Mathtruncxb

Mathtruncxb is a hypothetical mathematical function used in fixed-point arithmetic to truncate a real number to a fixed-point representation specified by two width parameters, x and b. The function is defined for applications where a real input must be represented as a scaled integer with a fixed number of fractional bits, while constraining the overall word width.

Definition and parameters: For integers x ≥ 2 and b ≥ 0, mathtruncxb(r, x, b) produces an integer that encodes the fixed-point value of r with b fractional bits. Formally, mathtruncxb(r, x, b) = sign(r) × floor(|r| × 2^b). The representable range of results is [-2^(x−1) × 2^b, (2^(x−1) − 1) × 2^b]. If the computed value lies outside this range, the result is saturated (clamped) to the nearest bound. This ensures the output fits in a fixed-point word with x integer bits and b fractional bits.

Semantics and use: The function truncates toward zero, yielding a fixed-point representation whose error is less than 2^(−b) in magnitude. It is distinct from rounding-to-nearest or from floor for negative inputs, which can produce different results. Mathtruncxb is commonly discussed in contexts such as embedded systems, digital signal processing, and simulations that rely on precise, predictable fixed-point behavior rather than floating-point dynamics.

Example: mathtruncxb(3.75, 4, 2) equals 15, representing 3.75 in a Q4.2 fixed-point format (since 15 / 2^2 = 3.75). mathtruncxb(-2.5, 4, 2) equals −10, representing −2.50 in the same format (−10 / 4 = −2.5).

Implementation notes: In practice, the operation is implemented as result = sign(r) × floor(|r| × 2^b) with subsequent saturation to the fixed-point range defined by x and b.