Better unit conversion strategy

[sharkdp]

The following problem has been resolved in the meantime (by defining imperial units through other imperial units, instead of always falling back to meters), but this is still relevant:

# https://xkcd.com/2585/
#
# I can ride my bike at 45 mph.
# If you round.

17 mph

ans -> meters/sec    // round
ans -> knots         // round
ans -> fathoms/sec   // round
ans -> furlongs/min  // round
ans -> fathoms/sec   // round
ans -> kph           // round
ans -> knots         // round
ans -> kph           // round
ans -> furlongs/hour // round
ans -> mi/h          // round       # TODO: replace mi/h with mph. see below for details
ans -> m/s           // round
ans -> furlongs/min  // round
ans -> yards/sec     // round
ans -> fathoms/sec   // round
ans -> m/s           // round
ans -> mph           // round
ans -> furlongs/min  // round
ans -> knots         // round
ans -> yards/sec     // round
ans -> fathoms/sec   // round
ans -> knots         // round
ans -> furlongs/min  // round
ans -> mph           // round

assert_eq(ans, 45 mph)


# Unfortunately, if we replace mi/h above with mph, the test fails.
# The reason lies in the '204 furlongs/hour -> mph' conversion.
# 204 furlongs are 25.5 miles (exact). 204 furlongs/hour -> mph is
# therefore supposed to yield 26 mph after rounding. However, in our
# primitive `Quantity::convert_to` implementation, we first convert
# to the corresponding base unit (m/s) and then back:
#
#    furlongs/hour -> m/s -> mph
#
# Minor floating point inaccuracies then lead to a result which is
# slightly less than 25.5 mph. After rounding, this becomes 25 mph.
#
# Improving the conversion algorithm slightly by removing common
# unit factors (here: 1/hour), improved the situation and now leads
# to the correct result when using mi/h (mile/hour). However, if we
# use the `mph` shorthand, our common-unit-factor computation fails
# and still leads to the wrong 25 mph result.
#
# A proper solution to this would probably build a full DAG for all
# unit definitions of a single physical dimension. The conversion
# algorithm could then be changed to use shortest paths in those
# graphs. For example: If both miles and furlongs were defined via
# the unit foot, we could use foot/hour as the 'base' unit for the
# conversion instead of meter/second:
#
#
#                meter       second
#
#                  ▲            ▲
#                  │            │
#                  │            │
#                  │            │
#
#                inch        minute
#
#                  ▲            ▲
#                  │            │
#                  │            │
#                  │            │
#
#         ┌────► foot         hour
#         │
#         │        ▲
#         │        │
#         │        │
#         │        │
#
#       mile    furlong
#
#
# Another, potentially simpler solution to this problem could be to
# introduce a proper type for rationals. This way, floating point
# inaccuracies would only appear for units with irrational defining
# conversion factors, like `unit degree = pi / 180 × radian`.