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.

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.

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.

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.

Same accuracy in vector mode