**Submitter:**J11

**Submission Date:**2006-07-07

**Source:**Fred J. Tydeman

**Reference Document:** N1181

**Version:** 1.4

**Date:** 2008-09-12

**Subject:**Math functions and directed rounding

**Summary:**

```
remainder(
DBL_MIN*(1.0+2.0*DBL_EPSILON), DBL_MIN*(1.0+DBL_EPSILON) )
```

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 though 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 though 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).

**Committee Discussion**

These look like new requirements on the math functions (they are not). They might break an existing implementation (possible). Therefore, only add these to annex F.

The above material for the rationale, which should be used, should merge the last two paragraphs.

It was noted that there is no rounding requirements for some of the cases mentioned

We believe that the proposed TC is correct, and nobody has found any additional rounding requirements.

**Technical Corrigendum**

In section 7.12.10.2, remove the sentence "Thus, the remainder is always exact." in footnote 204 in C99+TC1+TC2.

F.9.3.4 The `frexp`

functions add the following
sentence:

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.

F.9.3.5 The `ilogb`

functions, add the following
sentence:

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.

F.9.3.11 The `logb`

functions, add the following
sentence:

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

F.9.3.12 The `modf`

functions, add the following
sentence:

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

F.9.4.2 The `fabs`

functions, add the following
sentence:

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

F.9.4.5 The `sqrt`

functions, add the following
sentence:

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

F.9.6.1 The `ceil`

functions, add the following
sentence:

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

F.9.6.2 The `floor`

functions, add the following
sentence:

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

F.9.6.8 The `trunc`

functions, add the following
sentence:

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

F.9.7.1 The `fmod`

functions, add the following
sentence:

F.9.7.2 The `remainder`

functions, add the
following sentence:

F.9.7.3 The `remquo`

functions, add the following
sentence:

F.9.8.1 The `copysign`

functions, add the following
sentence:

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

F.9.8.2 The `nan`

functions, add the following
sentence:

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

F.9.8.3 The `nextafter`

functions, add the
following sentence:

Even though underflow or overflow can occur, the returned value is independent of the current rounding direction mode.

F.9.8.4 The `nexttoward`

functions, add the
following sentence:

Even though underflow or overflow can occur, the returned value is independent of the current rounding direction mode.

F.9.9.2 The `fmax`

functions, add the following
sentence:

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

F.9.9.3 The `fmin`

functions, add the following
sentence:

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