... As promised, my manual manipulation of FixedPoint (arbitrary length strings of digits) ...

<Reveal this spoiler or all in this thread>
__RESULTS__
------------------------------------------------------
ulp(1)   = 0.0000000000000002220446049250313080847263336181640625
1+ulp(1) = 1.0000000000000002220446049250313080847263336181640625
------------------------------------------------------

----------------------------------------
(1+ulp(1))*2**-1022 =
0.00000000000000000000000000000000000000000000000000000000000000000000
+000000000000000000000000000000000000000000000000000000000000000000000
+000000000000000000000000000000000000000000000000000000000000000000000
+000000000000000000000000000000000000000000000000000000000000000000000
+000000000000000000000000000000002225073858507201877155878558578948240
+788008848683704195613130031211968860399600696529790429221262885863903
+701367028190801717129607271191035512722741317515219905574004313880456
+780323337753988163917738732895924607422927011307805381339708165336129
+644744952978952121897909078385258336590185178961879988515042751478263
+607602168043622031129270045483207396484571310391222596393560832244062
+389690727689018671705454927517398658932481040173822832825124579506565
+573819103800864691161582871998970864729322144979697154670672039979199
+080916034762598038599542473984767886118009507251154376238960371621517
+172981601154460435953128432540644193864532490538913779568091580479240
+509922741385427494262054264040883983691918741817298779334027924276754
+4565229087538682506419718265533447265625

FixedPoint: 1                = Digits before the fixed decimal point
FixedPoint: 1074             = Total digits the fixed decimal point
FixedPoint: 308              = Location of first significant digit (nt
+h digit after decimal point)
FixedPoint: 767              = Number of significant digits

Double:           2.2250738585072019e-308
HexFloat:        +0x1.0000000000001p-1022
DecFloat:     +0d1.0000000000000002p-1022

Math:       308              = should start at ceil(-log10(x))-th digi
+t after decimal point
Math:       1074             = total number of digits after decimal sh
+ould be the lowest power of two = ceil(-log2(ulp(x)))
Math:       767              = Number of significant digits

[download]

So what I said poorly as "it takes n fixed-point decimal digits to exactly represent an n-bit fixed-point fractional binary number" could have been better phrased: "if the smallest power of two exactly represented in a floating-point number is -n, it takes n decimal digits after the fixed decimal point to exactly represent that same number." In the example of nextUp(1), the smallest power of two is -52, so it takes 52 decimal digits after the fixed decimal point to exactly represent it. In the example of nextUp(POS_NORM_SMALLEST) = 2**-1022 + 2**-1074, it takes 1074 digits after the fixed decimal point to exactly represent it.

Thanks for this interesting diversion.

update: removed duplicate end-code tag

In reply to Re^12: subtraction issue by pryrt
in thread subtraction issue by Anonymous Monk

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