The end of numerical error

Summary form only given, as follows. The full paper was not made available as part of this conference proceedings. It is time to overthrow a century of methods based on floating point arithmetic. Current technical computing is based on the acceptance of rounding error using numerical representations that were invented in 1914, and acceptance of sampling error using algorithms designed for a time when transistors were very expensive. By sticking to an antiquated storage format (now codified as an IEEE standard) well into the exascale area, we are wasting power, energy, storage, bandwidth, and programmer effort. The pursuit of exascale floating point is ridiculous, since we do not need to be making 10^18 sloppy rounding errors per second; we need instead to get provable, valid results for the first time, by turning the speed of parallel computers into higher quality answers instead of more junk per second. We introduce the 'unum' (universal number), a superset of IEEE Floating Point, that contains extra metadata fields that actually save storage, yet give more accurate answers that do not round, overflow, or underflow. The potential they offer for improved programmer productivity is enormous. They also provide, for the first time, the hope of a numerical standard that guarantees bitwise identical results across different computer architectures. Unum format is the basis for the 'ubox' method, which redefines what is meant by "high performance" by measuring performance in terms of the knowledge obtained about the answer and not the operations performed per second. Examples are given for practical application to structural analysis, radiation transfer, the n-body problem, linear and nonlinear systems of equations, and Laplace’s equation. This is a fresh approach to scientific computing that allows proper, rigorous representation of real number sets for the first time.