows compute the correct result. The square root is computed over a value between 1 and 2.  Finally, the multiplication by |''x''| cannot underflow, and overflows only when the result is too large to represent.
 
Pseudocode:
<pre>
double hypot(double x,double y)
{
    double t;
    x = abs(x);
    y = abs(y);
    t = min(x,y);
    x = max(x,y);
    t = t/x;
    return x*sqrt(1+t*t);
}
</pre>
 
== Programming language support==
The function is present in several programming languages:
* [[C99]]
* [[C++11]]
* [[Fortran 2008]]
* Python
* Apple's PowerPC Numerics <ref>http://developer.apple.com/DOCUMENTATION/mac/PPCNumerics/PPCNumerics-141.html</ref>
* MATLAB<ref>http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/hypot.html</ref>
* Pascal <ref>http://www.frameworkpascal.com/helphtml/hypot_func.htm</ref>
* PHP<ref>http://www.php.net/hypot</ref>
* Java (since version 1.5)<ref>http://java.sun.com/j2se/1.5.0/docs/api/java/lang/Math.html#hypot(double,%20double)</ref>
* Ruby <ref>http://www.ruby-doc.org/core/classes/Math.html#M001470</ref>
* Go <ref>http://golang.org/pkg/math/#Hypot</ref>
* Rust <ref>http://static.rust-lang.org/doc/std/num.html#function-hypot</ref>
* Javascript <ref>https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Math/hypot</ref>
 
Some C90 and C++ libraries have provided a hypot function.<ref>Single Unix Specification, Open Group, http://www.opengroup.org/onlinepubs/007908799/xsh/hypot.html</ref><ref>IBM, ILE C/C++ Run-Time Library Functions, http://publib.boulder.ibm.com/infocenter/iadthelp/v7r0/index.jsp?topic=/com.ibm.etools.iseries.langref.doc/rzan5mst144.htm</ref><ref>The GNU C Library, Mathematics, http://www.cs.utah.edu/dept/old/texinfo/glibc-manual-0.02/library_17.html</ref>
 
== See also ==
* [[Alpha max plus beta min algorithm]]
 
== References ==
<references/>
 
[[Category:Trigonometry]]
[[Category:Numerical analysis]]

Latest revision as of 05:56, 15 November 2014

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