Emulating Double-Precision Arithmetic with Pairs of Floats

In embedded systems with limited numerical capabilities, emulating double-precision data types becomes necessary for certain algorithms. This article explores the technique of emulating "double" data type using a tuple of two "float" values to achieve increased precision.

The comparison operation between two emulated doubles follows a straightforward lexicographic ordering. However, the addition operation presents challenges due to the need to detect carry-outs. The underlying question is, which base should be used for this operation? While FLT_MAX might be a potential candidate, it requires further consideration.

Emulating Addition

To emulate addition, we need to consider not only the addition of the individual components but also the potential for carry-outs. The base used for the operation should provide sufficient resolution to capture all possible carry-outs.

One approach is to use the sum of the two bounds of the float data type, FLT_MAX and -FLT_MAX, as the base. This ensures that any carry-out from the addition of the low components is accounted for in the addition of the high components.

Detecting Carry-outs

Detecting carry-outs requires monitoring the overflow or underflow status during the addition of the individual components. If an overflow occurs in the addition of the low components, a carry-out is indicated and should be added to the high component. Similarly, an underflow in the subtraction of the low components triggers a carry-down, which can be handled in the same manner.

Resources for Further Study

Additional insights can be gained from research in the field of double-float techniques. Two notable papers are:

• [Implementation of float-float operators on graphics hardware](https://hal.archives-ouvertes.fr/hal-00021443)
• [Extended-Precision Floating-Point Numbers for GPU Computation](http://andrewthall.org/papers/df64_qf128.pdf)

These resources provide valuable information on implementing float-float operators and optimizing their performance.