[Gnuplot-bugs] Function frac()

[Іван М. С. <iim...@gm...>]

Dear developers of GNUPLOT!

In my work, I use the GNUPLOT Version 5.0 patchlevel 7 in which the
function frac(x) is missing, and for the functions int(x) and floor(x)
there exists an Upper Bound UB = 2**31 = 2,147,483,648, which means
these functions do not work when abs(x) > UB.  Hence, simply defined
function frac(x)=x-int(x) and its analogue fraf(x)=x-floor(x) calculate
the fractional part of numbers with small errors only when abs(x) < UB.
In many cases, the accuracy of these functions across the entire
range of real numbers is extremely important.  For example, the
current Terrestrial Time frequently used by astronomores is about
1,792,400,000 seconds and is close to the UB number, while one often
measures time with accuracy of 0.00001 seconds.  Hence accurately working
function "frac" would be very useful.

I am attaching files illustrating the issue for two versions of Gnuplot:
Testing_results_frac_GNU_5_pl_7.txt and Testing_results_frac_GNU_6_pl_5.txt
for GNUPLOT 5.0 patchlevel 7 and GNUPLOT 6.0 patchlevel 5.
In the former case, the frac(x) strongly fails for x with abs(x) > UB.
In the latter case, the frac(x) in this newer Gnuplot operates better;
however, one can see that it sill has unacceptably large roundoff errors
for large x.

I have attempted to address these shortcomings by modernizing existing
functions and creating a new one.  The program script and an example of
its operation are provided at the end of this letter and illustrate these
changes.  The modernized and newly created functions - INT(x), FLOOR(x),
and FRAC() - have been comprehensively tested and operate without errors.
I would like to ask that you consider the above when fixing bugs in the
old version and creating a new version of Gnuplot.  Please let me know
about receiving this letter.

Sincerely,
Physicist-Astronomer Motrunich Ivan I.

===========================================================================================

# program for modernizing functions int(x) and floor(x) and creating new
function FRAC(x)
set macros
UB=2**31           # the new constant - Upper Bound of the functions int(x)
and floor(x)
frac(x)=x-int(x)   # old (bad) function frac(x) - does not work properly
when abs(x) >= UB
N="int(x/UB)*UB"
ain(x,y)=int(x-y)+y                    # the auxiliary function for new
function INT(x)
INT(x)=abs(x)<UB?int(x):ain(x,@N)      # the new modified function INT(x)
FLOOR(x)=INT(x)-(x<0?1:0)              # the new modified function FLOOR(x)
O="10.**int(log10(4e15/x))"
p(x,y)=INT(sgn(x)*.5+(x-INT(x))*y)/y   # the auxiliary function for new
function FRAC(x)
FRAC(x)=abs(x)<1?x:p(x,@O)             # the new function FRAC(x)
# The new functions operate without errors across the entire range of
numbers.

pr '          PROGRAM TEST FOR POSITIVE NUMBERS'
sn=1.06; mn=87654321.54; su=UB-4.37; bu=UB+4.37; bn=123456789012.23;
bp=pi*1e13
pr 'inp numb> ',sn,pi,mn,su,bu,bn,bp
pr '    int-> ',int(sn),int(pi),int(mn),int(su),int(bu),int(bn),int(bp)
pr '    INT-> ',INT(sn),INT(pi),INT(mn),INT(su),INT(bu),INT(bn),INT(bp)
pr '  floor->
',floor(sn),floor(pi),floor(mn),floor(su),floor(bu),floor(bn),floor(bp)
pr '  FLOOR->
',FLOOR(sn),FLOOR(pi),FLOOR(mn),FLOOR(su),FLOOR(bu),FLOOR(bn),FLOOR(bp)
pr '   frac->
',frac(sn),frac(pi),frac(mn),frac(su),frac(bu),frac(bn),frac(bp)
pr '   FRAC->
',FRAC(sn),FRAC(pi),FRAC(mn),FRAC(su),FRAC(bu),FRAC(bn),FRAC(bp)
pr '          PROGRAM TEST FOR NEGATIVE NUMBERS'
pr 'inp numb> ',-sn,-pi,-mn,-su,-bu,-bn,-bp
pr '    int->
',int(-sn),int(-pi),int(-mn),int(-su),int(-bu),int(-bn),int(-bp)
pr '    INT->
',INT(-sn),INT(-pi),INT(-mn),INT(-su),INT(-bu),INT(-bn),INT(-bp)
pr '  floor->
',floor(-sn),floor(-pi),floor(-mn),floor(-su),floor(-bu),floor(-bn),floor(-bp)
pr '  FLOOR->
',FLOOR(-sn),FLOOR(-pi),FLOOR(-mn),FLOOR(-su),FLOOR(-bu),FLOOR(-bn),FLOOR(-bp)
pr '   frac->
',frac(-sn),frac(-pi),frac(-mn),frac(-su),frac(-bu),frac(-bn),frac(-bp)
pr '   FRAC->
',FRAC(-sn),FRAC(-pi),FRAC(-mn),FRAC(-su),FRAC(-bu),FRAC(-bn),FRAC(-bp)