Peter Markstein

High-precision division and square root

We present division and square root algorithm for calculations with more bits than are handled by the floating-point hardware. These algorithms avoid the need to multiply two high-precision numbers, speeding up the last iteration by as much as a factor of 10. We also show how to produce the floating-point number closest to the exact result with relatively few additional operations.

Correctness proofs outline for Newton-Raphson based floating-point divide and square root algorithms

This paper describes a study of a class of algorithms for the floating-point divide and square root operations, based on the Newton-Raphson iterative method. The two main goals were. (1) Proving the IEEE correctness of these iterative floating-point algorithms, i.e. compliance with the IEEE-754 standard for binary floating-point operations. The focus was on software driven iterative algorithms, instead of the hardware based implementations that dominated until now. (2) Identifying the special cases of operands that require results. Assistance due to possible overflow, or loss of precision of intermediate This study was initiated in an attempt to prove the IEEE for a class of divide and square root based on the Newton-Rapshson iterative methods. As more insight into the inner workings of these algorithms was gained, it became obvious that a formal study and proof were necessary in order to achieve the desired objectives. The result is a complete and rigorous proof of IEEE correctness for floating-point divide and square root algorithms based on the Newton-Raphson iterative method. Even more, the method used in proving the IEEE correctness of the square root algorithm is applicable in principle to any iterative algorithm, not only based on the Newton-Raphson method. Conditions requiring Software Assistance (SWA) were also determined, and were used to identify cases when alternate algorithms are needed to generate correct results. Overall, this is one important step toward flawless implementation of these floating-point operations based on software implementations.

Accelerating sine and cosine evaluation with compiler assistance

Some software libraries add special entry points to enable both the sine and cosine to be evaluated with one call for performance purposes. We propose another method which does not involve new function names. By having the compiler front end recognize trigonometric function invocations, and replace them with a call to a common function which executes the code common to all the functions, followed by a short routine to produce the desired computation, it is possible to compute both the sine and cosine, when needed in about the same time as to compute only one of them.

A fast-start method for computing the inverse tangent

In a search for an algorithm to compute atan(x) which has both low latency and few floating point instructions, an interesting variant of familiar trigonometry formulas was discovered that allow the start of argument reduction to commence before any references to tables stored in memory are needed. Low latency makes the method suitable for a closed subroutine, and few floating-point operations make the method advantageous for a software-pipelined implementation.

Differential gene expression in the auditory system

Hearing disorders affect over 10% of the population and this ratio is dramatically increasing with age. Development of appropriate therapeutic approaches requires understanding of the auditory system, which remains largely incomplete. We have identified hearing-specific genes and pathways by mapping over 15000 cochlear expressed sequence tags (ESTs) to the human genome (NCBI Build 35) and comparing it to other EST clusters (Unigene Build 183). A number of novel potentially cochlear-specific genes discovered in this work are currently being verified by experimental studies. The software tool developed for this task is based on a fast bidirectional multiple pattern search algorithm. Patterns used for scoring and selection of loci include EST subsequences, cloning-process identifiers, and genomic and external contamination determinants. Comparison of our results with other programs and available annotations shows that the software developed provides potentially the fastest, yet reliable mapping of ESTs.

EST-based analysis of gene expression in the human cochlea

Hearing is one of the vital senses helping to perceive, reflect and communicate with the world around us. Genetics, developmental conditions, mechanical damage, infections, ototoxic medications, and aging are among the factors disabling or deteriorating this sense. The cochlea is a sensory organ responsible for hearing. Over 15,000 expressed sequence tags (ESTs) extracted from this organ had been previously clustered with other sequences and aligned to earlier versions of the human genome and known genes. This is especially important as more genomic and phenotypic data becomes available almost on a daily basis. Many transcripts corresponding to ESTs present in the dataset might not be expressed as proteins, but instead are degraded by nonsense-mediated mRNA decay or other cell surveillance mechanisms. Large-scale sequencing of tissue-specific genes and fast yet reliable mapping of sequences will help to identify key components of sound transduction and speed up progress in hearing research.