![]() This can be prevented if, for both the numerator and the denominator, a higher precision is simulated by representing the value as the sum of a floating-point number and an interval. The last digit of the mantissa may then be incorrect. The final division causes a relative error of less than ɛ. These can be evaluated with maximum accuracy. Any arithmetic expression can be transformed into a quotient in which the numerator and the denominator yield simultaneous linear equations. The method for polynomials is directly applicable to certain arithmetic expressions. In this case, additional computing time and storage are needed. The computing time of the new algorithm is of the same order as for conventional floating-point calculation, assuming the latter does not fail completely. ![]() ![]() If we let u1v1.x, u2v2.x, p1v1.x, p2v2.x (so each value is a scalar the result of a dot product of two vectors), then the equations are now: m1u12+m2u22m1p12+m2p22 m1u1+m2u2m1p1+m2p2 Its not a vector equation, and so its easily solvable. Let x 0 be our initial estimate of the root, and let x n be the n-th improved estimate. This algorithm uses floating-point operations with directed roundings and a scalar product of maximum accuracy in addition to the usual floating-point operations. So now we only have to consider the 1D collision in the direction of x. This chapter presents the evaluation of arithmetic expressions with maximum accuracy. ![]()
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