本文介绍了IEEE浮点数表示中指数部分的偏置形式，并详细阐述了如何在不改变实际值的情况下扩展偏置形式一位，以方便在计算中检测溢出和下溢。内容包括背景介绍、结论与证明、一般化过程以及如何缩小偏置形式的数字。
0. Background
The exponent part of the IEEE Floating-Point Representation is encoded in a biased form. During my design of the ALU for the floating point, I need to find a way to extend the biased form for one bit without change the actual value which it’s representing and make it easier to detect the overflow and underflow during the calculation.
By the way, a k k k-bit biased number’s bias in this article is B i a s = 2 k − 1 − 1 Bias = 2^{k - 1} - 1 Bias=2k−1−1.
1. Conclusion & Provement
Suppose a k k k-bit biased numer and the actual value of its representation : X k X k − 1 . . . X 1 = ∑ n = 1 k X n ∗ 2 n − 1 − 2 k − 1 + 1 X_kX_{k-1}...X_1 = \sum_{n=1}^{k}{X_n*2^{n-1}} - 2^{k-1} + 1 XkXk−1...X1=∑n=1kXn∗2n−1−2k−1+1
To extend it to a $ k+1$ -bit number and its actual value： X k X k ‾ X k − 1 . . . X 1 = ∑ n = 1 k + 1 X n ∗ 2 n − 1 − 2 k + 1 X_k\overline{X_k}X_{k-1}...X_1 = \sum_{n=1}^{k+1}{X_n*2^{n-1}} - 2^{k} + 1 XkXkXk−1...X1=∑n=1k+1Xn∗2n−1</
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