Convert from gnat's eye to astronomical unit

Unit Definition (gnat's eye)
The gnat's eye is an idiomatic "unit" of distance; it's common to hear that something is "as small as a gnat's eye". In fact, the eyes of typical gnats tend to have diameters similar in size to a hair's breadth: roughly 100-150 micrometers (0.10-0.15 millimeters).

Unit Definition (astronomical unit)
An Astronomical Unit is approximately the mean distance between the Earth and the Sun. It is a derived constant and used to indicate distances within the solar system. Its formal definition is the radius of an unperturbed circular orbit a massless body would revolve about the sun in 2*(pi)/k days (i.e., 365.2568983.... days), where k is defined as the Gaussian constant exactly equal to 0.01720209895. Since an AU is based on radius of a circular orbit, one AU is actually slightly less than the average distance between the Earth and the Sun (approximately 150 million km or 93 million miles).