Gawk/Settingtheroundingmode
Previous: Setting precision, Up: FP Math Caution [Contents][Index]
16.4.5 Setting the Rounding Mode
The ROUNDMODE
variable provides
programlevel control over the rounding mode.
The correspondence between ROUNDMODE
and the IEEE
rounding modes is shown in Table 16.5.
Rounding mode  IEEE name  ROUNDMODE


Round to nearest, ties to even  roundTiesToEven

"N" or "n"

Round toward positive infinity  roundTowardPositive

"U" or "u"

Round toward negative infinity  roundTowardNegative

"D" or "d"

Round toward zero  roundTowardZero

"Z" or "z"

Round away from zero  "A" or "a"

ROUNDMODE
has the default value "N"
, which
selects the IEEE 754 rounding mode roundTiesToEven
.
In Table 16.5, the value "A"
selects
rounding away from zero. This is only available if your version of the
MPFR library supports it; otherwise, setting ROUNDMODE
to "A"
has no effect.
The default mode roundTiesToEven
is the most preferred,
but the least intuitive. This method does the obvious thing for most values,
by rounding them up or down to the nearest digit.
For example, rounding 1.132 to two digits yields 1.13,
and rounding 1.157 yields 1.16.
However, when it comes to rounding a value that is exactly halfway between,
things do not work the way you probably learned in school.
In this case, the number is rounded to the nearest even digit.
So rounding 0.125 to two digits rounds down to 0.12,
but rounding 0.6875 to three digits rounds up to 0.688.
You probably have already encountered this rounding mode when
using printf
to format floatingpoint numbers.
For example:
BEGIN { x = 4.5 for (i = 1; i < 10; i++) { x += 1.0 printf("%4.1f => %2.0f\n", x, x) } }
produces the following output when run on the author’s system:^{101}
3.5 => 4 2.5 => 2 1.5 => 2 0.5 => 0 0.5 => 0 1.5 => 2 2.5 => 2 3.5 => 4 4.5 => 4
The theory behind roundTiesToEven
is that it more or less evenly
distributes upward and downward rounds of exact halves, which might
cause any accumulating roundoff error to cancel itself out. This is the
default rounding mode for IEEE 754 computing functions and operators.
Rounding Modes and Conversion It’s important to understand that, along with BEGIN { pi = 3.1416 OFMT = "%.f" # Print value as integer print pi # ROUNDMODE = "N" by default. ROUNDMODE = "U" # Now change ROUNDMODE print pi } Running this program produces this output: $ gawk M f roundmode.awk  3  4 
The other rounding modes are rarely used. Rounding toward positive infinity
(roundTowardPositive
) and toward negative infinity
(roundTowardNegative
) are often used to implement interval
arithmetic, where you adjust the rounding mode to calculate upper and
lower bounds for the range of output. The roundTowardZero
mode can
be used for converting floatingpoint numbers to integers. When rounding
away from zero, the nearest number with magnitude greater than or equal to
the value is selected.
Some numerical analysts will tell you that your choice of rounding style has tremendous impact on the final outcome, and advise you to wait until final output for any rounding. Instead, you can often avoid roundoff error problems by setting the precision initially to some value sufficiently larger than the final desired precision, so that the accumulation of roundoff error does not influence the outcome. If you suspect that results from your computation are sensitive to accumulation of roundoff error, look for a significant difference in output when you change the rounding mode to be sure.
Footnotes
(101)
It
is possible for the output to be completely different if the
C library in your system does not use the IEEE 754 evenrounding
rule to round halfway cases for printf
.
Previous: Setting precision, Up: FP Math Caution [Contents][Index]