By Dr. Ulrich W. Kulisch (auth.)
The number 1 requirement for machine mathematics has regularly been pace. it's the major strength that drives the know-how. With elevated pace higher difficulties should be tried. to realize pace, complicated processors and professional gramming languages supply, for example, compound mathematics operations like matmul and dotproduct. yet there's one other facet to the computational coin - the accuracy and reliability of the computed end result. growth in this facet is essential, if no longer crucial. Compound mathematics operations, for example, must always convey an accurate consequence. The person shouldn't be obliged to accomplish an mistakes research each time a compound mathematics operation, applied through the producer or within the programming language, is hired. This treatise bargains with machine mathematics in a extra common experience than ordinary. complex machine mathematics extends the accuracy of the straightforward floating-point operations, for example, as outlined by way of the IEEE mathematics usual, to all operations within the traditional product areas of computation: the advanced numbers, the true and complicated periods, and the genuine and intricate vectors and matrices and their period opposite numbers. The implementation of complicated desktop mathematics through quickly is tested during this ebook. mathematics devices for its easy parts are defined. it's proven that the necessities for velocity and for reliability don't clash with one another. complex machine mathematics is more advantageous to different mathematics with appreciate to accuracy, expenses, and speed.
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Additional info for Advanced Arithmetic for the Digital Computer: Design of Arithmetic Units
2 How much Local Memory should be Provided on a SPU? There are applications which make it desirable to provide more than one long accumulator on the SPU. If, for instance, the components of the two vectors a = (ai) and b = (b i ) are complex floating-point numbers, the scalar product a· b is also a complex floating-point number. It is obtained by accumulating the real and imaginary parts of the product of two complex floating-point numbers. The formula for the product of two complex floating-point numbers (x = Xl + iX2, Y = YI + iY2 X XY (Xl X YI - X2 X Y2) = '* + i (Xl X Y2 + X2 X YI)) shows that the real and imaginary part of ai and bi are needed for the computation of both the real part of the product ai x bi as well as the imaginary part.
The easiest way to solve this problem is to open a new LA for the program with higher priority. Of course, this can happen several times which raises the question how much local memory for how many long accumulators should be provided on a SPU. Three might be a good number to solve this problem. If a further interrupt requires another LA, the LA with the lowest priority could be mapped into the main memory by some kind of stack mechanism and so on. This technique would not limit the number of interrupts that may occur during a scalar product computation.
15 is shown directly beneath of the LA. The situation can be avoided by writing a dummy exponent into e" or by reading from the add/subtract unit with higher priority. This solution is not shown in Fig. 15. The product that arrives at the accumulation unit touches two consecutive words of the LA. A more significant third word absorbs the possible carry. The solution for the two pipeline conflicts work well, if this third word is the next more significant word of the LA. The probability that this is not the case is less than 10- 18 .
Advanced Arithmetic for the Digital Computer: Design of Arithmetic Units by Dr. Ulrich W. Kulisch (auth.)