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3.1 Floating Point

There is a vast body of literature concerning the implementation of floating point operators [22, 47, 29]; furthertmore, there is literature describing the implementation of correctly rounded floating-point arithmetics [35, 71, 53, 55]. Details concerning the implementation of the floating point system used are not relevant to the current discussion.

We assume that tex2html_wrap_inline32371 satisfies IEEE standard 754. The IEEE 754 standard imposes strict requirements on the rounding of the algebraic operations +, -, tex2html_wrap_inline32275 , tex2html_wrap_inline32277 , and tex2html_wrap_inline32435 . An implementation must return the nearest floating point number to the actual real result, when an operation is carried out with the rounding mode set to ``round to nearest''. This implies that the result is accurate to within 1/2 ULP, unless underflow or overflow occurs. Moreover, the algebraic operations must correctly round the result when the current rounding mode is ``round to tex2html_wrap_inline32377 '' or ``round to tex2html_wrap_inline33793 ''; this only requires 1 ULP accuracy.

Other operators, such as sine, are not as favoured. The IEEE 754 standard does not require tex2html_wrap_inline33795 return the nearest floating point number to the actual real result. No claims are made concerning tex2html_wrap_inline33797 or tex2html_wrap_inline33799 . Some systems assume the current rounding mode is ``round to nearest'' when sine is computed; using another rounding mode may adversely affect the trigonometric computation.

Some brief comments will illustrate how tex2html_wrap_inline33801 and tex2html_wrap_inline33803 may be constructed from tex2html_wrap_inline33805 , for various classes of functions.


next up previous notation contents
Next: 3.1.1 Exact Functions Up: 3 Arithmetic Previous: 3 Arithmetic
Jeff TupperMarch 1996