sqrt, pow, exp and log

An ore find lies 30 m east and 40 m north of your base. How far is that? One tank has 0.5 kPa, the next 50,000 kPa – how do you display both sensibly? For questions like these the chip has four arithmetic instructions that go beyond mul and div.

Root and power#

sqrt r a takes the square root, pow r a b computes a to the power of b. With Pythagoras you get the distance from two legs:

pow tmp dx 2        # dx * dx
pow d dz 2          # dz * dz
add d d tmp
sqrt d d            # 30 and 40 give 50

pow follows the floating point standard IEEE-754. A few results you should know:

There is no square root of a negative number. The chip reports no error, it returns NaN – and every calculation that works with the value afterwards becomes NaN as well. One value subtracted the wrong way round is enough.

exp and log#

exp r a computes e to the power of a (e ≈ 2.718). log r a goes the way back: it is the natural logarithm, to base e. You need both wherever something grows or shrinks by a fixed share per tick. Example: a value loses 10 % of its distance to the target every tick. After how many ticks is only 1 % left?

log n REST          # REST = 0.01, log(0.01)
log tmp FACTOR      # FACTOR = 0.9, log(0.9)
div n n tmp         # about 43.7 ticks

The other way round, exp works out how much is left after n ticks. exp and log cancel each other out:

log tmp FACTOR      # log(0.9)
mul tmp tmp n       # n * log(0.9)
exp rest tmp        # 0.9 to the power of n, like pow rest FACTOR n

There is no log10#

The chip does not know a base-10 logarithm, nor ln – log already is the natural one. You get log10 through a division:

log10(x) = log(x) / log(10)

log(10) is a fixed number. As a define it costs no register:

define LN10 2.302585092994046   # log(10)
log e x
div e e LN10                     # e = log10(x)
floor e e                        # power of ten

With that you split a big value into power of ten and mantissa, the way you write 50,000 as 5 · 10⁴. By the way, 5e4 is not a valid number in the code – you have to split it yourself.