"*I do not speak of fools, I speak of the wisest men; and it is among them that the imagination has the great gift of persuasion. Reason protests in vain; it cannot set a true value on things.*"

"*Absolutely amazing that the US Patent Office would grant a patent for an idea..."*

Absolutely true. The US Patent Office does not grant patents for ideas or algorithms. They issue patents for the embodiment of ideas: real, physical things. Not an idea like unums.

"... *published in a very well-received book (The End of Error: Unum Arithmetic) in February 2015."*

Published, yes. "Well-received book" is equivocal. William Kahan's review, "A Critique of John L. Gustafson’s THE END of ERROR — Unum Computation and his A Radical Approach to Computation with Real Numbers" and "Commentary on 'THE END of ERROR — Unum Computing' by John L. Gustafson, (2015) CRC Press" cite a number of deficiencies. The "Transcription of 'The Great Debate': John Gustafson vs. William Kahan on Unum Arithmetic, Held July 12, 2016, Moderated by Jim Demmel" poses many questions that are not answered with visible and extensive scientific proof.

"*All three forms of unum arithmetic are open source...*"

Three forms? I certainly have had difficulty identifying different forms of unums. It strikes me that unums are variable format with the normal sign, and variable length exponent, and variable length significand, with one ubit, and variable length fields identifying the exponent and significand lengths. Seems to me that the number of forms of unums could increase exponentially. Did I miss something here? Open source? I probably need to research this more. Google search for __open source unums __didn't produce anything useful. I'd certainly like to try it out and compare results.

". . . *and free of patent restrictions* . . ."

No problem there. His "concept" is probably not patentable. I would argue that his "concept" could be implemented as an algorithm and that as such, should be patentable, but I don't think an algorithm is, in general, patentable under current US law.

"*For Jorgensen to claim to be the inventor of this concept is pretty outrageous.*"

And here we are again with unprofessional personal attacks. I guess reason is insufficient to make the point. I have never claimed to invent the concept of bounding error. As a matter of point, one does not "invent" a concept, but rather has an idea that can be conceptualized. What I invented was a new, unique, and original device that computes bounds on real numbers stored in a fixed format and with a single instruction. None of these claims are made by Professor Gustafson. I would never say that he was outrageous to say that he had.

- A machine with registers of standard widths that contain bounded floating point (BFP) numbers as operands and results with fields for sign, exponent, significand, and a bound with subfields for lost bits and rounding error accumulation with subfields for the accumulated rounding error and bits lost due to rounding error.
- A machine that produces a BFP result that contains either a BFP number, a BFP zero (the result is significantly zero), or a signalling NaN when insufficient accuracy is retained.
- A computing device that calculates the "Dominant Bound" from the operands, being the bound after alignment with the least number of significant bits.
- A computing device that calculates the bound of the result.
- A computing device that adjusts the dominant bound for errors introduced in normalization.
- A processing device with BFP registers that accepts commands that control desired accuracy and performs arithmetic operations on BFP operands producing a BFP results.
- A processing device with commands to reset bound limit memory to default values and set a value in bound limit memory.
- BFP operates in conjunction with standard floating point so that existing software need not be discarded.
- BFP fully accommodates rounding and cancellation errors.

"*But this comparison requires a previous reflection, that is, a determination of the place to which the representations of the things which are compared belong, whether, to wit, they are cogitated by the pure understanding, or given by sensibility.*"