Rounding Mode
Rounding means replacing a number with an approximate value that has a shorter, simpler, or more explicit representation. For given \(x\), if the rounded result is \(y\), there are the following rounding modes for selection.
Half to Even
Rounding: when the decimal value is 0.5, round to the nearest even number. The enumeration value is
RM_HALF_TO_EVEN.
Half Away From Zero
Rounding: positive numbers are close to positive infinity, negative numbers are close to negative infinity. The enumeration value is
RM_HALF_AWAY_FROM_ZERO. The formula is\[\mathsf{y = \mathrm{sign}(x)\left\lfloor|x| + 0.5\right\rfloor = -\mathrm{sign}(x)\left\lceil-|x| - 0.5\right\rceil}\]
Towards Zero
Unconditionally round off decimals, close to zero point. The enumeration value is
RM_TOWARDS_ZERO. The formula is\[\begin{split}\mathsf{y = \mathrm{sign}(x)\left\lfloor|x|\right\rfloor = -\mathrm{sign}(x)\left\lceil-|x|\right\rceil} = {\begin{cases}\mathsf{\lfloor x\rfloor}&{\text{if}}\mathsf{\ \ x > 0,}\\ \mathsf{\lceil x\rceil}&{\text{otherwise}}.\end{cases}}\end{split}\]
Down
Rounds toward negative infinity. The enumeration value is
RM_DOWN. The formula is\[\mathsf{y = \lfloor x\rfloor = -\lceil-x\rceil}\]
Up
Rounds toward positive infinity. The enumeration value is
RM_UP. The formula is\[\mathsf{y = \lceil x\rceil = -\lfloor-x\rfloor}\]
Half Up
Rounded to the nearest positive infinity. The enumeration value is
RM_HALF_UP. The formula is\[\mathsf{y = \lceil x + 0.5\rceil = -\lfloor-x - 0.5\rfloor = \left\lceil\frac{\lfloor 2x\rfloor}{2}\right\rceil}\]
Half Down
Rounds to the nearest negative infinity. The enumeration value is
RM_HALF_DOWN. The formula is\[\mathsf{y = \lfloor x - 0.5\rfloor = -\lceil-x + 0.5\rceil = \left\lfloor\frac{\lceil 2x\rceil}{2}\right\rfloor}\]
Example
The following table lists the correspondence between x and y under different rounding modes.