Dynamic range — where it appears
Named by 9 essays across 2 fields — each of them below, with the objects they name alongside it.
Where the hardware went
bfloat16 carries eight mantissa bits, which puts its refinement threshold at a condition number of 256. That is not an exotic matrix. It is an ordinary one, and past it the method still improves the answer by a factor of four hundred while getting nowhere near a usable one.
The other half of a format
fp16 and tf32 have the same eleven significand bits and their largest numbers are 65,504 and 3.4·10³⁸. For two phases this site simulated the significand alone, so it was obliged to report them as the same format — which is a claim, and a false one.
A norm that overflows before it is a norm
The vector of sixteen thousands has a Euclidean norm of 4,000, which fp16 represents exactly. Written as the square root of the sum of squares it returns infinity, because squaring doubles the exponent — and the expression costs half the format's range on the one computation every iterative method performs at every step.
One exponent for thirty-two numbers
Share the exponent across a block and the cost per value drops from eight bits to 6.25, and the accuracy improves — up to about three octaves of spread inside a block. Past that a single outlier deletes the thirty-one values beside it, and the 2-norm barely notices.
The definition asks for more of what defeats it
The rank of a p × p Hankel matrix of Markov parameters resolves a degree of p, and it needs 2p parameters to do it. Those parameters grow like the norm of the state matrix raised to their index, so the count that buys resolution is the same count that buys dynamic range. One model, four run lengths, and a spread that runs from 10¹¹ to 10¹⁰².
A bit buys an octave
The outlier a block survives is exactly two raised to its significand width — 8 at three bits, 32 at five, 128 at seven, 512 at nine. Each extra bit doubles the range the block tolerates and halves the ordinary entry's error. Reordering the same numbers buys every octave at once and costs nothing.
Room for the step, not the value
Starting a recurrence at the top of an eight-bit format's range spends its headroom once. An exponent counter beside the stored value was predicted to spend it at every step, remove every range failure and leave only the precision. Rescaled on the value — back to one whenever it nears the bottom or the top — it does not: E4M3 still loses its far well on seven or more chains of nine, because one climbing step multiplies the value by forty and E4M3 clips silently at 448. Rescaled into a window read off the chain's own rates, it reproduces the unbounded exponent bit for bit on all 531 runs of every format, and what is left of E4M3's limit is nothing.
An octave to land in
A window read off a chain's own rates kept an eight-bit recurrence exact to sixty octaves, and its width, falling towards zero, was predicted to mark where it would stop being exact, to within an octave. It stops two to seven octaves earlier, as soon as the window is narrower than the octave a power-of-two rescale moves by. The rule that aims every step, predicted to outlast it, fails too, once the value it aims at no longer fits. A window computed for each step separately never narrows below 4.8 octaves, reproduces the unbounded exponent on every depth at which the format can still store the chain, and makes the fewest rescales of the three.
Twice the block, half the depth
An eight-bit chain kept exact by a window per step was predicted to keep exactness with one exponent shared by two states, and to lose it with four from the depth where four of its steepest steps span more than the narrowest window, 4.8 octaves. Blocks of two fail on four chains of five, at 66 to 114 octaves. Blocks of four fail at 35 to 66, two to three times deeper than the predicted 13 to 26. Every block size fails within three octaves of the depth where its block's window empties, and that depth is 4.0 to 4.6 octaves for each state a block holds.
Named alongside it
The objects these essays reach for when they reach for this one.
SignificandOverflowExponent rangeFp8bfloat16Half-precisionScalingShared exponentSilent failureStationary distributionSubnormal numbersBlock floating point