#2514 Taylor result inconsistent T/x^2 vs. T/x/x

[Stavros Macrakis]

taylor(x,x,0,5)/x^2 => 1/x+...
Internally: ((MRAT SIMP ($X) (#:X1942) (($X ((5 . 1)) 0 NIL #:X1942 . 1)) TRUNC) PS
(#:X1942 . 1) ((3 . 1)) ((-1 . 1) 1 . 1))

(taylor(x,x,0,5)/x)/x => 1/x (without ellipses)
Internally: ((MRAT SIMP ($X) (#:X1945) (($X ((5 . 1)) 0 NIL #:X1945 . 1)) TRUNC) PS
(#:X1945 . 1) * (NIL) * ((-1 . 1) 1 . 1))

taylorinfo of both expressions remains [[x,0,5]]

Versions: 5.27.0 2012-04-30 11:59:06 i686-apple-darwin11.3.0 SBCL 1.0.55.0-abb03f9

[Rupert Swarbrick]

When dividing twice by x the result is, as you say:

((MRAT SIMP ($X) (#:X1461) (($X ((5 . 1)) 0 NIL #:X1461 . 1)) TRUNC) PS
 (#:X1461 . 1) (NIL) ((-1 . 1) 1 . 1))

Printing is done by FORM-MRAT, which decides the result is "exact"
(because of the NIL) and intentionally doesn't print a trailing ...

Another way to get the same result is:

taylor(1, x, 0, 5) / x;

[Rupert Swarbrick]

Something that definitely appears wrong shows up when you trace
TAYLOR2 in the division taylor(1, x, 0, 5) / x;:

0: (TAYLOR2
    ((MTIMES)
     ((MRAT SIMP ($X) (#:X1480) (($X ((5 . 1)) 0 NIL #:X1480 . 1)) TRUNC) 1
      . 1)
     ((MEXPT SIMP) $X -1)))
0: TAYLOR2 returned (PS (#:X1481 . 1) (NIL) ((-1 . 1) 1 . 1))

The (NIL) is "<trunc-lvl>" as explained in the comment at the top of
hayat.lisp and I think it should probably be set to ((4 . 1)) not
NIL!

Digging into TAYLOR2, and tracing TSTIMES, this looks pretty bogus:

1: (TSTIMES
    (((MRAT SIMP ($X) (#:X1480) (($X ((5 . 1)) 0 NIL #:X1480 . 1)) TRUNC) 1
      . 1)
     ((MEXPT SIMP) $X -1)))
  2: (TAYLOR2 ((MEXPT SIMP) $X -1))
  2: TAYLOR2 returned (PS (#:X1485 . 1) (NIL) ((-1 . 1) 1 . 1))
  2: (TAYLOR2
      ((MRAT SIMP ($X) (#:X1480) (($X ((5 . 1)) 0 NIL #:X1480 . 1)) TRUNC)
       1 . 1))
    3: (TAYLOR2 1)
    3: TAYLOR2 returned (1 . 1)
  2: TAYLOR2 returned (1 . 1)
  2: (TAYLOR2 ((MEXPT SIMP) $X -1))
  2: TAYLOR2 returned (PS (#:X1485 . 1) (NIL) ((-1 . 1) 1 . 1))
1: TSTIMES returned (PS (#:X1485 . 1) (NIL) ((-1 . 1) 1 . 1))

Basically, I think the second call to TAYLOR2 should probably result
in something like

(PS (#:X1485 . 1) ((5 . 1)) (1 . 1))

which would say "This is a power series in X, which is 1 up to
x^5". If something like this doesn't get returned, there's no way for
the following code to work out the truncation order.