# Database Math 101¶

DEF defines the units it uses with the units command.

```UNITS DISTANCE MICRONS 1000 ;
```

Typically the units are 1000 or 2000 database units (DBU) per micron. DBUs are integers, so the distance resolution is typically 1/1000u or 1nm.

OpenDB uses an `int` to represent a DBU, which on most hardware is 4 bytes. This means a database coordinate can be +/-2147483647, which is about 2 billion, or 2000 microns or 2 meters.

Since chip coordinates cannot be negative, it would make sense to use an `unsigned int` to represent a distance. This conveys the fact that it can never be negative and doubles the maximum possible distance that can be represented. The problem is doing subtraction with unsigned numbers is dangerous because the differences can be negative. An unsigned negative number looks like a very very big number. So this is a very bad idea and leads to bugs.

Note that calculating an area with `int` values is problematic. An `int * int` does not fit in an `int`. My suggestion is to use `int64_t` in this situation. Although `long` “works”, it’s size is implementation dependent.

Unfortunately I have seen multiple instances of programs using a `double` for distance calculations. A double is 8 bytes, with 52 bits used for the mantissa. So the largest possible integer value that can be represented without loss is 5e+15, 12 bits less than using a `int64_t`. Doing an area calculation on a large chip that is more than `sqrt(5e+15) = 7e+7 DBU` will overflow the mantissa and truncate the result.

Not only is a `double` less capable than an `int64_t`, using it the tells any reader of the code that the value can be real number, such as 104.23. So it is extremely misleading.

Circling back to LEF, we see that unlike DEF the distances are real numbers like 1.3 even though LEF also has a distance unit statement. I suspect this is a historical artifact of a mistake made in the early definition of the LEF file format. The reason it is a mistake is because decimal fractions cannot be represented exactly in binary floating point. For example, 1.1 = 1.00011001100110011…, a continued fracion.

OpenDB uses `int` to represent LEF distances, just like DEF. This solves the problem by multiplying distances by a decimal constant (distance units) to convert the distance to an integer. In the future I would like to see OpenDB use a `dbu` typedef instead of `int` everywhere.

Unfortunately, I see RePlAce, OpenDP, TritonMacroPlace and OpenNPDN all using `double` or `float` to represent distances and converting back and forth between DBUs and microns everywhere. This means they also need to `round` or `floor` the results of every calculation because the floating point representation of the LEF distances is a fraction that cannot be exactly represented in binary. Even worse is the practice of reinventing round in the following idiom.

`(int) x_coord + 0.5`

Even worse than using a `double` is using `float` because the mantissa is only 23 bits, so the maximum exactly representable integer is 8e+6. This makes it even less capable than an `int`.

When a value has to be snapped to a grid such as the pitch of a layer the calculation can be done with a simple divide using `int`s, which `floor`s the result. For example, to snap a coordinate to the pitch of a layer the following can be used.

```int x, y;
inst->getOrigin(x, y);
int pitch = layer->getPitch();
int x_snap = (x / pitch) * pitch;
```

The use of rounding in existing code that uses floating point representations is to compensate for the inability to represent floating point fractions exactly. Results like 5.99999999992 need to be “fixed”. This problem does not exist if fixed point arithmetic is used.

The only place that the database distance units should appear in any program should be in the user interface, because humans like microns more than DBUs. Internally code should use `int` for all database units and `int64_t` for all area calculations.

James Cherry, 2019