Rounding

When converting or quantizing numbers (like converting a high-precision Fix_18_17 signal down to Fix_8_7 in FPGA design, or floating-point rounding), rounding modes define how you resolve values that fall between two integers or quantum steps.The main difference between these modes lies in how they handle exact midpoints (like $1.5$ or $-1.5$) and whether they introduce a DC bias.

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Value, Round to Even, Round to Odd, Round to Zero, Round to +∞, Round Half Up, Round Half Down +1.4, +1, +1, +1, +2 (if ceiling) / +1, +1, +1

+1.5, +2,+1,+1,+2,+2,+1

+2.5, +2,+3,+2,+3,+3,+2

−1.5, −2,−1,−1,−1,−1,−2

−2.5, −2,−3,−2,−2,−2,−3

1. Round to Even (Convergent Rounding / Banker's Rounding)How it works: Rounds to the nearest number. If the value is exactly halfway (e.g., $.5$), it rounds to the nearest even integer.Key Feature: Zero DC Bias. Because midpoints end in an even integer 50% of the time and an odd integer 50% of the time, the rounding errors cancel out statistically over large datasets.Best Use Case: DSP, Financial Applications, IEEE 754 Floating-Point standard default. Highly recommended for DSP/FPGA datapath quantization to prevent cumulative offset errors.

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2. Round to OddHow it works: Rounds to the nearest number. If the value is exactly halfway, it rounds to the nearest odd integer.Key Feature: Also bias-free across large datasets, but less common in general computing than Round to Even.Best Use Case: Specialized arithmetic or systems designed to avoid zero-valued outputs in specific boundary conditions.

3. Round to Zero (Truncation / Directed Rounding) How it works: Simply drops the fractional part (chops off the LSBs toward zero). positive numbers move down toward zero, negative numbers move up toward zero.Key Feature: Extremely cheap hardware implementation (requires no adder logic). However, it squeezes values toward zero, shrinking magnitude.Best Use Case: Fixed-point code conversions, integer division in C/C++ ((int)x).

4. Round to $+\infty$ (Ceiling / Directed Rounding) How it works: Always rounds UP toward positive infinity, regardless of the fractional value.Key Feature: Moves all numbers to the right on the number line. Introduces a positive DC bias.Best Use Case: Interval arithmetic, buffer allocation sizing (ensuring you allocate enough space/memory)."Nearest Neighbor" Tie-Breaking ModesThese three modes round to the nearest number for non-midpoints, but differ in how they break the tie at exact $.5$ midpoints:

5. Round Half Up (Nearest, Ties Away From Zero / Commercial Rounding) How it works: If halfway, rounds away from zero toward the larger magnitude ($+1.5 \rightarrow +2$, $-1.5 \rightarrow -2$).Key Feature: This is the standard "schoolbook" rounding taught in elementary math.Drawback: Introduces a slight positive magnitude bias because midpoints always push away from zero.

6. Round Half Down (Nearest, Ties Toward Zero) How it works: If halfway, rounds toward zero ($+1.5 \rightarrow +1$, $-1.5 \rightarrow -1$).Key Feature: The exact inverse of Round Half Up. Shrinks magnitude at midpoints.

7. Refer to casper's library block, "convert".