Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Exponent range — where it appears

The span of magnitudes a floating-point format can represent before it overflows or underflows. It is set by the number of exponent bits and is the property that separates bfloat16 from half precision at equal width.

Named by 9 essays across one field — each of them below, with the objects they name alongside it.

3456789101100.250.50.751bits in the exponent fieldeach curve as a fraction of its own maximumbfloat16fp16rangeprecisionto 617 decadesto 3.9 digitswhat the split buysbfloat16: largest number3.4·10³⁸fp16: largest number6.6·10⁴a bit of exponent doubles the rangea bit of significand adds a third of a digit

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.

arithmetic · Floating point 10⁻³10⁻¹10¹10³01magnitude448NaN — no ∞0.0156 — the smallest normaldrawn from the format's own rulespositive finite values126largest finite value448worst round-trip error0the subnormals are the evenly spaced ticks at the lefteverything a byte can be

Eight bits, and a format that breaks the rules

E4M3 reuses the exponent code IEEE reserves for infinities, so it reaches 448 where the same bits under IEEE's rules would reach 240 — and has no infinity left to signal an overflow with. The same computation is a NaN on one conforming device and 448 on another.

arithmetic · Floating point half a stepthe 32 values of one block, in the order they arrivestep 0.0625one scale, thirty-two valuesoctaves inside the block1.9entries rounded to zero0worst error over its bound131 levels either side of zerothe largest entry chose the 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.

arithmetic · Block formats 02468101201change of units, by exponent10⁰10¹10²10³10⁴10⁶10¹⁰10¹⁶10¹⁹10²⁰10⁴⁰10¹⁵⁰as writtenafter scalinga range questionlargest finite value3.4·10³⁸predicted boundary γ1.8·10¹⁹last γ that forms10¹⁹stops where scaling fails0not a poor answerno answer at all

The units that overflow before the answer does

A change of variable that is exact in the algebra requires γ² times a matrix to be a number the format can hold. In binary64 that is a bound nobody meets by accident. In binary32 it arrives at 10¹⁹ and in fp16 at 256, and past it there is no answer rather than a poor one.

arithmetic · Overflow deepest barrier, far well right to within halfno exponent limit40gradual, started at 102433fp16, gradual underflow23fp16, flush to zero13051015202530354010⁻⁵10⁻⁴10⁻³10⁻²10⁻¹110¹barrier depth, octaves below the near wellrelative error in the far well's masssmallest normalsmallest subnormalno exponent limitgradual, started at 1024fp16, gradual underflowfp16, flush to zeroan error of one: the far well is emptysubnormals bought nine octaves

The well on the far side of the band

Gradual underflow was said to buy a predicate and not an answer, because a quantity that has decayed into the subnormal range is already lost. A quantity that passes through the band on its way somewhere else is not. The stationary distribution of a two-well chain, computed in half precision across a barrier whose top is two to the minus sixteen of the near well, keeps its far well's probability of 0.2454 to three digits with subnormals and returns exactly zero without them. The normwise backward error calls both answers exact.

arithmetic · Subnormals 0510152025303540first barrier depth at which the far well is wrong by half, octavesfp16the band adds 11 octaves · starting at the top adds 15E4M3the band adds 2 octaves · starting at the top adds 10E5M2the band adds 3 octaves · starting at the top adds 14flush to zero13gradual underflow24flush, started at the top29gradual, started at the top39flush to zero5gradual underflow7flush, started at the top13gradual, started at the top17flush to zero13gradual underflow16flush, started at the top27gradual, started at the top30dot: median of nine chains · bar: their rangethe headroom is worth more than the band

The far well in a byte

Half precision's subnormals carried a two-well chain's far well ten octaves deeper than flush to zero, and the essay that measured it predicted that on the eight-bit formats the band would be worth almost nothing and starting high almost everything. Measured on nine chains, the band is worth what it always was, two or three octaves; starting at the top of the range is worth ten and fourteen; and at three significant bits the far well is lost on eight chains of nine with no exponent limit at all, because the precision fails before the range does.

arithmetic · Subnormals 0102030405060first barrier depth at which the far well is wrong by half, octavesfp16E4M3E5M2gradual underflow, from onegradual, from the topcounter, floor and ceiling9 of 9counter, window from the rates9 of 9gradual underflow, from onegradual, from the topcounter, floor and ceiling1 of 9counter, window from the rates8 of 9gradual underflow, from onegradual, from the topcounter, floor and ceiling1 of 9counter, window from the rates1 of 9ring: chains that keep the far well past sixty octavesthe window is the unbounded exponent

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.

arithmetic · Subnormals 020406080100120barrier depth, octaves32 statesa rate is zero from 9140 statesa rate is zero from 11548 states56 states64 stateswindow under one octavewindow width below zerothe window rule first failsthe aiming rule first failsthe window per step: exacta stored rate is zeroE4M3, five chain lengths, every depth from 2 to 120a window needs an octave to hold a power of two

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.

arithmetic · Subnormals blocks of 232 states×40 states×48 states×56 states×64 states×blocks of 432 states×40 states×48 states×56 states×64 states×blocks of 832 states×40 states×48 states×56 states×64 states×020406080100120barrier depth, octaves×the predicted lineblock window under an octaveblock window emptyfirst stops being exactexacta stored rate is zeroE4M3, every depth from 2 to 120the block fails where its window empties

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.

arithmetic · Subnormals

Named alongside it

The objects these essays reach for when they reach for this one.

SignificandFp8OverflowDynamic rangeStationary distributionSubnormal numbersScalingSilent failureUnit roundoffFlush-to-zeroGradual underflowHalf-precision

All concepts