東京大学 情報理工学系研究科 創造情報学専攻 2026年2月実施 筆記試験 第1問

标签：

• Tokyo-University
• Number-Representation

Author

itsuitsuki

Description

Memorized version (English)

Consider an 8-bit binary number (e.g., 0b10110111, where underscores _ can be freely added as separators without affecting the meaning).

• [7:4] (Bits 7, 6, 5, 4): represent the exponent E.
• [3:0] (Bits 3, 2, 1, 0): represent the mantissa M.

The floating-point value represented by this binary number is calculated using the formula: $\text{Value} = \left(1 + \frac{M}{16}\right) \times 2^{(E-7)}$ This is called EM notation.

Answer the following questions.

(1) How is 1.0 represented in EM notation?

(2) What is the decimal value of the binary number 0b1000_1000?

(3) What are the decimal values of the largest and the second largest numbers that can be represented?

We define functions as follows:

• $R(x)$: Input a decimal number $x$; output the largest decimal number strictly less than $x$ that can be exactly represented in EM notation.
• $F(y)$: Input a binary EM representation $y$; output the corresponding decimal value.
• $I(z)$: Input a decimal number $z$; output the EM representation (binary) of the largest number strictly less than $z$ that can be exactly represented in EM notation.

(4) Prove: For two binary numbers $X_1$ and $X_2$, if $X_1 > X_2$, then $F(X_1) > F(X_2)$.

(5) Prove: Among all numbers representable in EM notation, no two numbers have the same EM representation (i.e., the mapping is unique).

(6) Calculate the value of $R(1.1) + R(1.1)$.

(7) Calculate the value of $R( F(\text{0b0111\_1000}) + F(\text{0b0001\_1000}) )$.

(8) Let $e$ and $h$ be numbers that can be exactly represented in EM notation, with $e \le 15$. Let $L(e)$ be the integer part of the decimal value of $e$. Find a value for $h$ (provide the decimal value) such that the lower 4 bits (the Mantissa part, $[3:0]$) of $I(e+h)$ correspond to the value $L(e)$ for every $e$.

题目描述

本页原文标为“英文回忆版”。考虑 8 位二进制数（如 0b10110111，可任意加入下划线作分隔，不影响数值）：高 4 位 [7:4] 表示指数 $E$，低 4 位 [3:0] 表示尾数 $M$。其浮点值定义为

$$\left(1+\frac M{16}\right)2^{E-7},$$

称为 EM 表示。

1. 数值 1.0 的 EM 二进制表示是什么？
2. 0b1000_1000 的十进制值是多少？
3. 可表示的最大数与第二大数分别是多少？

定义：

• $R(x)$：对十进制数 $x$，返回严格小于 $x$ 的、可被 EM 精确表示的最大十进制数；
• $F(y)$：把 EM 二进制表示 $y$ 转成对应十进制值；
• $I(z)$：对十进制数 $z$，返回严格小于 $z$ 的最大可精确表示数之 EM 二进制表示。

4. 证明对两个 8 位二进制编码 $X_1>X_2$，有 $F(X_1)>F(X_2)$。

5. 证明 EM 表示具有唯一性，即不会有两个数对应同一个 EM 表示。

6. 计算 $R(1.1)+R(1.1)$。

7. 计算

   $$R\!\left(F(\texttt{0b0111\_1000})+F(\texttt{0b0001\_1000})\right).$$

8. $e,h$ 均可由 EM 精确表示，且 $e\le15$。令 $L(e)$ 为 $e$ 十进制值的整数部分。求一个 $h$（给十进制值），使对每个这样的 $e$，$I(e+h)$ 的低 4 位尾数字段 [3:0] 所表示的值都等于 $L(e)$。