Mrs. Hudson was busy serving breakfast in the dining
room when Sherlock Holmes strode in. A worried look sullied his noble countenance
and he was clutching in his hand what appeared to be an important letter
bearing an impressive seal.

"Momentous news Watson, my dear friend, this
letter is a top secret report just arrived by special messenger from our old
friend Inspector Lestrade of Scotland Yard. He tells me that some fiendish devil has
been counterfeiting the Error function with a shoddy imitation and there's a
grave danger it could damage the environment if it spreads. It's vital that we put a stop to it as soon
as we can."

I looked at my friend's troubled features as he
extracted from the envelope two sheets of tracing paper which he closely
perused for a while before carefully placing them side-by-side in front of me
on the tablecloth. "It's alright
Watson" he sighed with relief after a few moments study, "The Yard
appear to have overstated the case as usual.
As you can see (Figure 1) the graph on the left is the real error
function and the one on the right is nothing more than our old friend the
hyperbolic tangent. Not even a rank
amateur could get those two mixed up, although I must admit there's more than a
passing resemblance to the untrained eye now I come to look at it though.
Perhaps I'd better investigate it a bit closer just to quantify
matters".

Fig. 1.The graph on the left
is _{} and on the right *y = tanh(x)*

Sherlock tapped away on the PC
keyboard for a few minutes before turning to me with an excited look on his
face. "It's really quite
interesting Watson, but the integral of the squared difference between Erf(x) and tanh(x) over the
infinite range is surprisingly small. I
make it just 0.01488. Mind you, when the
curves are superimposed (Figure 2) you could drive a hansom cab through those
gaps. Fortunately it's not a good enough
forgery to be really useful to anybody and our friends who print the Erf tables needn't worry themselves too much on that score
I feel sure."

Fig.2. The graphs of *y = Erf(x*) and
*y = tanh(x)*
when superimposed. Note the big gaps
referred to by Holmes.

At that very moment the door sprang open with a loud
bang and Mrs. Hudson bustled in followed closely at her heels by a flushed and
excited Inspector Lestrade. "Thank goodness I've caught you in
Holmes, the case has taken a decided turn for the worse, Look!"

Dramatically Lestrade drew
from his pocket another sheet of tracing paper similar to the two already on
the table and handed it to Holmes with a flourish. In silence Sherlock
took it from him and superimposed it on the real Error function (figure 3).
Holding them both up to the light he whistled softly and then declared
excitedly "My oath, Watson, but
it's the dead spit." (In the heat
of the moment this coarse lapse into the vernacular was surely understandable
and presumably due to a recent case that necessitated frequent trips to Southend-On-Sea in disguise as a common applied
mathematician). We all gazed in awe at
the uncanny statement scrawled in an unknown hand on the top of the new arrival

_{}

Fig. 3. The superimposed functions *y = Erf(x)* and
*y = tanh(ax+bx ^{3})*. Notice that the gaps have now disappeared!

Sherlock chewed thoughtfully on a
mouthful of the Pythagorean toast before exclaiming "By Harry! I see it
all clearly now Watson, the cunning devil has minimised
the squared difference by using two adjustable parameters, a and b, and he's
closed those gaps quite convincingly. In
fact I suspect that this imposter function will never be more than plus or
minus 0.06% in error! Clearly some
further investigation is urgently called for."

Several tense
minutes passed before Sherlock Holmes finally stood
up to leave the PC and stride over to the mantlepiece
in the direction of the Persian slipper.
Lighting his pipe Holmes looked up to the ceiling and said " The
clever devil has realised that there's no point in
going beyond the cubic power. The fit is
actually worsened if the fifth or the seventh powers are included. We're obviously dealing with a master here
Watson. It can be none other than that
devilish Napoleon of crime Professor Moriarty
himself. His treatise on the Binomial
Theorem at the age of 21 presaged a brilliant career as a mathematician before
he turned his fiendish mind to dastardly crime. I'm already hot on his trail over that Nongentium Dome business and this latest piece of villainy
is clearly an attempt to divert me from it.

Leaning forward and with a worried and thoughful look Lestrade said at
last "This is terrible news, Holmes, if this gets out it means anybody
with a non-programmable calculator will be able to evaluate the error function
quite accurately just by tapping on a few keys without doing any numerical
integration or even a power series summation.
Every skinflint in town with a tanh key will
be a threat, even kids in the classroom will have it. The only thing we can be thankful for is that
there's still no way of easily getting at the inverse."

For the second time that morning the door flew open
with a bang, this time to admit an increasingly flustered Mrs. Hudson bearing a
silver tray on top of which was perched a letter. "This has just come for you Mr. Holmes,
it was delivered by a rough looking fellow.
He refused to leave his name but I noticed he was wearing one of those
IMA ties you warned me about."

At this latest news and with trembling fingers Sherlock inspected the envelope, closely scrutinising the hand writing and holding the letter up to
his nose. " This envelope has few minor points of interest, Watson. It was clearly written by a left-hander
during a train journey, see here the way the lamdas
in the address are done backwards and notice those jerking breaks in the
otherwise smooth hand-writing. Moriarty is left-handed and I happen to know he is
currently travelling abroad on the Continent in a sly scheme to promote the
Euro; the smell of chocolate on the envelope is quite unmistakable. "

Holmes
then opened the envelope and withdrew from it a single sheet of paper headed *"The Inverse*" followed by the
stark lines of text

_{}

Written underneath were the chilling words *"I'll send it all to 'Mathematics
Today' if you don't lay off .'*"
A stunned silence descended on the room as we digested this latest piece
of dramatic news. The silence was broken
only by Holmes saying "We're done for Watson. If this is what I think it is then the inverse
error function can also be closely got by just a few more taps on the keyboard,
I'd better test it out right away, there's not a moment to lose."

The PC in the corner was our old *Babbage** Three (cwt.) CDLXXXVI* with the initials "VR" burned into
its monitor display. This had occurred
accidentally some time back when that popular screen saver had unavoidably been
left switched continuously on for several months when Sherlock
was away in Switzerland on the heels of Moriarty
during a previous adventure chronicled elsewhere^{1}. Impatiently Holmes fired up his latest
version of *Mathcad** 1900 Professional* and tapped away on
the keyboard. After a few moments he
turned to Lestrade to say thoughtfully. "This proxy inverse is pretty accurate
but there's hope for us yet. If Moriarty's inverse error function is any good and we give
it for an argument the error function itself we should always come back to
where we started. You can see (Figure
4) it's only really good between x equals
plus and minus two where the maximum error is (illegible). Unfortunately for us though, both the
imposter and its inverse also seem to work very well even with the complex
arguments *z = x + jy.*
With x running from 0 to 3 and y from 0 to 2 the maximum error is only
(illegible percent ) for the forward function and (illegible percent) for the
inverse. If this gets out we're in for
it alright, bad functions drive out good, the Error function market will
collapse and the Java programmers will soon have it wrapped up in a simple
one-liner method before you can say 'ArrayIndexOutOfBoundsException' "

Fig. 4.. If Moriarty's inverse error function were perfect both graphs should exactly coincide with the straight line y = x.

Pulling himself up to his full height and with a
somber tread Holmes walked over to his foot-locker to pull out the bulls-eye
lantern and his life-preserver. "The game's afoot Watson! There's nothing for it but for me to find
that scoundrel Moriarty as soon as possible. I'll
simply have to find a way to steal this dangerous formula from him and put a
stop to his evil mischief once and for all.
I leave for Brussels immediately.
If you don't hear from me in three weeks you'd better check the pages of
"Mathematics Today. Farewell dear friend." With those portentous words and a last bow he
turned to leave. The door slowly closed behind his tall figure and he was gone,
forever?

With sincere apologies to the memory of Sir
Arthur Conan Doyle.

1.
" The Memoirs of Sherlock Holmes: The Final
Problem", Doyle, Arthur Conan, Sir,