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Interval2.html
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<link rel="stylesheet" href="style.css" type="text/css">
<meta content="text/html; charset=iso-8859-1" http-equiv="Content-Type">
<meta name="viewport" content="width=device-width, initial-scale=1">
<link rel="Start" href="index.html">
<link rel="previous" href="Interval1.html">
<link rel="Up" href="index.html">
<link title="Index of types" rel=Appendix href="index_types.html">
<link title="Index of values" rel=Appendix href="index_values.html">
<link title="Index of modules" rel=Appendix href="index_modules.html">
<link title="Interval1" rel="Chapter" href="Interval1.html">
<link title="Interval2" rel="Chapter" href="Interval2.html"><link title="Interval operations" rel="Section" href="#6_Intervaloperations">
<link title="Floating-point operations with directed rounding" rel="Section" href="#6_Floatingpointoperationswithdirectedrounding">
<title>Interval2</title>
</head>
<body>
<div class="navbar"><a class="pre" href="Interval1.html" title="Interval1">Previous</a>
<a class="up" href="index.html" title="Index">Up</a>
</div>
<h1>Module <a href="type_Interval2.html">Interval2</a></h1>
<pre><span class="keyword">module</span> Interval2: <code class="code">sig</code> <a href="Interval2.html">..</a> <code class="code">end</code></pre><div class="info module top">
A simple OCaml interval library.
<p>
This interval library needs the OCaml <code class="code">Num</code> module.
<p>
It is assumed that all floating-point operations are IEEE 754
compatible and the rounding mode is to nearest.
<p>
It is also assumed that OCaml functions <code class="code">exp</code>, <code class="code">log</code>, <code class="code">atan</code> compute results with
less than 1 ulp error.
<p>
Intervals computed with this library are optimal floating-point
intervals for basic arithmetic operations.
<p>
<a href="Interval1.html"><code class="code">Interval1</code></a> provides faster interval functions which are only
slightly less optimal.<br>
</div>
<hr width="100%">
<pre><code><span id="TYPEinterval"><span class="keyword">type</span> <code class="type"></code>interval</span> = {</code></pre><table class="typetable">
<tr>
<td align="left" valign="top" >
<code> </code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTinterval.low">low</span> : <code class="type">float</code>;</code></td>
</tr>
<tr>
<td align="left" valign="top" >
<code> </code></td>
<td align="left" valign="top" >
<code><span id="TYPEELTinterval.high">high</span> : <code class="type">float</code>;</code></td>
</tr></table>
}
<div class="info ">
The interval type<br>
</div>
<pre><span id="VALempty_interval"><span class="keyword">val</span> empty_interval</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
The empty interval<br>
</div>
<pre><span id="VALentire_interval"><span class="keyword">val</span> entire_interval</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
The entire interval representing (-infinity, infinity)<br>
</div>
<pre><span id="VALzero_interval"><span class="keyword">val</span> zero_interval</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
<code class="code">[0., 0.]</code><br>
</div>
<pre><span id="VALone_interval"><span class="keyword">val</span> one_interval</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
<code class="code">[1., 1.]</code><br>
</div>
<br>
<h6 id="6_Intervaloperations">Interval operations</h6><br>
<pre><span id="VALmake_interval"><span class="keyword">val</span> make_interval</span> : <code class="type">float -> float -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Creates an interval from given endpoints<br>
</div>
<pre><span id="VALis_empty"><span class="keyword">val</span> is_empty</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> bool</code></pre><div class="info ">
Tests if an interval is empty<br>
</div>
<pre><span id="VALis_entire"><span class="keyword">val</span> is_entire</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> bool</code></pre><div class="info ">
Tests if an interval is the entire interval<br>
</div>
<pre><span id="VALis_valid"><span class="keyword">val</span> is_valid</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> bool</code></pre><div class="info ">
Tests if an interval is valid. A valid interval is either empty
or <code class="code">[a, b]</code> with <code class="code">a <= b</code>, <code class="code">a < infinity</code>, <code class="code">-infinity < b</code>.<br>
</div>
<pre><span id="VALmid_i"><span class="keyword">val</span> mid_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> float</code></pre><div class="info ">
Computes a midpoint of an interval. This function returns finite
values for all valid non-empty intervals.<br>
</div>
<pre><span id="VALneg_i"><span class="keyword">val</span> neg_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval negation <b>(optimal)</b><br>
</div>
<pre><span id="VALabs_i"><span class="keyword">val</span> abs_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval absolute value <b>(optimal)</b><br>
</div>
<pre><span id="VALmax_ii"><span class="keyword">val</span> max_ii</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval maximum <b>(optimal)</b><br>
</div>
<pre><span id="VALmin_ii"><span class="keyword">val</span> min_ii</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval minimum <b>(optimal)</b><br>
</div>
<pre><span id="VALadd_ii"><span class="keyword">val</span> add_ii</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval addition <b>(optimal)</b><br>
</div>
<pre><span id="VALadd_id"><span class="keyword">val</span> add_id</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> float -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Addition of an interval and a number <b>(optimal)</b><br>
</div>
<pre><span id="VALadd_di"><span class="keyword">val</span> add_di</span> : <code class="type">float -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Addition of a number and an interval <b>(optimal)</b><br>
</div>
<pre><span id="VALsub_ii"><span class="keyword">val</span> sub_ii</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval subtraction <b>(optimal)</b><br>
</div>
<pre><span id="VALsub_id"><span class="keyword">val</span> sub_id</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> float -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Subtraction of an interval and a number <b>(optimal)</b><br>
</div>
<pre><span id="VALsub_di"><span class="keyword">val</span> sub_di</span> : <code class="type">float -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Subtraction of a number and an interval <b>(optimal)</b><br>
</div>
<pre><span id="VALmul_ii"><span class="keyword">val</span> mul_ii</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval multiplication <b>(optimal)</b><br>
</div>
<pre><span id="VALmul_id"><span class="keyword">val</span> mul_id</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> float -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Multiplication of an interval and a number <b>(optimal)</b><br>
</div>
<pre><span id="VALmul_di"><span class="keyword">val</span> mul_di</span> : <code class="type">float -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Multiplication of a number and an interval <b>(optimal)</b><br>
</div>
<pre><span id="VALdiv_ii"><span class="keyword">val</span> div_ii</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval division <b>(optimal)</b><br>
</div>
<pre><span id="VALdiv_id"><span class="keyword">val</span> div_id</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> float -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Division of an interval by a number <b>(optimal)</b><br>
</div>
<pre><span id="VALdiv_di"><span class="keyword">val</span> div_di</span> : <code class="type">float -> <a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Division of a number by an interval <b>(optimal)</b><br>
</div>
<pre><span id="VALinv_i"><span class="keyword">val</span> inv_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval reciprocal <b>(optimal)</b><br>
</div>
<pre><span id="VALsqrt_i"><span class="keyword">val</span> sqrt_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval square root <b>(optimal)</b><br>
</div>
<pre><span id="VALsqr_i"><span class="keyword">val</span> sqr_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval square <b>(optimal)</b><br>
</div>
<pre><span id="VALpown_i"><span class="keyword">val</span> pown_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> int -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval integer power. This function returns an optimal interval
but this behavior may change in the future.<br>
</div>
<pre><span id="VALexp_i"><span class="keyword">val</span> exp_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval exponential function. It is assumed that the standard
function <code class="code">exp:float->float</code> has less than 1 ulp error.<br>
</div>
<pre><span id="VALlog_i"><span class="keyword">val</span> log_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval natural logarithm. It is assumed that the standard
function <code class="code">log:float->float</code> has less than 1 ulp error.<br>
</div>
<pre><span id="VALatan_i"><span class="keyword">val</span> atan_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval arctangent. It is assumed that the standard
function <code class="code">atan:float->float</code> has less than 1 ulp error.<br>
</div>
<pre><span id="VALsin_i"><span class="keyword">val</span> sin_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval sine (not implemented yet)<br>
</div>
<pre><span id="VALcos_i"><span class="keyword">val</span> cos_i</span> : <code class="type"><a href="Interval2.html#TYPEinterval">interval</a> -> <a href="Interval2.html#TYPEinterval">interval</a></code></pre><div class="info ">
Interval cosine (not implemented yet)<br>
</div>
<br>
<h6 id="6_Floatingpointoperationswithdirectedrounding">Floating-point operations with directed rounding</h6><br>
<pre><span id="VALfsucc"><span class="keyword">val</span> fsucc</span> : <code class="type">float -> float</code></pre><div class="info ">
Computes a successor of a floating-point number <b>(optimal)</b><br>
</div>
<pre><span id="VALfpred"><span class="keyword">val</span> fpred</span> : <code class="type">float -> float</code></pre><div class="info ">
Computes a predecessor of a floating-point number <b>(optimal)</b><br>
</div>
<pre><span id="VALfadd_low"><span class="keyword">val</span> fadd_low</span> : <code class="type">float -> float -> float</code></pre><div class="info ">
Returns a lower bound of the sum of two floating-point numbers <b> (optimal)</b><br>
</div>
<pre><span id="VALfadd_high"><span class="keyword">val</span> fadd_high</span> : <code class="type">float -> float -> float</code></pre><div class="info ">
Returns an upper bound of the sum of two floating-point numbers <b> (optimal)</b><br>
</div>
<pre><span id="VALfsub_low"><span class="keyword">val</span> fsub_low</span> : <code class="type">float -> float -> float</code></pre><div class="info ">
Returns a lower bound of the difference of two floating-point
numbers <b>(optimal)</b><br>
</div>
<pre><span id="VALfsub_high"><span class="keyword">val</span> fsub_high</span> : <code class="type">float -> float -> float</code></pre><div class="info ">
Returns an upper bound of the difference of two floating-point
numbers <b>(optimal)</b><br>
</div>
<pre><span id="VALfmul_low"><span class="keyword">val</span> fmul_low</span> : <code class="type">float -> float -> float</code></pre><div class="info ">
Returns a lower bound of the product of two floating-point numbers
<b>(optimal)</b><br>
</div>
<pre><span id="VALfmul_high"><span class="keyword">val</span> fmul_high</span> : <code class="type">float -> float -> float</code></pre><div class="info ">
Returns an upper bound of the product of two floating-point
numbers <b>(optimal)</b><br>
</div>
<pre><span id="VALfdiv_low"><span class="keyword">val</span> fdiv_low</span> : <code class="type">float -> float -> float</code></pre><div class="info ">
Returns a lower bound of the ratio of two floating-point numbers
<b>(optimal)</b><br>
</div>
<pre><span id="VALfdiv_high"><span class="keyword">val</span> fdiv_high</span> : <code class="type">float -> float -> float</code></pre><div class="info ">
Returns an upper bound of the ratio of two floating-point numbers
<b>(optimal)</b><br>
</div>
<pre><span id="VALfsqr_low"><span class="keyword">val</span> fsqr_low</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns a lower bound of <code class="code">x^2</code> <b>(optimal)</b><br>
</div>
<pre><span id="VALfsqr_high"><span class="keyword">val</span> fsqr_high</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns an upper bound of <code class="code">x^2</code> <b>(optimal)</b><br>
</div>
<pre><span id="VALfsqrt_low"><span class="keyword">val</span> fsqrt_low</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns a lower bound of <code class="code">sqrt x</code> <b>(optimal)</b><br>
</div>
<pre><span id="VALfsqrt_high"><span class="keyword">val</span> fsqrt_high</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns an upper bound of <code class="code">sqrt x</code> <b>(optimal)</b><br>
</div>
<pre><span id="VALfexp_low"><span class="keyword">val</span> fexp_low</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns a lower bound of <code class="code">exp x</code><br>
</div>
<pre><span id="VALfexp_high"><span class="keyword">val</span> fexp_high</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns an upper bound of <code class="code">exp x</code><br>
</div>
<pre><span id="VALflog_low"><span class="keyword">val</span> flog_low</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns a lower bound of <code class="code">log x</code><br>
</div>
<pre><span id="VALflog_high"><span class="keyword">val</span> flog_high</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns an upper bound of <code class="code">log x</code><br>
</div>
<pre><span id="VALfatan_low"><span class="keyword">val</span> fatan_low</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns a lower bound of <code class="code">atan x</code><br>
</div>
<pre><span id="VALfatan_high"><span class="keyword">val</span> fatan_high</span> : <code class="type">float -> float</code></pre><div class="info ">
Returns an upper bound of <code class="code">atan x</code><br>
</div>
<pre><span id="VALfpown_low"><span class="keyword">val</span> fpown_low</span> : <code class="type">float -> int -> float</code></pre><div class="info ">
Returns a lower bound of <code class="code">x^n</code><br>
</div>
<pre><span id="VALfpown_high"><span class="keyword">val</span> fpown_high</span> : <code class="type">float -> int -> float</code></pre><div class="info ">
Returns an upper bound of <code class="code">x^n</code><br>
</div>
</body></html>