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PT vectors are a method for approximately storing large numbers based on power towers. It is employed in Hypercalc.

## Description Edit

A PT vector [p,v] consists of a nonnegative integer p and a floating-point number v. The numerical interpretation of this vector is

$[p,v] = \text{sgn}(v) \cdot (10 \uparrow 10 \uparrow \cdots \uparrow 10 \uparrow |v|)$

where there are p copies of 10.

PT vectors are normalized so that |v| is in a comfortable range. Hypercalc restricts $$\log_{10} 300 \leq |v| \leq 10^{300}$$, allowing $$|v| < \log_{10} 300$$ if $$p = 0$$.

From here, standard operations on PT vectors may be defined. The general principle is that PT vector operations use casework to deal with small values and estimation to deal with large values. It helps to look at an example — addition of two positive PT vectors. One very useful assumption we can make: when a and b are large enough, a + b = max(a, b). For smaller a and b, the cases can be handled more exactly:

• $$[0, v] + [0, w] = [0, v + w]$$
• $$[0, v] + [1, w] = [1, \log_{10}(10^{\log_{10}(v) - w} + 1) + w]$$
• $$[1, v] + [1, w] = [1, \log_{10}(10^{v - w} + 1) + w]$$

This same principle applies to other operators. A factorial implementation, for example, would use direct computation for small values, Stirling's approximation for medium sized values, and a rough approximation such as x! = 10x for very large values.