#!perl -l

# Looking at function F = pi ^ y
#
#             F = pi ^ y
#    error = dF = y * pi ^ (y-1) * dpi
# =>        dpi = error / ( y * x ^ (y-1) )
#
# So if we want to limit the error in F to 0.1:

use constant ERROR => 0.1;
use constant PI    => 4 * atan2(1,1);

warn "Check: pi is ", PI, "\n";

my @y = (1..9, map(10 * $_, 1..9), map(100*$_,1..10));

foreach my $y (@y) {
    printf "y=%d  dpi=%g\n", $y, dpi($y);
}

sub dpi {
    my $y = $_[0];
    my $denom = $y * PI ** ($y-1);
    return ERROR / $denom;
}

__END__
Check: pi is 3.14159265358979
y=1  dpi=0.1
y=2  dpi=0.0159155
y=3  dpi=0.00337737
y=4  dpi=0.000806288
y=5  dpi=0.00020532
y=6  dpi=5.44627e-05
y=7  dpi=1.48594e-05
y=8  dpi=4.13867e-06
y=9  dpi=1.171e-06
y=10  dpi=3.35468e-07
y=20  dpi=1.79111e-12
y=30  dpi=1.27507e-17
y=40  dpi=1.02116e-22
y=50  dpi=8.72341e-28
y=60  dpi=7.76258e-33
y=70  dpi=7.10495e-38
y=80  dpi=6.6385e-43
y=90  dpi=6.30114e-48
y=100  dpi=6.05568e-53
y=200  dpi=5.8364e-103
y=300  dpi=7.5001e-153
y=400  dpi=1.08428e-202
y=500  dpi=1.67203e-252
y=600  dpi=2.68581e-302
y=700  dpi=0
y=800  dpi=0
y=900  dpi=0
y=1000  dpi=0
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Update: yes, for example if we're calculating pi^y for @y=5..15, and we introduce an error to pi of 1e-7:

#!perl -l

use strict;
use warnings;
use constant PI    => 4 * atan2(1,1);
use constant DPI   => 1e-7;

my @y = (5..15);

foreach my $y (@y) {
    my $f1 = F(PI,$y);
    my $f2 = F(PI+DPI,$y);
    my $delta = $f2 - $f1;
    printf "y=%-2d  delta=%g\n", $y, $delta;
}

# F = pi ^ y
sub F {
    my ($pi, $y) = @_; 
    return $pi ** $y,
}

__END__
y=5   delta=4.87045e-05
y=6   delta=0.000183612
y=7   delta=0.000672972
y=8   delta=0.00241623
y=9   delta=0.00853968
y=10  delta=0.0298091
y=11  delta=0.103013
y=12  delta=0.353045
y=13  delta=1.20155
y=14  delta=4.06515
y=15  delta=13.6833
[download]