Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

IEEE 754 — where it appears

The standard that fixes what a floating-point number is, how it rounds, and what happens at the edges. Every claim about arithmetic on this site is stated against it, including the directed rounding modes that make interval arithmetic possible.

Named by 5 essays across 2 fields — 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 10⁻⁷10⁻⁵10⁻³10⁻¹⁰10⁻⁸10⁻⁶magnitudespacing 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 2022242628303210⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²k, where the entries are near 2ᵏrelative error of the determinantas writtenfused: exactproducts need 54 bitsone rounding, the whole answertrue determinant1naive, k = 300fused, k = 301first wrong at k27sizes returning 06both forms conformand the source does not say which

One multiply the compiler removed

A determinant whose value is exactly 1, computed as exactly 0 by the expression that is written down, and exactly 1 by the same expression with the multiply and the add fused. Both forms conform to IEEE-754, both are legal compilations of the same source, and nothing in the program says which one you have.

machine · Fma contraction 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 10²10³10⁴10⁻⁸10⁻⁷10⁻⁶10⁻⁵10⁻⁴terms summedrelative error+∞, −∞ 1.01zero 1.00stochastic 0.50nearest 0.47√n against n, fittednearest, fitted exponent0.47stochastic, fitted exponent0.5toward +∞, fitted exponent1twelve seeds averaged at each sizethe slope is the bias, not the precision

The direction the error leans

The size of one rounding error is set by the precision. How ten thousand of them combine is set by something else entirely — the rounding mode — and the fitted exponents are 0.47 for round-to-nearest and 1.01 for round-toward-infinity, on identical data at identical precision.

arithmetic · Rounding

Named alongside it

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

SignificandHalf-precisionOverflowUnit roundoffExponent rangebfloat16Bitwise reproducibilityCatastrophic cancellationDefiniteness testDeterminantDirected roundingDynamic range

All concepts