## Advanced Arithmetic for the Digital Computer: Design of Arithmetic UnitsThe number one requirement for computer arithmetic has always been speed. It is the main force that drives the technology. With increased speed larger problems can be attempted. To gain speed, advanced processors and pro gramming languages offer, for instance, compound arithmetic operations like matmul and dotproduct. But there is another side to the computational coin - the accuracy and reliability of the computed result. Progress on this side is very important, if not essential. Compound arithmetic operations, for instance, should always deliver a correct result. The user should not be obliged to perform an error analysis every time a compound arithmetic operation, implemented by the hardware manufacturer or in the programming language, is employed. This treatise deals with computer arithmetic in a more general sense than usual. Advanced computer arithmetic extends the accuracy of the elementary floating-point operations, for instance, as defined by the IEEE arithmetic standard, to all operations in the usual product spaces of computation: the complex numbers, the real and complex intervals, and the real and complex vectors and matrices and their interval counterparts. The implementation of advanced computer arithmetic by fast hardware is examined in this book. Arithmetic units for its elementary components are described. It is shown that the requirements for speed and for reliability do not conflict with each other. Advanced computer arithmetic is superior to other arithmetic with respect to accuracy, costs, and speed. |

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### Contents

Bibliography and Related Literature | 63 |

Rounding Near Zero 71 | 70 |

Interval Arithmetic Revisited | 81 |

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32 bit data 64 bits accumulation unit accuracy accurate scalar product additive inverse advanced computer arithmetic algorithm arith arithmetic expression arithmetic operations arithmetic unit automatic differentiation basic carry resolution complete lattice complex intervals computer representable subsets contains zero cycle data format data types defined denote digits division dotprecision double precision element elementary functions empty set enclosures executed fast flag floating-point arithmetic floating-point numbers floating-point operations function values hardware IEEE arithmetic standard implementation infimum interval arithmetic interval evaluation interval mathematics Interval Newton Method interval operation Kulisch long adder mantissa mathematical matrix memory metic operands PASCAL-XSC performed pipeline conflict processors produce a carry programming languages real and complex real function real numbers requires result rounding mode scalar product computation scientific computing Section semimorphism shift shifter significant solution speed step subtraction summand unique additive inverse Universität Karlsruhe variable vector vector processors