![]() ![]() |-+-+-|įor all the values of es above, a 16-bit posit number has a smaller epsilon than either FP16 or BF16. The dynamic range and epsilon values for 16-bit posits with es ranging from 1 to 4 are given in the table below. Posits represent large numbers with low precision and small numbers with high precision, but this trade-off is often what you’d want.įor an n-bit posit, the number of fraction bits near 1 is n – 2 – es and so epsilon is 2 to the exponent es – n – 2. Posit numbers can achieve more precision and more dynamic range than IEEE-like floating point numbers with the same number of bits. (Note “maximum.” The number of exponent bits varies for different numbers.) This post explains the anatomy of a posit number. The precision and dynamic range of posit numbers depends on how many bits you allocate to the maximum exponent, denoted es by convention. (Python code to calculate dynamic range is given here.) |-+-| The dynamic ranges of the numeric formats are given below. Range is mostly determined by the number of exponent bits, though not entirely.ĭynamic range in decades is the log base 10 of the ratio of the largest to smallest representable positive numbers. Relative to FP32, BF16 sacrifices precision to retain range. The dynamic range of bfloat16 is similar to that of a IEEE single precision number. BF16 has much less precision near 1 than the other formats. The epsilon value, the smallest number ε such that 1 + ε > 1 in machine representation, is 2 – e where e is the number of fraction bits. To got up from BF16 to FP32, pad with zeros, except for some edge cases regarding denormalized numbers. To go from FP32 to BF16, just cut off the last 16 bits. This makes conversion between FP32 and BF16 simple. The rest of the bits in each of the formats are allocated as in the table below. The BF16 format is sort of a cross between FP16 and FP32, the 16- and 32-bit formats defined in the IEEE 754-2008 standard, also known as half precision and single precision.īF16 has 16 bits like FP16, but has the same number of exponent bits as FP32. It is supported by several deep learning accelerators (such as Google’s TPU), and will be supported in Intel processors two generations from now. BF16 is becoming a de facto standard for deep learning. Here I want to look at bfloat16, or BF16 for short, and compare it to 16-bit number formats I’ve written about previously, IEEE and posit. Together these can benefits can give significant speedup. Also, low-precision circuits are far less complex. Lower precision makes it possible to hold more numbers in memory, reducing the time spent swapping numbers in and out of memory. Algorithms often don’t need as much precision as standard IEEE-754 doubles or even single precision floats. Deep learning has spurred interest in novel floating point formats. ![]()
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