Theory of linear and integer programming
Theory of linear and integer programming
Integer Multiplication and Division on the HP Precision Architecture
IEEE Transactions on Computers - Special issue on architectural support for programming languages and operating systems
Scanning polyhedra with DO loops
PPOPP '91 Proceedings of the third ACM SIGPLAN symposium on Principles and practice of parallel programming
The Omega test: a fast and practical integer programming algorithm for dependence analysis
Proceedings of the 1991 ACM/IEEE conference on Supercomputing
Improving locality and parallelism in nested loops
Improving locality and parallelism in nested loops
Division by invariant integers using multiplication
PLDI '94 Proceedings of the ACM SIGPLAN 1994 conference on Programming language design and implementation
SUIF: an infrastructure for research on parallelizing and optimizing compilers
ACM SIGPLAN Notices
Data and computation transformations for multiprocessors
PPOPP '95 Proceedings of the fifth ACM SIGPLAN symposium on Principles and practice of parallel programming
Design issues in high performance floating point arithmetic units
Design issues in high performance floating point arithmetic units
Parallelizing compiler techniques based on linear inequalities
Parallelizing compiler techniques based on linear inequalities
Maps: a compiler-managed memory system for raw machines
ISCA '99 Proceedings of the 26th annual international symposium on Computer architecture
Concrete Math
Strength reduction for division and modulo with application to accessing a multilevel store
IBM Journal of Research and Development
A step towards unifying schedule and storage optimization
ACM Transactions on Programming Languages and Systems (TOPLAS)
Constructing application-specific memory hierarchies on FPGAs
Transactions on high-performance embedded architectures and compilers III
Exploiting Task- and Data-Level Parallelism in Streaming Applications Implemented in FPGAs
ACM Transactions on Reconfigurable Technology and Systems (TRETS)
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Integer division, modulo, and remainder operations are expressive and useful operations. They are logical candidates to express complex data accesses such as the wrap-around behavior in queues using ring buffers. In addition, they appear frequently in address computations as a result of compiler optimizations that improve data locality, perform data distribution, or enable parallelization. Experienced application programmers, however, avoid them because they are slow. Furthermore, while advances in both hardware and software have improved the performance of many parts of a program, few are applicable to division and modulo operations. This trend makes these operations increasingly detrimental to program performance. This paper describes a suite of optimizations for eliminating division, modulo, and remainder operations from programs. These techniques are analogous to strength reduction techniques used for multiplications. In addition to some algebraic simplifications, we present a set of optimization techniques that eliminates division and modulo operations that are functions of loop induction variables and loop constants. The optimizations rely on algebra, integer programming, and loop transformations.