PARI real precision is broken in many ways

[jdemeyer]

The following do not work as they should (try these examples with a freshly started copy of Sage, such that everything is default).

This is definately a bug with the initialization of the precision:

# Default: 2 significant words (while we really should get only 1)
sage: pari('Pi').debug()
[&=0000000004fc9620] REAL(lg=4):0400000000000004 (+,expo=1):6000000000000001 c90fdaa22168c234 c4c6628b80dc1cd1

# Change precision and then change it back: we get 1 word
sage: n = pari.get_real_precision(); pari.set_real_precision(100); pari.set_real_precision(n);
sage: pari('Pi').debug()
[&=00000000012bf200] REAL(lg=3):0400000000000003 (+,expo=1):6000000000000001 c90fdaa22168c235

set_real_precision() seems to affect essentially only the precision for evaluating strings in PARI and not much else:

sage: pari.set_real_precision(50);
sage: pari('Euler')   # Precision changes
0.57721566490153286060651209008240243104215933593992
sage: pari.euler()    # Precision does NOT change
0.5772156649015328607

This last behaviour is actually documented in sage/libs/pari/gen.pyx:

Unless otherwise indicated in the docstring, most Pari functions that return inexact objects use the precision of their arguments to decide the precision of the computation. However, if some of these arguments happen to be exact numbers (integers, rationals, etc.), an optional parameter indicates the precision (in bits) to which these arguments should be converted before the computation. If this precision parameter is missing, the default precision of 53 bits is used.

In other words, the default precision is an unchangable 53 bits. I would expect set_real_precision() to change this. This is also trivial to implement since the global variable prec is already there (and initialized once and for all to 53).

[cremona]

There's a lot of relevant information written by Alex Ghitza and me a couple of years ago in the file gen.pyx. Yes, it is counterintuitive; but not undocumented.

[jdemeyer]

Replying to cremona:

There's a lot of relevant information written by Alex Ghitza and me a couple of years ago in the file gen.pyx. Yes, it is counterintuitive; but not undocumented.

I know it is documented (although not too clearly), but the question is: does it make sense? (In my opinion: no).

[jdemeyer]

More precisely: this is counter-intuitive:

sage: pari.set_real_precision(100);
sage: pari('Euler')   # Precision changes
0.5772156649015328606065120900824024310421593359399235988057672348848677267776646709369470632917467495
sage: pari.euler()    # Precision does NOT change
0.5772156649015328607