Half Unit Biased Approach

It optimizes the hardware implementation of computation with real numbers


To Start

                                                           HUB in a nutshell (clik on the image to enlarge it)

                    wmv (15 MB)   short video introducing HUB                       mp4 (5 MB)Short video introducing HUB (mp4)


                        wmv (33 MB)video with basic technical explanation                     mp4 (14 MB)


Reaseach Publications

URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7349231&isnumber=7476004

URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7465754&isnumber=4358609

URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7270998&isnumber=7486166

URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7177822&isnumber=7177764
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7177797&isnumber=7177764

URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7094576&isnumber=7094373


Spanish Patent Granted

·         Hormigo, J.; Villalba, J. Multiplicadores coma flotante y conversores asociados, ES2546895B2, 2014. (Link)

·         Hormigo, J.; Villalba, J. Dispositivos coma flotante y conversores, ES2546898B2, 2014. (Link)

·         Hormigo, J.; Villalba, J. Sumadores coma flotante y conversores, ES2546916B2, 2014. (Link)

·         Hormigo, J.; Villalba, J. Unidades aritméticas en coma fija y conversores asociados, ES2546915B2, 2014. (Link)

·         Hormigo, J.; Villalba, J. Dispositivos para operaciones de multiplicación-suma fusionadas en coma flotante y conversores asociados, ES2546899B2, 2014. (Link)

 International patent application (PCT)

PCT 30-month deadline is 28th September 2016.


(Please send your own questions to fjhormigo@uma.es)

Yes, of course. Conversion to HUB at the input and rounding could be fused in one operation without additional error, and similarly, conversion from HUB to conventional at the output. 

The conventional input values could be regularly operated and only when a rounding is required, a truncation is used to produce a HUB number rounded-to-nearest.  Similarly, the last intermediate results (which is a conventional value although it came from an operation with HUB numbers), could be rounded in a conventional way to generate the output in conventional format. Using this procedure, no extra rounding error is introduced by conversions.     

No, it doesn't, but it provides the same precision. This means that, although the value of the result is different, the bound and other statistical properties of the rounding errors are the same. This point has been proved theoretically and experimentally as it is shown in references [4],[5] and [6].


This example does not prove that HUB format has less precision than IEEE standard.

It is absolutely true that if a value can be exactly represented by IEEE format (that is an ERN), HUB will cause an error which is actually the higher possible, 0.5 ULP. Since the ERN of IEEE format has been shifted 0.5 ULP on HUB format by definition, ERNs on IEEE format will be represented with maximum error under HUB format.

However, that does not mean that HUB format is less accurate than IEEE format since that also works the other way around, i.e., the ERNs under HUB format will be represented with the maximum error (0.5 ULP) under IEEE format. For single precision, the values in the form 2^(-n)+2^(-n-24) are represented under IEEE standard with an error of 0.5 ULP whereas they are exactly represented under HUB format. Indeed, as it is said in the paper, both errors are always complementary, i.e.,|eIEEE|+|eHUB|=0.5 ULP. Therefore, the better a value is represented under one format, the worst it is represented under the other.

I cannot imagine any floating-point application where values in the form 2^(-n) appear more likely than values in the form 2^(-n)+2^(-n-24). However, there are lots of applications where the probability to have numbers better represented under IEEE is the same that the  probability to have numbers better represented under HUB format, such as, DSP, physics simulation, neural networks, computer graphics…   


Conversions between HUB-FP  and conventional FP numbers is addressed in references [4].  In both cases this conversion implies rounding, but no explicit operation is required for tie-to-away  or just forcing the value of the LSB for tie-to-even-like rounding.   

Yes, It can.Actually, the tie case is not possible rounding a HUB number and no sticky bit computation is required in floating-point computation. However, under certain circumstances the intermediate result of an FP addition may be the tie case, but some simple hardware can be utilized to guarantee an unbiased rounded results.   The general theory of unbiased rounding is explained in [7] and the unbiased rounding for conversion between HUB and conventional FP is explained in [4].