Arbitrary precision type design

When a numerical computation is required to deliver a binary [yes/no|in/out] result via a floating-point computation, the required computational precision is unknown a-priori. In such cases, especially during exploration, an adaptive, arbitrary precision arithmetic can be used to study the results and the computational dynamics that lead to that result.

There are two adaptive arithmetic floating-point types that have been developed:

1. a value modeled as a triple, < sign, exponent, adaptive multi-limb significant >
2. a value modeled as a composite sum of floating-point values

The first construction is called a multi-digit format, while the second type is referred to as multi-component. The benefit of a multi-component format is the sparse representation of complex values, but a disadvantage is computational results need to be normalized before storage or continued calculations. In contrast, a multi-digit format can be used without normalization but tends to be verbose.

Another benefit of the multi-component format is that it can be executed on existing floating-point hardware resulting in a relatively high-performance (tens of MOPS on GOPS hardware). However, the fixed exponent of the hardware will limit the attainable precision.

The Universal library supports both representations, in static and elastic (=adaptive) incarnations.

• multi-digit

  • static

    areal: faithful format with an uncertainty bit

    cfloat: a classic binary floating-point

    dfloat: decimal floating-point

    hfloat: hexadecimal floating-point

  • elastic

    efloat: elastic binary floating-point

• multi-component

  • static

    dd (double-double)

    qd (quad-double)

  • elastic

    ereal: elastic real, adaptive precision approximation to exact

    elreal: exact lazy real