@EVAL's precision and significant digits

#1
The first result below is correct; the second is close but has fewer significant digits. The strings have 211 characters. I can't figure out what my precision specification has to do with the output.
upload_2017-9-4_23-8-2.png
 
#2
For N<103, @EVAL[2**N] comes out right. The first precision spec that makes @EVAL[2**103] come out right is "0.11".
Code:
v:\> echo %@eval[2**103=0.10]
10141204801825835211973625643010

v:\> echo %@eval[2**103=0.11]
10141204801825835211973625643008
I can't figure out exactly how my precision spec is related to the output.
 
#4
Maybe it's a bug, or a lack of documentation, (or me) but when computing base**exponent, (MAPM::pow() or m_apm_pow()) and base>1, "decimal_places" actually specifies the number of significant digits (which certainly includes what's to the left of the decimal point) and not the number of places after the decimal point.

If you add the number below (instead of 20) to the desired number of decimal places (when base>1) you should be safe. If you wish, I'll translate it into the non-C++ version (i.e., using m_apm_func() instead of MAPM::func()).

Code:
// this is before the decimal point; add the desired number of decimal places
INT extra (MAPM base, MAPM exponent)
{
   MAPM u, mm1 = MM_One;
   INT x;
   u = ++((base.log10() * exponent).ceil());
   CHAR *szdp = (CHAR*) AllocMem(u.significant_digits()+1);
   u.toIntegerString(szdp);
   x = atoi(szdp);
   FreeMem(szdp);
   return x;
}
I reproduced (and fixed, as above) the problem with 2**103. But I don't need the fix for .5**(-103), which is the same number. @EVAL, however, has the same problem with both.
 
#5
... or, more appropriately for @EVAL, the minimum of that number and 15,000.

Another thing ... 10**10000 works, but 10**10001 evokes an overflow message. Does that reflect an older limit? Isn't it 15000 these days?
 
#7
10,000 digits to the left of the decimal, and 10,000 to the right. It is not 15,000.
The help says
The maximum size is 30,000 digits (15,000 digits to the left of the decimal point and 15,000 decimal places).
And I was wrong about my .5**(-103) not needing the extra() digits. If you're going to use that, you should use it for any base.
 
#8
FWIW, extra() is a pretty tight estimate of what's needed. Below, extra is 46 when the number has 45 digits before the decimal point. I think there are special cases when extra()-1 won't be enough.
Code:
v:\> echo %@pow[1.01 10310.99]^r^n%_ruler
extra = 46
361111948879059149982803812474402661922976059.62768572890231244276
....+....1....+....2....+....3....+....4....+....5....+....6....+....7....+....8....+....9....+....0....+....1...
 
#10
It needs absolute value of base.log10() (found out the hard way) :-)
Code:
INT extra (MAPM base, MAPM exponent)
{
   MAPM mm1 = MM_One, u;
   INT x;
   u = ++(((base.log10()).abs() * exponent).ceil());
   CHAR *szdp = (CHAR*) AllocMem(u.significant_digits() + 1);
   u.toIntegerString(szdp);
   x = atoi(szdp);
   FreeMem(szdp);
   return x;
}
 
#11
FWIW, this (below) is more than 20 times faster than extra() and should work in reasonable cases (intergalactic distances and the national debt being possible exceptions).
Code:
    double   b = ... ,   // base
           e = ... ,   // exponent
           extra = 1 + ceil(e * abs(log10(b)));