2. floating point number

待处理

1. Scale e.g. E8M0

2. 类型转换

2.1. Microscaling Formats

OCP 8-bit Floating Point Format

	E4M3	E5M2
Exponent bias	7	15
Infinities	N/A	S 1111 002
NaN	S 1111 1112	S 11111 {01, 10, 11}2
Zeros	S 0000 0002	S 00000 002
Max normal	S 1111 1102 = ± 28 × 1.75 = ± 448	S 11110 112 = ± 215 × 1.75 = ± 57,344
Min normal	S 0001 0002 = ± 2−6	S 00001 002 = ± 2−14
Max subnormal	S 0000 1112 = ± 2−6 × 0.875	S 00000 112 = ± 2−14 × 0.75
Min subnormal	S 0000 0012 = ± 2−9	S 00000 012 = ± 2−16

FP6 Format

	E2M3	E3M2
Exponent bias	1	3
Infinities	N/A	N/A
NaN	N/A	N/A
Zeros	S 00 0002	S 000 002
Max normal	S 11 1112 = ± 22 × 1.875 = ± 7.5	S 111 112 = ± 24 × 1.75 = ± 28
Min normal	S 01 0002 = ± 20 = ± 1	S 001 002 = ± 2−2 = ± 0.25
Max subnormal	S 00 1112 = ± 20 × 0.875 = ± 0.875	S 000 112 = ± 2−2 × 0.75 = ± 0.1875
Min subnormal	S 00 0012 = ± 2−3 = ± 0.125	S 000 012 = ± 2−4 = ± 0.0625

FP4 Format

	E2M1
Exponent bias	1
Infinities	N/A
NaN	N/A
Zeros	S 00 02
Max normal	S 11 12 = ± 22 × 1.5 = ± 6
Min normal	S 01 02 = ± 20 × 1.0 = ± 1
Subnormal	S 00 12 = ± 20 × 0.5 = ± 0.5

2.2. IEEE 浮点数

浮点数最新标准为IEEE 754-2019

浮点数格式如下：

S(sign)	E (biased exponent)	T (trailing significand field)
1 bit	w bits	t bits, t = p -1

具有如下关系：

\[\begin{aligned} e & = E - bias \\ e_{max} & = bias = 2^{w-1} - 1 \\ e_{min} & = 1 - e_{max} \end{aligned}\]

关于biased E的说明:

1. normal number: [1 , \(2^w - 2\)]， 值为 \((-1)^s \times 2^{E-bias} \times (1+ 2^{1-p} \times T)\)

2. 0, 当T=0表示 \(\pm 0\); 当T!=0 表示 subnormal number, 值为 \((-1)^s \times 2^{e_{min}} \times (0+ 2^{1-p} \times T)\)

3. \(2^w − 1\) (二进制全部为1), 当T=0, 表示 \(\pm \infty\); 当T != 0, 表示 NaN.

2.2.1. ieee 规定的16, 32, 64, 128比特的浮点数格式列表

参数	binary16	binary32	binary64	binary128
指数位数	5	8	11	15
emax/bias	15	127	1023	16383
小数位数	10	23	52	112