Decimal fractions to signed binary

Suppose we wish to compute the signed binary representation U of the decimal fraction $(0.u_{-1}u_{-2}u_{-3} \dots u_{-m})$ using signed binary operations, we simply evaluate

$$U = u_{-1}\!\cdot\!10^{-1} + u_{-2}\!\cdot\!10^{-2} + u_{-3}\!\cdot\!10^{-3} + \cdots + u_{-m}\!\cdot\!10^{-m}$$

To compute this more efficiently, we observe

$$U = (((u_{-m}/10 + u_{-(m-1)})/10 + \cdots + u_{-2})/10 +u_{-1})/10$$

We can now use a simple recursive algorithm using the average and division by an integer operations which evaluates to zero on an empty input, but otherwise computes:

$$\mathrm{decSB\_frac}(a::x) = \frac{a + \mathrm{decSB\_frac}(x)}{10} = \frac{a \oplus \mathrm{decSB\_frac}(x)}{5}$$

Because we cannot represent decimal digits in signed binary streams, we replace a in the right hand side of the expression by a stream which represents a/2⁴, and multiply the entire result by 2⁴. Multiplication and division by powers of two can be performed easily using shift operations.