What is the difference between fixed point and floating point representation? Fixed point keeps the decimal point in a fixed position, leading to faster but less precise calculations. Floating point allows the decimal point to move, offering more precision and a wider range of values but with more complexity.

We introduce the syntax of fixed-point sorts and operations in the SMT-LIB format similar to that of the SMT floating-point theory. Then, we formalize the semantics of the fixed-point theory, including rounding and overflow, based on the exact rational arithmetic.

The automation of floating-point to fixed-point transformation is a key step for high-level synthesis. Based on Extreme Value Theory, this paper presents a novel approach to find both the optimal integer and fractional bit-widths for signals.

典型的比如相对于浮点数的定点数（Fixed Point Number）。 在这种表达方式中，小数点固定的位于实数所有数字中间的某个位置。 货币的表达就可以使用这种方式，比如 99.00 或者 00.99 可以用于表达具有四位精度（Precision），小数点后有两位的货币值。

2 Fixed-Point Binary Representations A collection of N (N a positive integer) binary digits (bits) has 2Npossible states. This can be seen from elementary counting theory, which tells us that there are two possibilities for the first bit, two possibilities for the next bit, and so on until the last bit, resulting in 2|×2×{z... ×2} Ntimes =2N