Given an arbitrary floating point value (eg. 1.9041105342991877e+258 or -8.2727285363069939e-293) how can I determine the smallest representable value that can be added or subtracted from that FP value and cause it to actually change?

To demonstrate the problem:

$fp1 =  1.9041105342991877e+258;;

$change = $fp1 - 1.9041105342991876e+258;                  ## note cha
+nge from 7 to 6 in the last digit of the number being subtracted
printf "% 25.17g\n% 25.17g\n", $fp1, $change;;
  1.9041105342991886e+258
                        0                                  ## no chang
+e

$change = $fp1 - 1.9041105342991875e+258; 
printf "% 25.17g\n% 25.17g\n", $fp1, $change;;             ## note cha
+nge from 7 to 5 in last digit
  1.9041105342991886e+258
                        0                                  ## no chang
+e

$change = $fp1 - 1.9041105342991874e+258;                  ## note cha
+nge from 7 to 4 in last digit
printf "% 25.17g\n% 25.17g\n", $fp1, $change;;
  1.9041105342991886e+258
  2.1337646185215534e+242                                  ## A change
+!                                     ### 1 ###

## Now try the increment instead of decrement 
$change = $fp1 - 1.9041105342991878e+258;                  ## note cha
+nge from 7 to 8 in last digit
printf "% 25.17g\n% 25.17g\n", $fp1, $change;;
  1.9041105342991886e+258
 -2.1337646185215534e+242                                  ## A change
+ straight away!                      ### 2 ###

##### Note that whilst the sign of the change at ### 1 ### & ### 2 ###
+ is different, the absolute value is the same.

[download]

So a restatement of the problem might be: Given 1.9041105342991877e+258 find the smallest number that can be both added to and subtracted from that number and will cause it to change value. Ie. 2.1337646185215534e+242

But the problem gets more interesting.

Now I try to work with the second value in the (eg.) above: -8.2727285363069939e-293

$fp2 =  -8.2727285363069939e-293;;
printf "% 25.17g\n", $fp2;;
 -7.9999999999999948e-293                          ### WTF? ###

[download]

Please note: that number (-8.2727285363069939e-293) was output from a perl program. Though it is hard to reproduce, as there were random numbers involved.

My only explanation for why it comes out differently to what was entered is that -8.2727285363069939e-293 must be a "denormal number which can be produced as a result of calculations, and will be displayed, but when you initialise a variable with that value, it will normalise it; hence what comes out is different to what goes in.

All of which makes the problem I'm looking for a solution to -- that of finding the minimal representable change -- more complicated.

At this point someone is going to point me at this. Please don't, I've read it.

And someone is going to ask what is this for, which I could answer, but it would take us way off topic and doesn't change the posed problem.

With the rise and rise of 'Social' network sites: 'Computers are making people easier to use everyday'
Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.
"Science is about questioning the status quo. Questioning authority". I knew I was on the right track :)
In the absence of evidence, opinion is indistinguishable from prejudice. Not understood.

In reply to Determining the minimum representable increment/decrement possible? by BrowserUk

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