You still don't legally get a patent license; if you don't have one and HP decides to drop a patent suit on you (say their printer business flops and they need cash), you're toast.
You don't get a patent license, I know. What I meant was that if a company is willing to license a patent by putting a so-patented bit of code under the Apache license, then they are likely to be willing to explicitly give a patent license to reimplementors as well.
However, can you incorporate a minimal portion of the HP code to get the patent license? Maybe just use one function or something, just to make it a derived work?