数值 / 格式Numbers / Formats

浮点数位拆解Float Bit Inspector

输入任意数字，看它如何被塞进 8 / 16 / 32 / 64 个格子：符号、指数、尾数各占几位，存进去的到底是谁，误差有多大。 Type any number and see how it lands in 8 / 16 / 32 / 64 cells: how many bits go to sign, exponent, and mantissa — and what actually gets stored.

选择格式Format

十进制数值Decimal value 支持 0.1 / -3.14 / 1e8 / π / infsupports 0.1 / -3.14 / 1e8 / π / inf

快速填入Presets

位分布（点击格子可手动翻转每一位） Bit layout (click any cell to flip that bit)

输入的数Input —

实际存储的数（精确）Actually stored (exact) —

绝对误差Absolute error —

相对误差Relative error —

状态Class —

同一个数，在不同格式下 The same number, in every format 点击行可切换主格式click a row to switch format

格式Format	位分布Bits	存储值（精确）Stored (exact)	绝对误差Abs error

* 整数格式（INT2 / INT4 / INT8）：小数会截断取整，超出表示范围会饱和。E4M3 没有无穷大，溢出时饱和到 ±448。 * Integer formats (INT2 / INT4 / INT8): fractions truncate, values beyond range saturate. E4M3 has no infinity — overflow saturates to ±448.

原理速览：定点数 → 浮点数 → FP8 家族 Primer: fixed-point → floating-point → the FP8 family

定点数：小数点不动Fixed-point: the point never moves

8 个格子直接存二进制、约定小数点位置，就是定点数。前 4 位整数、后 4 位小数——范围和精度在编码前就锁死了，二者不可兼得。 Store raw binary and fix the radix point somewhere — that's fixed-point. Four integer bits, four fraction bits: range and precision are frozen at design time, and you can't have both.

浮点数：让小数点浮动Floating-point: let the point float

(−1)^s × 1.m × 2^e —— 用指数控制小数点位置。调小指数可以逼近 0.000000001，调大指数可以够到十亿。代价是：大数的绝对精度被牺牲了，保住的是相对精度。 (−1)^s × 1.m × 2^e — the exponent moves the radix point. Small exponents reach 0.000000001; large ones reach billions. The price: absolute precision of huge numbers is traded away, while relative precision survives.

FP8 家族：同一套格子，不同分法The FP8 family: same cells, different splits

E4M3（±448，精度优先）、E5M2（±57344，范围优先）是英伟达 GPU 硬件支持的两种 FP8。UE8M0 干脆砍掉尾数和符号，只留 8 位指数——每个值都是 2 的幂，乘法退化成位移，硬件最好优化，DeepSeek 用它存缩放因子。 E4M3 (±448, precision-first) and E5M2 (±57344, range-first) are the two FP8 formats NVIDIA GPUs implement. UE8M0 drops sign and mantissa entirely — every value is a power of two, multiplication becomes a bit shift, which is why DeepSeek uses it for scaling factors.

为什么一亿不需要绝对精度Why 100 million doesn't need absolute precision

对一亿来说，「一亿」和「一亿点零零零零一」的差别完全可以忽略。两个相邻浮点数的间隔随大小增长，所以既装得下极大的范围，又没损失相对精度——这就是浮点数的既要又要。 At 100 million scale, the difference between 100000000 and 100000000.00001 is negligible. The gap between adjacent floats grows with magnitude, so you get enormous range without losing relative precision — the best of both worlds, at a price you don't feel.