Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Flush-to-zero — where it appears

A floating-point mode that replaces any result below the smallest normal number with exactly zero instead of a subnormal. It saves hardware cost on tiny operands, and in exchange a quantity that is small but meaningful can vanish outright rather than lose precision gradually.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as gradual underflow — the same set of essays touches all of them, so they are one junction rather than several.

10⁻⁷10⁻⁵10⁻³10⁻¹⁰10⁻⁸10⁻⁶magnitudespacing to the next numberthe smallest normalgradualflush to zerosmallest normal6.1·10⁻⁵smallest subnormal6·10⁻⁸octaves of subnormals10pairs that lie under FTZ10the spacing stops halving and stays putwhich is what makes x − y = 0 mean x = y

The numbers below the smallest one

Below the smallest normal number the spacing stops halving and stays put, all the way to zero. That is what gradual underflow is, and the thing it buys is the sentence every algorithm assumes without being told — x minus y is zero only when x equals y.

arithmetic · Subnormals 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

Named alongside it

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

Gradual underflowSubnormal numbersBackward errorBidiagonal matrixComponentwise condition numberExponent rangeHalf-precisionIEEE 754OverflowRelative errorSignificandSkeel condition number

All concepts