... now that my memory has been jogged...

I believe Kahan actually stated "The language of pairs is insufficient in
the complex domain." (Cryptic enough for you?)

His demonstration was called "Borda's Mouthpiece" and was a simple conformal
mapping problem from linear rays in the right half complex plain to the
streamlines associated with a fluid flow at hypersonic speeds in a tube.

Specifically, the problem in other languges is the following:

Borda's Mouthpiece requires taking the square root of the square of values
along the negative imaginary axis. When you square (0 - 1i) you get (-1).
...but actually you get (-1 + 0-). Most languages simply accept the square
as (-1) = (-1 + 0+).

Taking the subsequent square root when you have made this incorrect
assumption gives the value sqrt(-1) -> (0+1i). And so the stream line mapped
from the negative imaginary axis ends up cutting across all the other stream
lines when this condition is encountered. An obvious error!

The correct answer is obtained by noting that (0 - 1i)^2 -> (-1 + 0-) and
sqrt(-1 + 0-) -> (0 -1i) again. Only by properly considering the nature of
floating point zero (i.e., which of the two you really have) can you perform
this computation correctly.

Whew! That was a bloody difficult two weeks to go from the cryptography of
Kahan to a correctly working implementation... And BTW, you DON'T want to
use polar representation -- that eats away seriously at your arithmetic
precision, dropping from 15 digits to something like 6-8 digits in double
precision math. You have to stay in the rectangular representation to
preserve arithmetic precision -- and that requires a full understanding of
the nature of 0+ and 0-.

- DM


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