If you don't care if how much less than 1% it is, how about using Math.max()
to just start at one percent and then start making meaningful progress after that.
Demo in Stack Snippets
<!-- begin snippet: js hide: false -->
<!-- language: lang-js -->
var $traveled = $("#traveledMiles"),
$total = $("#totalMiles"),
$progressBar = $('.progress-bar');
$($traveled).add($total).keyup(function(){
var percent = $traveled.val() / $total.val() * 100;
// at least 1%
percent = Math.max(percent,1);
$progressBar
.css('width', percent+'%')
.attr('aria-valuenow', percent)
.children(".sr-only").html(percent+'%');
}).keyup();
<!-- language: lang-html -->
<link href="//cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.2/css/bootstrap.css" rel="stylesheet"/>
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.js"></script>
<script src="//cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.2/js/bootstrap.js"></script>
<div class="form-group">
<label for="traveledMiles">Traveled Miles</label>
<input type="text" class="form-control" id="traveledMiles"
placeholder="Enter Traveled Miles" value="5">
</div>
<div class="form-group">
<label for="totalMiles">Total Miles</label>
<input type="text" class="form-control" id="totalMiles"
placeholder="Enter Total Miles" value="23872">
</div>
<div class="progress">
<div class="progress-bar" role="progressbar" style="width: 60%;"
aria-valuenow="60" aria-valuemin="0" aria-valuemax="100">
<span class="sr-only">60%</span>
</div>
</div>
<!-- end snippet -->
Since you have a label, you could even follow Bootstrap's suggestion:
To ensure that the label text remains legible even for low percentages, consider adding a min-width to the progress bar.
Like this:
<!-- language: lang-html -->
<div class="progress-bar" style="min-width: 2em;">
<!-- begin snippet: js hide: true -->
<!-- language: lang-js -->
var $traveled = $("#traveledMiles"),
$total = $("#totalMiles"),
$progressBar = $('.progress-bar');
$($traveled).add($total).keyup(function(){
var percent = $traveled.val() / $total.val() * 100;
// at least 1%
percent = Math.max(percent,1);
// round off decimals
percent = Math.round(percent);
// max of 100%
percent = Math.min(percent,100);
$progressBar
.css('width', percent+'%')
.attr('aria-valuenow', percent)
.html(percent+'%');
}).keyup();
<!-- language: lang-html -->
<link href="//cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.2/css/bootstrap.css" rel="stylesheet"/>
<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.3/jquery.js"></script>
<script src="//cdnjs.cloudflare.com/ajax/libs/twitter-bootstrap/3.3.2/js/bootstrap.js"></script>
<div class="form-group">
<label for="traveledMiles">Traveled Miles</label>
<input type="text" class="form-control" id="traveledMiles"
placeholder="Enter Traveled Miles" value="5">
</div>
<div class="form-group">
<label for="totalMiles">Total Miles</label>
<input type="text" class="form-control" id="totalMiles"
placeholder="Enter Total Miles" value="23872">
</div>
<div class="progress">
<div class="progress-bar" role="progressbar" style="min-width: 2em;width: 60%;"
aria-valuenow="60" aria-valuemin="0" aria-valuemax="100">
60%
</div>
</div>
<!-- end snippet -->