Float-point basic

1. Storage Format （IEEE 754-2008）

Format	half-precision	single-precision	double-precision
width	16-bit	32-bit	64-bit
Sign	1-bit	1-bit	1-bit
Exponent	5-bit	8-bit	11-bit
Mantissa	10-bit	23-bit	52-bit

Half-precision floating-point number (16-bit)

The IEEE 754 half-precision format packs 16 bits as: 1 sign bit | 5 exponent bits (bias 15) | 10 mantissa bits.

16-bit half-precision floating-point number layout:

Example of 16-bits half-precision floating-point number calculation process

2. Formula for Floating-Point Number

half-precision floating-point number = (-1)^sign * 2^(exp-15)* (1 + mantissa/1024)

single-precision floating-point number = (-1)^sign * 2^(exp-127)* (1 + mantissa/(2^23))

double-precision floating-point number = (-1)^sign * 2^(exp-1023)* (1 + mantissa/(2^52))

3. Half-precision Encoding Rules for Special Values

Type	Sign	Exponent	Mantissa	Hexadecimal expression
Positive Zero	0	00000	0000000000	0x0000
Negative Zero	1	00000	0000000000	0x8000
Positive Denormal	0	00000	non-zero	0x0001-0x03FF
Negative Denormal	1	00000	non-zero	0x8001-0x83FF
Positive Normal	0	00001~11110	any value	0x0400-0x7BFF
Negative Normal	1	00001~11110	any value	0x8400-0x9BFF
Positive Infinity	0	11111	0000000000	0x7C00
Negative Infinity	1	11111	0000000000	0xFC00
QNaN	0/1	11111	1xxxx	0x7C01-0x7DFF
SNaN	0/1	11111	0xxxx(non-zero)	0x7E00-0x7FFF

Explanation: QNaN is a quiet NaN, SNaN is a signaling NaN

1. Zero

• 正零和负零在数值上相等
• 但某些运算会保留符号
• 1/0 = +∞, 1/-0 = -∞

2. Denormal / Subnormal

• 指数位全0，尾数位非0

• 用于表示接近0的极小数

• 公式：(-1)^sign × 2^(-14) × (0 + mantissa/1024)

• 没有隐含的整数部分1

  • 最小的正非规格化数

  16'h0001 : sign=0, exp=0, man=1
  数值 = (-1)^0 × 2^(-14) × (0 + 1/1024) ≈ 5.96e-8

  • 最大的正非规格化数

  16'h03FF : sign=0, exp=0, man=1023
  数值 = (-1)^0 × 2^(-14) × (0 + 1023/1024) ≈ 6.10e-5

3. Normal

• 指数位不全0也不全1
• 隐含整数部分1
  • 最小正规格化数

  16'h0400 // sign=0, exp=1, man=0
  数值 = (-1)^0 × 2^(1-15) × (1 + 0/1024) = 2^(-14) ≈ 6.10e-5

  • 最大正规格化数

  16'h7BFF // sign=0, exp=30, man=1023
  数值 = (-1)^0 × 2^(30-15) × (1 + 1023/1024) ≈ 65504

4. Infinity

• 任何有限数 / 0 = ±∞
• ∞ + ∞ = ∞
• ∞ - ∞ = NaN
• ∞ × 0 = NaN
• ∞ / ∞ = NaN

5. NaN

• QNaN (Quiet NaN)
  • 尾数最高位为1（man[9]=1）
  • 运算传播时不会触发异常
  • 用于表示无效运算结果
• SNaN (Signaling NaN)
  • 尾数最高位为0（man[9]=0），但尾数不全0
  • 运算时会触发异常
  • 用于初始化未初始化的变量