With ordinary integer arithmetic the block floating-point logic chooses exponents and operand shifts to prevent integer overflow with worst-case input values. However, the XS3 VPU utilizes symmetrically saturating integer arithmetic.

Saturating arithmetic is that where partial results of the applied operation use a bit depth greater than the output bit depth, and values that can't be properly expressed with the output bit depth are set to the nearest expressible value.

For example, in ordinary C integer arithmetic, a function which multiplies two 32-bit integers may internally compute the full 64-bit product and then clamp values to the range (INT32_MIN, INT32_MAX) before returning a 32-bit result.

Symmetrically saturating arithmetic also includes the property that the lower bound of the expressible range is the negative of the upper bound of the expressible range.

One of the major troubles with non-saturating integer arithmetic is that in a twos complement encoding, there exists a non-zero integer (e.g. INT16_MIN in 16-bit twos complement arithmetic) value \(x\) for which \(-1 \cdot x = x\). Serious arithmetic errors can result when this case is not accounted for.

One of the results of symmetric saturation, on the other hand, is that there is a corner case where (using the same exponent and shift logic as non-saturating arithmetic) saturation may occur for a particular combination of input mantissas. The corner case is different for different operations.

When the corner case occurs, the minimum (and largest magnitude) value of the resulting vector is 1 LSb greater than its ideal value (e.g. -0x3FFF instead of -0x4000 for 16-bit arithmetic). The error in this output element's mantissa is then 1 LSb, or \(2^p\), where \(p\) is the exponent of the resulting BFP vector.

Of course, the very nature of BFP arithmetic routinely involves errors of this magnitude.