Error bounded multiplication by invariant rationals
A hardware logic representation of a circuit to implement an operation to perform multiplication by an invariant rational is generated by truncating an infinite single summation array (which is represented in a finite way). The truncation is performed by identifying a repeating section and then discarding all but a finite number of the repeating sections whilst still satisfying a defined error bound. To further reduce the size of the summation array, the binary representation of the invariant rational is converted into canonical signed digit notation prior to creating the finite representation of the infinite array.
1. A hardware logic multiplier configured to multiply an input value by a predetermined invariant rational that satisfies a defined error bound, wherein the hardware logic multiplier is synthesized using a process comprising:
truncating a binary expansion of the predetermined invariant rational;
converting the truncated binary expansion into canonical signed digit notation and expanding the canonical signed digit representation into a finite representation of an infinite expansion;
generating a truncated single summation array from the infinite expansion by discarding one or more repeating sections of the array based upon the defined error bound; and
synthesizing a representation of the hardware logic multiplier from the truncated single summation array, wherein the representation comprises a computer-readable dataset description which is usable in an integrated circuit manufacturing system to fabricate said hardware logic multiplier.
2. The hardware logic multiplier according to claim 1 , wherein the process further comprises:
determining the binary expansion of the predetermined invariant rational.
3. The hardware logic multiplier according to claim 1 , wherein generating a truncated single summation array from the infinite expansion by discarding one or more repeating sections of the array comprises:
determining a maximum binary value of a repeating section in a finite representation of an infinite single summation array generated from the infinite expansion;
weighting the maximum binary value by a weight of each section; and
calculating a minimum number of whole repeating sections to be retained within the truncated single summation array to satisfy the defined error bound.
4. The hardware logic multiplier according to claim 3 , wherein generating a truncated single summation array from the infinite expansion by discarding one or more repeating sections of the array further comprises:
identifying one or more bits from a retained whole repeating section that can be discarded from the truncated single summation array whilst satisfying the defined error bound.
5. The hardware logic multiplier according to claim 1 , wherein the hardware logic multiplier is fabricated in silicon.
6. A non-transitory computer readable storage medium having stored thereon a computer-readable dataset representation of a hardware logic multiplier configured to multiply an input value by a predetermined invariant rational that satisfies a defined error bound, wherein the representation is usable in an integrated circuit manufacturing system to fabricate a hardware logic multiplier and is generated by:
truncating a binary expansion of the predetermined invariant rational; converting the truncated binary expansion into canonical signed digit notation and expanding the canonical signed digit representation into a finite representation of an infinite expansion;
generating a truncated single summation array from the infinite expansion by discarding one or more repeating sections of the array based upon the defined error bound; and
synthesizing the representation of the hardware logic multiplier from the truncated single summation array.
7. The non-transitory computer readable storage medium according to claim 6 , wherein generating the hardware logic multiplier further comprises:
determining the binary expansion of the predetermined invariant rational.
8. The non-transitory computer readable storage medium according to claim 6 , wherein generating a truncated single summation array from the infinite expansion by discarding one or more repeating sections of the array comprises:
determining a maximum binary value of a repeating section in a finite representation of an infinite single summation array generated from the infinite expansion;
weighting the maximum binary value by a weight of each section; and
calculating a minimum number of whole repeating sections to be retained within the truncated single summation array to satisfy the defined error bound.
9. The non-transitory computer readable storage medium according to claim 8 , wherein generating a truncated single summation array from the infinite expansion by discarding one or more repeating sections of the array further comprises:
identifying one or more bits from a retained whole repeating section that can be discarded from the truncated single summation array whilst satisfying the defined error bound.
10. An integrated circuit manufacturing system comprising:
the non-transitory computer readable storage medium as set forth in claim 6 ;
a layout processing system configured to process said computer-readable dataset representation so as to generate a circuit layout description of an integrated circuit embodying hardware logic multiplier; and
an integrated circuit generation system configured to manufacture the hardware logic multiplier according to the circuit layout description.
11. The integrated circuit manufacturing system according to claim 10 , wherein the hardware logic multiplier is fabricated in silicon.