Database Math 101#

Introduction#

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 0.001 um or 1nm.

OpenDB uses an int to represent a DBU, which on most hardware is 4 bytes. This means a database coordinate can be \(\pm 2147483647\), which is about \(2 \cdot 10^9\) units, corresponding to \(2 \cdot 10^6\) or \(2\) meters.

Datatype Choice#

This section is important as we cover important math considerations for your datatype choice when dealing with large numbers.

Why not pure int?#

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, however, is that 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. Our suggestion is to use int64_t in this situation. Although long “works”, its size is implementation-dependent.

Why not double?#

It has been noticed that some programs use double to calculate distances. This can be problematic, as double have a mantissa of 52 bits, which means that the largest possible integer value that can be represented without loss is \(5\cdot 10^{15}\). This is 12 bits less than the largest possible integer value that can be represented by an int64_t. As a result, if you are doing an area calculation on a large chip that is more than \(\sqrt{5\cdot 10^{15}} = 7\cdot 10^7\ DBU\) on a side, the mantissa of the double will overflow and the result will be truncated.

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

Use int only for LEF/DEF Distances#

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. We 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 fraction.

OpenDB uses int to represent LEF distances, just as with 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.

Why not float?#

We have also noticed RePlAce, OpenDP, TritonMacroPlace and OpenNPDN all using double or float to represent distances. This can be problematic, as floating-point numbers cannot always represent exact fractions. As a result, these tools need to round or floor the results of their calculations, which can introduce errors. Additionally, some of these tools reinvent the wheel by implementing their own rounding functions, as we shall see in the example below. This can lead to inconsistencies and is highly discouraged.

(int) x_coord + 0.5

Worse than using a double is using a float, because the mantissa is only 23 bits, so the maximum exactly representable integer is \(8\cdot 10^6\). 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, which floor 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.

Database Math 101