The Planck units are not good natural units, but any good system of natural units will result in very large and very small values (i.e. in the range 10^10 to 10^50 or their reciprocal values) for most physical quantities describing properties of things close in size to a human.

Therefore double precision, which accepts values even over 10^300, is good enough to store any values measured with natural units, while single precision (range only up to around 10^38) would be overflown by many values measured with natural units, and overflow would be even more likely in intermediate values of computations, e.g. products or ratios.

For those not familiar with the term, a system of natural units for the physical quantities is one that attempts to eliminate as many as possible of the so-called universal constants, which appear in the relationships between the physical quantities only as a consequence of choosing arbitrary units to measure some of them.

While the Planck units form one of the most notorious systems of natural units, the Planck units are the worst imaginable system of units and they will never be useful for anything. The reason is that the Newtonian constant of gravity can be measured only with an extremely poor uncertainty in comparison with any other kind of precise measurement.

Because of that, if the Newtonian constant of gravity is forced to have the exact value 1, as it is done in the system of Planck units, then the uncertainty of its measurement becomes an absolute uncertainty of all other measured values, for any physical quantities.

The result is that when the Planck units are used, the only precise values are the ratios of values of the same physical quantity, e.g. the ratio between the lengths of 2 objects, but the absolute values of any physical quantity, e.g. the length of an object, have an unacceptably high uncertainty.

There are many other possible choices that lead to natural systems of units, which, unlike the Planck units, can simplify symbolic theoretical work or improve the accuracy of numeric simulations, but the International System of Units is too entrenched to be replaced in most applications.

All the good choices for natural units have 2 remaining "universal constants", which must be measured experimentally. One such "universal constant" must describe the strength of the gravitational interaction, i.e. it must be either the Newtonian constant of gravity, or another constant equivalent to it.

The second "universal constant" must describe the strength of the electromagnetic interaction. There are many possible choices for that "universal constant", depending on which relationships from electromagnetism are desired to not contain any constant. The possible choices are partitioned in 2 groups, in one group the velocity of light in vacuum is chosen to be exactly one (or another constant related to the velocity of light is defined to be 1), which results in a natural system of units more similar to the International System of units, while in the second group of choices some constant related to the Coulomb electrostatic constant is chosen to be exactly 1, in which case the velocity of light in vacuum becomes an experimentally measured constant that describes the strength of the electromagnetic interaction (and the unit of velocity is e.g. the speed of an electron in the fundamental state of a hydrogenoid atom).

I have experimented with several systems of natural units and, in my opinion, the best for practical applications, i.e. which lead to the simplest formulas for the more important physical relationships, are those in which the Coulomb law does not include "universal constants" and the speed of light is a constant measured experimentally, i.e. the opposite choice to the choice made in the International System of Units.

The Planck units are always suggested only by people who have never tried to use them.

The choice from the International System of Units, to have the speed of light as a defined constant while many other "universal constants" must be measured, was not determined by any reasons having anything to do with what is more appropriate for modern technology.

This choice is a consequence of a controversy from the 19th century, between physicists who supported the use of the so called "electrostatic units" and the physicists who supported the use of the so called "electromagnetic units". Eventually the latter prevailed (which caused the ampere to be a base unit in the older versions of the SI, instead of the coulomb), because with the technology of the 19th century it was easier to compare a weight with the force between 2 conductors passing a fixed current than to compare a weight with the force between 2 conductors carrying a fixed electrical charge. There is a long history about how SI evolved during the last century, but the original choice of the "electromagnetic units" instead of the "electrostatic units" made SI more compatible with the later successive changes in the meter definition, which eventually resulted in the speed of light being a defined constant, not a measured constant.

Nowadays that does not matter any more, but few people remember how the current system has been established and most who have grown learning the International System of Units have the wrong impression that having an exact value for the speed of light is somehow more "natural" than having for it an experimentally measured value.

The truth is that there are many systems of natural units, and each of them is exactly as natural as any other of them, because all have a single experimentally measured electromagnetic constant. When the velocity of light is removed from some equations, an equivalent "universal constant" is introduced in other equations, so which choice is best depends on which equations are more frequently used in applications.