Abstract
In this work, we present a novel method for directly computing functions of two real numbers using logic circuits without decoding; the real numbers are mapped to a particularlychosen set of integer numbers. We theoretically prove that this mapping always exists and that we can implement any kind of binary operation between real numbers regardless of the encoding format. While the real numbers in the set can be arbitrary (rational, irrational, transcendental), we find practical applications to lowprecision posit™ number arithmetic. We finally provide examples for decodingfree 4bit Posit arithmetic operations, showing a reduction in gate count up to a factor of \(7.6\times \) (and never below \(4.4\times \)) compared to a standard twodimensional tabulation.
Research supported by Horizon H2020 projects EPISGA2 and TextaRossa.
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Work partially supported by H2020 projects EPI2 (grant no. 101036168, https://www.europeanprocessorinitiative.eu/ and TextaRossa (grant no. 956831, https://textarossa.eu/).
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Appendix: How to Build an Initial Feasible Solution
Appendix: How to Build an Initial Feasible Solution
An initial feasible solution (useful to speedup Matlab intlinprog function) can be constructed as shown below. We will focus on the positive values in X, different both from NaR (notice that we are excluding the zero as well). Let us call this set \(\mathcal {X}\). Let us indicate with \(x^{*}_i \in \mathcal {X}\) the corresponding bistring (in the next we will refer to the \(Posit\left\langle 4,0\right\rangle \) case, as an example). Therefore, the bistrings will be the ones of \(Posit\left\langle 4,0\right\rangle \) without its most significant bit (see Table 1).

Each \(x_i \in \mathcal {X}\) is mapped to the natural number \(L^x_i = x_i^* \cdot 2^n,\) n being the maximum number of bits needed for representing the \(x_i\) (in the case of \(Posit\left\langle 4,0\right\rangle \), \(n = 3\) )

Each \(y_j \in \mathcal {Y}\) is mapped to the natural number \(L^y_j = y_j^*\)
Therefore, we obtain the \(L^x, L^y\) sets: \(L^x: \{x_1^* \cdot 2^n, \dots , x_{\mathcal {X}}^* \cdot 2^n \}\) and \(L^y: \{ y_1^*, \dots , y_{\mathcal {Y}}^* \}\). Each \(L_{i,j}^z \in L^z\) is obtained by the concatenation of the bit strings \(x_i^*, y_j^*\) (or equivalently, as \(L_{i,j}^z = L_i^x + L_j^y\), as shown in Table 12).
We now prove that this solution satisfies the constraint given in Eq. (1):

Since there are no conflicting encodings of the real numbers in \(\mathcal {X}\) and \(\mathcal {Y}\), we can guarantee that different real numbers have different bitstrings that digitally encode them. Therefore, \(L_i^x \ne L_j^x, \forall i \ne j\) and \(L_p^y \ne L_l^y, \forall k \ne l\).

Since all the encodings in \(L^x, L^y\) are different from each other, also the concatenation of any pair \(L_i^x, L_j^y\) is unique. Therefore, \(L_{i,j}^z \ne L_{k,q}^z, \) if \(x_i \triangledown y_j \ne X_j \triangledown Y_q \) (with \(\triangledown \) we indicate the generic operation for which we are finding the mapping).

Being the values \(L_{i,j}^z\) unique (no duplicates), we can easily obtain the ordered set \(L^z\), by sorting them.

the kelement vector \(\boldsymbol{w}\) can be trivially obtained as cast\((x_i \triangledown y_j)\), when \(i\cdot 2^n + j = k\).
An example of feasible solution for the multiplication of \(Posit\left\langle 4,0\right\rangle \) numbers is reported in Table 12.
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Gustafson, J.L., Cococcioni, M., Rossi, F., Ruffaldi, E., Saponara, S. (2023). DecodingFree TwoInput Arithmetic for LowPrecision Real Numbers. In: Gustafson, J., Leong, S.H., Michalewicz, M. (eds) Next Generation Arithmetic. CoNGA 2023. Lecture Notes in Computer Science, vol 13851. Springer, Cham. https://doi.org/10.1007/9783031321801_4
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