A laser rangefinder measures a distance and the reading is stored in the binary32…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Numerical and computational mathematics · Numerical and computational mathematics

Error, Chance and the Machine, Ch Rounding, backward error and the sensitivity of a computation

Problem

A laser rangefinder measures a distance

L = 12345.6\ \mathrm{m}

and the reading is stored in the binary32 (single-precision) floating-point format, for which the significand (including the implicit leading bit) has

t = 24

bits. ((a)) State

\epsilon_{\rm M}

and

u

for binary32. ((b)) Using the rounding model

\mathrm{fl}(L) = L(1+\delta)

with

|\delta|\le u

, give the bound on the absolute error committed in storing

L

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