  Deep learning has grown rapidly thanks to its state-of-the-art performance
across a wide range of real-world applications. While neural networks have been
trained using IEEE-754 binary32 arithmetic, the rapid growth of computational
demands in deep learning has boosted interest in faster, low precision
training. Mixed-precision training that combines IEEE-754 binary16 with
IEEE-754 binary32 has been tried, and other $16$-bit formats, for example
Google's bfloat16, have become popular. In floating-point arithmetic there is a
tradeoff between precision and representation range as the number of exponent
bits changes; denormal numbers extend the representation range. This raises
questions of how much exponent range is needed, of whether there is a format
between binary16 (5 exponent bits) and bfloat16 (8 exponent bits) that works
better than either of them, and whether or not denormals are necessary.
  In the current paper we study the need for denormal numbers for
mixed-precision training, and we propose a 1/6/9 format, i.e., 6-bit exponent
and 9-bit explicit mantissa, that offers a better range-precision tradeoff. We
show that 1/6/9 mixed-precision training is able to speed up training on
hardware that incurs a performance slowdown on denormal operations or
eliminates the need for denormal numbers altogether. And, for a number of fully
connected and convolutional neural networks in computer vision and natural
language processing, 1/6/9 achieves numerical parity to standard
mixed-precision.
