Homomorphic computational extensions

Introduction

We consider homomorphisms which are based on computational concerns which are used to transform inefficient or lossy computations over some original domain $T$ into a conceptually equivalent group $T^{\ast }$ over a restricted set of operations.

If the original problem can be solved using these restricted operations, then we may transform $T$ into $T^{\ast }$ and efficiently perform the computations. Sometimes, the entire solution cannot be transformed back to $T$, but the restricted set of functions or operations may still be sufficient, e.g., evaluating $a+c<b+c$ even though $a+c$ or $b+c$ may not be in the domain of $T$.

Motivation

Suppose you have a value type $T$ (e.g., $\mathtt{\text{double}}$ or something more fancy), but when you apply some operations to it, undesirable behavior is produced, e.g., $\mathtt{\text{double}}$ overflows due to multiplications or loses too much precision due to round-off error.

In the case of a numeric type like $\mathtt{\text{double}}$, one option is to use a big number library. However, this approach has some disadvantages:

1. You may not want to have a dependency on some external library
2. The big number implementation may be too inefficient.

Assume the computational basis of $T$ is $F(T)=\{\ast ,/,+,-,<,==\}$, which can be used to relatively efficiently implement other operations like $\sin$.

Suppose we map $T$ to a modified type $\hat{T}$ such that the computational basis $F(\hat{T})$ is a proper subset of $F(T)$, say $F(\hat{T})=\{\ast ,/,<,==\}$. In exchange for this restriction, the undesired behavior may be avoidable, say multiplications and divisions almost never overflow or underflow. If we do not require the operations in basis $F(T)$, this is a good exchange.

Alternatively, if the computational basis $F(T)$ is needed, it may or may not be possible to map $\hat{T}$ back to $T$ without any loss of information.

Example

Suppose we try to compute $a!/b!$ (ratio of factorials). The end result may be a value in the domain of $T$, but intermediate values (e.g., $a!$) may only be in the domain of $\hat{T}$. In this case, we can map the final result back to $T$ without any loss.

The logarithm of the data is a natural candidate for this is mapping $\mathtt{\text{double}}$ to $\widehat{\mathtt{\text{double}}}$ with the restricted basis

$$\{\ast ,/,<,==,=\}$$

such that any of these operations almost never fail and can be performed very efficiently, both in terms of time and space complexity.

Code

template <typename T = double>
struct lg
{
    T k;

    lg(lg const &) = default;

    // default constructs the multiplicative identity
    lg() : k(T(0)) {}

    lg(T x) : k(log(x)) { assert(0 < x); };

    // operator to convert (back) to type T.
    operator T() const { return exp(k); }
};

template <typename T>
auto operator*(lg<T> x, lg<T> y) { return lg<T>{x.k + y.k}; }

template <typename T>
auto operator/(lg<T> x, lg<T> y) { return lg<T>{x.k - y.k}; }

template <typename T>
auto operator<(lg<T> x, lg<T> y) { return x.k < y.k; }

template <typename T>
auto operator==(lg<T> x, lg<T> y) { return x.k == y.k; }