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 method of synthesizing a 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 method is performed by at least one processor, and the method comprising, by at least one processor:
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 using said truncated single summation array.
2. The method according to claim 1 , further comprising:
determining the binary expansion of the predetermined invariant rational.
3. The method 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 method 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 method according to claim 1 , further comprising:
fabricating the hardware logic multiplier.
6. The method according to claim 5 , wherein the hardware logic multiplier is fabricated in silicon.
7. A non-transitory computer readable storage medium having stored thereon computer executable instructions to synthesize a representation of a hardware logic multiplier, the computer executable instructions being arranged, when executed by a processor, to cause the processor to:
truncate a binary expansion of a predetermined invariant rational, convert the truncated binary expansion into canonical signed digit notation and expand the canonical signed digit representation into a finite representation of an infinite expansion;
generate a truncated single summation array from the infinite expansion by discarding one or more repeating sections of the array based upon a defined error bound; and
synthesize the representation of the hardware logic multiplier using said truncated single summation array.
8. The non-transitory computer readable storage medium according to claim 7 , wherein the hardware logic multiplier is a hardware logic implementation of an operation to multiply an input value by the predetermined invariant rational that satisfies the defined error bound.
9. 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 the hardware logic multiplier using said truncated single summation array.
10. The hardware logic multiplier according to claim 9 , wherein the process further comprises:
determining the binary expansion of the predetermined invariant rational.
11. The hardware logic multiplier according to claim 9 , 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.
12. The hardware logic multiplier according to claim 11 , 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.
13. The hardware logic multiplier according to claim 9 , wherein the hardware logic multiplier is fabricated in silicon.