Writing xml tags with SimpleXml

I have this xml file:

<?xml version="1.0"?>
<products>
<product>
  ... some nodes 
</product>

<product>
  ... some nodes 
</product>

<product>
  ... some nodes 
</product>
</products>

And i wrote this php code:

<?php
....
....
$xml=simplexml_load_file("myfile.xml") or die("Error: Cannot create object");

foreach ($xml->xpath("/products/product") as $prod) 
{
    $xmlname = $xml->addChild("name",$name);
    $xmlid = $xml->addChild("id",$id);
    $xmlprice = $xml->addChild("price",$price);
}
$xml->asXML ("result_file.xml");
?>

This works, but not as I want. It creates the file 'result_file.xml' with new nodes: $name,$id and $price. The problem is that all these nodes are inserted at the end of the xml file like this:

<?xml version="1.0"?>
<products>
<product>
  ... some nodes 
</product>

<product>
  ... some nodes 
</product>

<product>
  ... some nodes 
</product>
  <name>$name</name>
  <id>$id</id>
  <price>$price</price>
  <name>$name</name>
  <id>$id</id>
  <price>$price</price>
  <name>$name</name>
  <id>$id</id>
  <price>$price</price>
</products>

But I'd like to insert all these nodes after their product node. For example, i want to obtain:

<?xml version="1.0"?>
<products>
<product>
  <name>$name</name>
  <id>$id</id>
  <price>$price</price>
  ... some nodes 
</product>

<product>
  <name>$name</name>
  <id>$id</id>
  <price>$price</price>
  ... some nodes 
</product>

<product>
  <name>$name</name>
  <id>$id</id>
  <price>$price</price>
  ... some nodes 
</product>
</products>