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I would like to know if there is any way to convert decimal numbers to floating point, using the IEEE-754 standard, specifying whether it is simple or double. And vice versa, floating point to decimal specifying whether it is double or single.

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Yes. You should be able to import and export singles, doubles, or quad-precision numbers in the IEEE standard. See documentation here.

Let's check. We start with some nice 140 digit numbers.

(someNumbers = RandomReal[{1, 200}, 4, WorkingPrecision -> 140])

yielding:

{131.57843910491117542009357067044305298919523687461972590508545252797246656713683959501545442109954305145360196695641268407818906794518467693, 36.777212169625076229757244904535982444480287433912465096550209184759013144280916569005201142526738998958909685743061158332762603357972081895, 32.211986718696311559869777490390932918154744943543676822285668781237395283963457422161288449125141442835469928797914730615636279609237747203, 91.455311969662025855984807531478321762889779722600757410998496792433355267117887298097053278192108027867375906636235367744938744286139195293}

Let's export and import them as floats, doubles, and quads:

Export["/tmp/quads.dat", someNumbers, "Real128"];
Export["/tmp/doubles.dat", someNumbers, "Real64"];
Export["/tmp/floats.dat", someNumbers, "Real32"];
(floats = Import["/tmp/floats.dat", "Real32"]);
(doubles = Import["/tmp/doubles.dat", "Real64"]);
(quads = Import["/tmp/quads.dat", "Real128"]);

Now look at the accuracy relative to the original 140 digit Reals:

someNumbers - SetPrecision[floats, 140] // N

yields:

{-6.32966*10^-6, -1.88067*10^-6, 1.76948*10^-7, -2.66657*10^-6}

someNumbers - SetPrecision[doubles, 140] // N

yields:

{-7.80161*10^-15, 2.67301*10^-15, 2.41998*10^-15, 5.82303*10^-15}

and

someNumbers - SetPrecision[quads, 140] // N

yields:

{2.27275*10^-32, 4.51408*10^-33, 3.39809*10^-33, 1.31565*10^-33}

Let's convert within Mathematica going from doubles to floats:

floatsFromDoubles = ImportString[ExportString[doubles, "Real32"], "Real32"];

Do we land on the same thing as converting our 140 digit reals?

Yes: floatsFromDoubles===floats $\mapsto$ True.

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