Thiele's interpolation formula

Implemented to parallel the generalized formula, making for clearer, but slower, code. Offsetting that, the use of Promise allows concurrent calculations, so running all three types of interpolation should not take any longer than running just one (presuming available cores).

# reciprocal difference:
multi sub ρ(&f, @x where * < 1) { 0 } # Identity
multi sub ρ(&f, @x where * == 1) { &f(@x[0]) }
multi sub ρ(&f, @x where * > 1) {
    ( @x[0] - @x[* - 1] )       # ( x - x[n] )
    / (ρ(&f, @x[^(@x - 1)])     # / ( ρ[n-1](x[0], ..., x[n-1])
    - ρ(&f, @x[1..^@x]) )       # - ρ[n-1](x[1], ..., x[n]) )
    + ρ(&f, @x[1..^(@x - 1)]);  # + ρ[n-2](x[1], ..., x[n-1])
}
 
# Thiele:
multi sub thiele($x, %f, $ord where { $ord == +%f }) { 1 } # Identity
multi sub thiele($x, %f, $ord) {
  my &f = {%f{$^a}};                # f(x) as a table lookup
 
  # must sort hash keys to maintain order between invocations
  my $a = ρ(&f, %f.keys.sort[^($ord +1)]);
  my $b = ρ(&f, %f.keys.sort[^($ord -1)]);
 
  my $num = $x - %f.keys.sort[$ord];
  my $cont = thiele($x, %f, $ord +1);
 
  # Thiele always takes this form:
  return $a - $b + ( $num / $cont );
}
 
## Demo
sub mk-inv(&fn, $d, $lim) {
  my %h;
  for 0..$lim { %h{ &fn($_ * $d) } = $_ * $d }
  return %h;
}
 
sub MAIN($tblsz = 12) {

  my ($sin_pi, $cos_pi, $tan_pi);
  my $p1 = Promise.start( { my %invsin = mk-inv(&sin, 0.05, $tblsz); $sin_pi = 6 * thiele(0.5, %invsin, 0) } );
  my $p2 = Promise.start( { my %invcos = mk-inv(&cos, 0.05, $tblsz); $cos_pi = 3 * thiele(0.5, %invcos, 0) } );
  my $p3 = Promise.start( { my %invtan = mk-inv(&tan, 0.05, $tblsz); $tan_pi = 4 * thiele(1.0, %invtan, 0) } );
  await $p1, $p2, $p3;
 
  say "pi = {pi}";
  say "estimations using a table of $tblsz elements:";
  say "sin interpolation: $sin_pi";
  say "cos interpolation: $cos_pi";
  say "tan interpolation: $tan_pi";
}

Output:

pi = 3.14159265358979
estimations using a table of 12 elements:
sin interpolation: 3.14159265358961
cos interpolation: 3.1387286696692
tan interpolation: 3.14159090545243