Precision preset Mantissa bits m = 3

Linear number line

Same number of representable values per binade; twice the spacing in the next binade.

representable fp value real value rounding error

Jump to binade

Zoomed active binade

Drag or click inside the plot. The gold band is the rounding interval for the selected floating point value.

Current numbers

binade [1, 2) All values here share exponent e.

ULP 0.125 2^{e - m}, the local spacing.

real value 1.3125 The value before rounding.

nearest fp 1.25 Nearest discrete value in this toy format.

absolute error 0.0625 At most 0.5 ULP for round-to-nearest.

ULP error 0.5 How far the real value is from the nearest fp slot.

relative error 4.76e-2 Relative error is largest near the left edge of the binade because the same absolute error is divided by a smaller value.

ULP = 2^{e − m}

max absolute rounding error = 0.5 × ULP

relative error = |real − rounded| / |real|

For small precisions every representable value is drawn. For float32/float64, the plot samples markers because a single binade contains millions or quadrillions of values.

Note: this toy visualization only shows normalized floating point values; it does not model subnormals.