The result is DBL_MIN*DBL_EPSILON, a subnormal number. But, if the implementation does not support subnormal numbers, such as IBM S/360 hex floating-point, then it is either zero or DBL_MIN, depending upon the current rounding direction mode. Hence, the sentence "Thus, the remainder is always exact." in footnote 204 in C99+TC1+TC2 (N1124) is wrong. This problem also applies to remquo and fmod.

After finding that flaw, I looked at the other math functions and their relationship to directed rounding. That search found several areas where things could be improved and one area (nextafter) that took an informal request for interpretation to the IEEE-754 committee to find the answer.

7.12.6.4 The frexp functions should be updated along the lines of:

When the radix of the argument is a power of 2, the returned value is exact and is independent of the current rounding direction mode.

7.12.6.5 The ilogb functions should be updated along the lines of:

When the returned value is representable in the range of the return type, the returned value is exact and is independent of the current rounding direction mode.

7.12.6.11 The logb functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

7.12.6.12 The modf functions should be updated along the lines of:

The returned values are exact and are independent of the current rounding direction mode.

7.12.7.2 The fabs functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

7.12.9.1 The ceil functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

7.12.9.2 The floor functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

7.12.9.8 The trunc functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

7.12.10.1 The fmod functions should be updated along the lines of:

When subnormal results are supported, the returned value is exact and is independent of the current rounding direction mode.

7.12.10.2 The remainder functions should be updated along the lines of:

When subnormal results are supported, the returned value is exact and is independent of the current rounding direction mode.

7.12.10.3 The remquo functions should be updated along the lines of:

When subnormal results are supported, the returned value is exact and is independent of the current rounding direction mode.

7.12.11.1 The copysign functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

7.12.11.2 The nan functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

7.12.11.3 The nextafter functions should be updated along the lines of:

Even thought underflow or overflow may happen, the returned value is independent of the current rounding direction mode.

7.12.11.4 The nexttoward functions should be updated along the lines of:

Even thought underflow or overflow may happen, the returned value is independent of the current rounding direction mode.

7.12.12.2 The fmax functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

7.12.12.3 The fmin functions should be updated along the lines of:

The returned value is exact and is independent of the current rounding direction mode.

F.9.4.5 The sqrt functions could be updated along the lines of:

The returned value is dependent on the current rounding direction mode.

Consider adding the following to section 7.12.1 (or make it its own section) of the Rationale.

There are several functions that are independent of the current rounding direction. Some are documented as such: round, lround, llround, remainder (when subnormal results are supported), remquo (when subnormal results are supported), nextafter (as per IEEE-754), and nexttoward (as per C99 and nextafter). Note, even though nextafter and nexttoward can raise underflow+inexact and overflow+inexact, they are not affected by the rounding direction.

Some are independent because they are exact: frexp (when radix is power of 2), logb, modf, ilogb, fabs, ceil, floor, trunc, fmod (when subnormal results are supported), copysign, nan, fmax, and fmin.

There are several functions that are dependent on the current rounding direction: sqrt (as per IEEE-754), nearbyint, rint, lrint, llrint, and fma.

There are many functions (it is implementation defined as to which ones) that may honor the current rounding direction. First are functions that are inexact for most arguments: acos, asin, atan, atan2, cos, sin, tan, acosh, asinh ,atanh, cosh, sinh, tanh, exp, exp2, expm1, frexp (when radix is not a power of 2), ldexp (when radix is not 2), log, log10, log1p, log2, hypot, pow, cbrt, erf, erfc, tgamma, lgamma, and fdim.

Second are functions that are exact for most arguments (but are inexact when they overflow or underflow): ldexp (when radix is 2), scalbn, scalbln, fmod (when subnormal results are not supported), remainder (when subnormal results are not supported), and remquo (when subnormal results are not supported).