CBO: (semi-open, semi-closed) range predicate cardinality

Now let's examine the case of semi-open, semi-closed ranges:

select x from t where x >  low_x and x <= high_x; -- (open, closed)
select x from t where x >= low_x and x <  high_x; -- (closed, open)

Instead of repeating the same steps, that would be a rather tedious process, let's see how we can apply "formal method of ranges decomposition" to re-use the results for open and closed ranges already discussed.

The strategy is:

Choose a point X between low_x and high_x;
decompose in two subranges, one closed, and one open, eg for x > low_x and x <= high_x:
```
 x >  low_x and x <  X
 x >= X     and x <= high_x 
```
Compute the cardinalities (now on completely open, or completely closed, subranges), and add them;
subtract num_rows / num_distinct since we have added the undelined ">=" operator.

Similarly for three, four subranges.

We'll experiment (script range_sel_semi.sql) on the case of num_distinct=4 only; remember that for this case we have:

 left band     : 0   < x < 2.5
 central region: 2.5 < x < 7.5
 right band    : 7.5 < x < 10

For ranges completely contained in the left band:

For x > low_x and x <= high_x, choose X at random, eg X = 1.0:

 SQL> select x from t where x >  0.1 and x <= 2.4;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  1920K|  5625K|  1440   (6)| 00:00:18 | (ORIGINAL)
 |*  1 |  TABLE ACCESS FULL| T    |  1920K|  5625K|  1440   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >  0.1 and x <  1.0;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |   960K|  2812K|  1436   (6)| 00:00:18 | (LEFT)
 |*  1 |  TABLE ACCESS FULL| T    |   960K|  2812K|  1436   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >= 1.0 and x <= 2.4;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  1960K|  5742K|  1440   (6)| 00:00:18 | (RIGHT)
 |*  1 |  TABLE ACCESS FULL| T    |  1960K|  5742K|  1440   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 960K + 1960K - 1000k = 1920K, as requested.

Note that the choice of X is irrelevant, since (LEFT) is open and so the right extreme(X) is ignored, and (RIGHT) is closed and so the left extreme (X again) is ignored as well.
Analitically:

    (LEFT) + (RIGHT) - num_rows / num_distinct
  = height_ramp - height_ramp * (low_x  - min_x) / B 
  + height_ramp + height_ramp * (high_x - min_x) / B
  - num_rows / num_distinct 
  = (num_rows / num_distinct) (high_x - low_x) / B +  num_rows / num_distinct
  = 1000000 * (2.4 - 0.1) / 2.5 + 1000000 = 1920K.

For x >= low_x and x < high_x, choose X at random, eg X = 1.0:

 SQL> select x from t where x >= 0.1 and x <  2.4;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 -------------------------------------------------------------------------- 
 |   0 | SELECT STATEMENT  |      |  1000K|  2929K|  1436   (6)| 00:00:18 | (ORIGINAL)
 |*  1 |  TABLE ACCESS FULL| T    |  1000K|  2929K|  1436   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >= 0.1 and x <= 1.0;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  1400K|  4101K|  1437   (6)| 00:00:18 | (LEFT)
 |*  1 |  TABLE ACCESS FULL| T    |  1400K|  4101K|  1437   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >  1.0 and x <  2.4;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |   600K|  1757K|  1436   (6)| 00:00:18 | (RIGHT)
 |*  1 |  TABLE ACCESS FULL| T    |   600K|  1757K|  1436   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 1400K + 600K - 1000K = 1000K, as requested.

Note that the choice of X is irrelevant, but also that the actual values of low_x and high_x do not matter(!).
Analitically:

    (LEFT) + (RIGHT) - num_rows / num_distinct
  = height_ramp + height_ramp * (X - min_x) / B 
  + height_ramp - height_ramp * (X - min_x) / B
  - num_rows / num_distinct 
  = num_rows / num_distinct = 1000K.

It's funny that just moving the equality from one extreme to the other can have such a big impact on the cardinality formula.

For ranges partially contained in the left band, and partially in the central region:
choose X = min_x + B

For x > low_x and x <= high_x:

 SQL> select x from t where x >  0.1 and x <= 6.4;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  3520K|    10M|  1447   (7)| 00:00:18 | (ORIGINAL)
 |*  1 |  TABLE ACCESS FULL| T    |  3520K|    10M|  1447   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >  0.1 and x <  2.5;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |   960K|  2812K|  1436   (6)| 00:00:18 | (LEFT)
 |*  1 |  TABLE ACCESS FULL| T    |   960K|  2812K|  1436   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >= 2.5 and x <= 6.4;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  3560K|    10M|  1447   (7)| 00:00:18 | (CENTRAL)
 |*  1 |  TABLE ACCESS FULL| T    |  3560K|    10M|  1447   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 960K + 3560K - 1000K = 3520K, as requested.

Note that the standard formula applies:
4000000 * (6.4 - 0.1) / (10-0) + 1000000 = 3520K.

For x >= low_x and x < high_x:

 SQL> select x from t where x >= 0.1 and x <  6.4;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  2560K|  7500K|  1443   (6)| 00:00:18 | (ORIGINAL)
 |*  1 |  TABLE ACCESS FULL| T    |  2560K|  7500K|  1443   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >= 0.1 and x <= 2.5;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  2000K|  5859K|  1440   (6)| 00:00:18 | (LEFT)
 |*  1 |  TABLE ACCESS FULL| T    |  2000K|  5859K|  1440   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >  2.5 and x <  6.4;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  1560K|  4570K|  1443   (6)| 00:00:18 | (CENTRAL)
 |*  1 |  TABLE ACCESS FULL| T    |  1560K|  4570K|  1443   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 2000K + 1560K - 1000K = 2560K, as requested.

Note that the standard formula applies if you chop the "<=" extreme to the band boundary (2.5):
4000000 * (6.4 - 2.5) / (10-0) + 1000000 = 2560K

For ranges partially contained in the left band, in the central region, and partially in the right band:

For x > low_x and x <= high_x:

 
 SQL> select x from t where x >  0.1 and x <= 9.8;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  3960K|    11M|  1449   (7)| 00:00:18 | (ORIGINAL)
 |*  1 |  TABLE ACCESS FULL| T    |  3960K|    11M|  1449   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >  0.1 and x <  2.5;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |   960K|  2812K|  1436   (6)| 00:00:18 | (LEFT)
 |*  1 |  TABLE ACCESS FULL| T    |   960K|  2812K|  1436   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >  2.5 and x <  7.5;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  2000K|  5859K|  1444   (7)| 00:00:18 | (CENTRAL)
 |*  1 |  TABLE ACCESS FULL| T    |  2000K|  5859K|  1444   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >= 7.5 and x <= 9.8;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  2000K|  5859K|  1440   (6)| 00:00:18 | (RIGHT)
 |*  1 |  TABLE ACCESS FULL| T    |  2000K|  5859K|  1440   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 960K + 2000K + 2000K - 1000k = 3960K, as requested.

Note that the standard formula applies if you chop the "<=" extreme to the band boundary (7.5):
4000000 * (7.5 - 0.1) / (10-0) + 1000000 = 3960K

For x >= low_x and x < high_x:

 SQL> select x from t where x >= 0.1 and x <  9.8;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  3920K|    11M|  1449   (7)| 00:00:18 | (ORIGINAL)
 |*  1 |  TABLE ACCESS FULL| T    |  3920K|    11M|  1449   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >= 0.1 and x <= 2.5;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  2000K|  5859K|  1440   (6)| 00:00:18 | (LEFT)
 |*  1 |  TABLE ACCESS FULL| T    |  2000K|  5859K|  1440   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >  2.5 and x <  7.5;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 -------------------------------------------------------------------------- 
 |   0 | SELECT STATEMENT  |      |  2000K|  5859K|  1444   (7)| 00:00:18 | (CENTRAL)
 |*  1 |  TABLE ACCESS FULL| T    |  2000K|  5859K|  1444   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >  7.5 and x <  9.8;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |   920K|  2695K|  1436   (6)| 00:00:18 | (RIGHT)
 |*  1 |  TABLE ACCESS FULL| T    |   920K|  2695K|  1436   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 2000K + 2000K + 920K - 1000k = 3920K, as requested.

Note that the standard formula applies if you chop the ">=" extreme to the band boundary (2.5):
4000000 * (9.8 - 2.5) / (10-0) + 1000000 = 3920K

Special case for x > low_x = min_x (or x < high_x = max_x for the right ramp):

 SQL> select x from t where x > 0         and x <= 1;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |   400K|  1171K|  1437   (6)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |   400K|  1171K|  1437   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x > 0 + 1e-12 and x <= 1;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  1400K|  4101K|  1437   (6)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  1400K|  4101K|  1437   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x > 0         and x <= 3;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  1200K|  3515K|  1441   (6)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  1200K|  3515K|  1441   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x > 0 + 1e-12 and x <= 3;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  2200K|  6445K|  1441   (6)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  2200K|  6445K|  1441   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x > 0         and x <= 9;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  3000K|  8789K|  1444   (7)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  3000K|  8789K|  1444   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x > 0 + 1e-12 and x <= 9;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  4000K|    11M|  1449   (7)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  4000K|    11M|  1449   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x > 0         and x < 10;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  2000K|  5859K|  1444   (7)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  2000K|  5859K|  1444   (7)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x > 0 + 1e-12 and x < 10-1e-12;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  4000K|    11M|  1449   (7)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  4000K|    11M|  1449   (7)| 00:00:18 |
 --------------------------------------------------------------------------

The rule seems to be "compute as normal, then subtract num_rows/num_distinct for each range extreme matching the corresponding interval extreme".

Note that for x >= low_x = min_x instead, we don't see anything special:

 SQL> select x from t where x >= 0         and x <= 1;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  1400K|  4101K|  1437   (6)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  1400K|  4101K|  1437   (6)| 00:00:18 |
 --------------------------------------------------------------------------
 SQL> select x from t where x >= 0 + 1e-12 and x <= 1;
 --------------------------------------------------------------------------
 | Id  | Operation         | Name | Rows  | Bytes | Cost (%CPU)| Time     |
 --------------------------------------------------------------------------
 |   0 | SELECT STATEMENT  |      |  1400K|  4101K|  1437   (6)| 00:00:18 |
 |*  1 |  TABLE ACCESS FULL| T    |  1400K|  4101K|  1437   (6)| 00:00:18 |
 --------------------------------------------------------------------------

Let's move to the conclusion.

For corrections / feedback:
alberto.dellera@gmail.com