Four Rounding Modes

Normalized result has the form: “1.f1f2...fn r s”. The round bit r and sticky bit s appear after the last fraction bit fn. IEEE 754 standard specifies four modes of rounding:
Round to nearest even:
It is the default rounding mode. Increment result if: (rs = “11”) or (rs = “10” and fn = ‘1’). Otherwise, truncate result significand to 1.f1f2...fn.

Round toward +∞ (rounding up):
Increment result if sign is positive and r or s = ‘1’.

Round toward -∞ (rounding down):
Increment result if sign is negative and r or s = ‘1’.

Round toward 0:
It always truncates the result.
For example, round the following result by using IEEE 754 rounding modes: 
   –1.11111111111111111111111 0 1 × 2-7
where the round and sticky bits are in blue and red colors, respectively.
Round to nearest even:
Truncate it to –1.11111111111111111111111×2-7 since r = ‘0’.

Round toward +∞:
Truncate the result since it is negative.

Round toward -∞:
Increment it to –10.00000000000000000000000×2-7 since negative and s = ‘1’. Renormalize and increment exponent (because of carry). Final rounded result is –1.00000000000000000000000×2-6.

Round toward 0:
Always truncate.



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