Thomas and Fred Tydeman (
Reply to: Jim Thomas <firstname.lastname@example.org>
Subject: contractions and expression evaluation methods
This paper notes an ambiguity in the specification of contractions and the effect of expression evaluation methods on them, and it proposes text changes for resolution. The relevant portions of text in C1x draft N1336 are
A floating expression may be contracted, that is, evaluated as though it were an atomic operation, thereby omitting rounding errors implied by the source code and the expression evaluation method.79)
79) A contracted expression might also omit the raising of floating-point exceptions.
F.6 Contracted expressions
A contracted expression treats infinities, NaNs, signed zeros, subnormals, and the rounding directions in a manner consistent with the basic arithmetic operations covered by IEC 60559.
A contracted expression should raise floating-point exceptions in a manner generally consistent with the basic arithmetic operations. A contracted expression should deliver the same value as its uncontracted counterpart, else should be correctly rounded (once).
Problem: The definition in 6.5 #8 implies that contraction entails omitting rounding errors. This is the case when contractions are implemented by machine instructions that round just once. However, the second sentence of the recommended practice in F.6, implies that contractions might not omit rounding errors and, alternatively, might round intermediate values in unspecified ways. This latter interpretation would introduce more potential variability for the user to cope with. Also, it’s unclear how wide expression evaluation might fit with this latter interpretation of contractions.
Resolving the ambiguity in favor of contractions that round just once results in a more manageable model for users and allows a natural fit with wide expression evaluation. It appears (without thorough investigation) that instructions that might support contractions on surviving and new hardware round just once. The fused multiply add and so-called formatOf operations (which round results to formats different from the operand formats) required by 754-2008 are suitable for contractions that round just once.
Footnote 79: Insert at the beginning “The intermediate operators in the contraction are evaluated as if to infinite precision and range, while the final operator is rounded to the format determined by the expression evaluation method.”
F.6: change to “A contracted expression is correctly rounded (once) and treats infinities, NaNs, signed zeros, subnormals, and the rounding directions in a manner consistent with the basic arithmetic operations covered in IEC 60559.”
F.6, Recommended practice: remove the second sentence.