How far can you trust your computer?

[AlanMiller]

The equation

y = e^x ln(1 + e^-x)

is one of those functions which asymptotes towards a fixed value (y=1) as x gets bigger and bigger: y ® 1 as x ® ¥

This computer-generated graph (or any graphing calculator) shows this behaviour as expected for x ranging from 0 to 30.

Now, change the x-range maximum to 37. What's happened? Bonus points if you can identify the unexpected "artefacts".

Mathematicians/ numerical analysts will readily pick up on this ... so maybe put your answers in spoiler tags.

Alan

[Rudi]

Now, change the x-range maximum to 37. What's happened?

Spoiler

My heart took a beat.

Not a math(s) answer, but it's what happened on the graph I saw.

[HansV]

Excel shows the same behavior:

S1178.png

Spoiler

Many applications, including Excel, compute floating numbers with a precision of about 15 significant digits.
e^-x becomes very small. For example, e^-37 ~ 8.5*10^-17. That in itself is not a problem.
But that means that 1+e^-x is so close to 1 that the difference is beyond the 15th digit.
Effectively, 1+e^-x becomes indistinguishable from 1, and its logarithm becomes 0.
And although e^x becomes very large, multiplying with 0 still yields 0.
In the range between x=30 and x=37, rounding errors cause the chart to 'wobble'.

[AlanMiller]

This may provide some insight as to what those "wobbles" actually are, and what their root cause is.

Alan

[AlanMiller]

I should have really signed this topic off. The key to the behaviour is a limitation of computers called machine epsilon e.
It is essentially the smallest number a computer can store, about 2 x 10^-16 as shown on this graph of e^-x.

So if you follow the original plot from right to left, the value of the function (as calculated by the computer) is actually
y = e^x ln(1 + ne) where n = 0, 1, 2, 3, ....

For n=0 (for x > 36.7 or so) y = 0
For n=1 (the first "step") y = e^x ln(1 + e)
For n=2 (second step) y = e^x ln(1 + 2e) etc.

The strange curves or "wobbles" are all just exponentials of x multiplied by constants over small ranges of constant ne.

The whole phenomenon can create "all hell done break loose" when it occurs deep internally in iterative models, unbeknownst to the modeller e.g. like in the original "puzzle", where the inputs and outputs are just "ordinary-sized" numbers - neither excessively small or large for the computer capabilities.

Alan