TITLE: How do pocket calculators calculate exponents?
QUESTION [17 upvotes]: I'd like to know specifically how a pocket calculator (TI calculators also apply) calculates $e^{0.1}$, and what methods or algorithms pocket calculators use in order to produce their answer.

REPLY [12 votes]: I would be surprised if they actually used Taylor series.  For example, the basic 80x87 "exponential" instruction is F2XM1 which computes $2^x-1$ for $0 \le x \le 0.5$.  I don't think the implementation is documented, but if I were
programming this, I might use a minimax polynomial or rational approximation: the following polynomial of degree $9$ approximates $2^x - 1$ on this interval
with maximum error approximately $1.57 \times 10^{-17}$:
-0.15639e-16+(.69314718055995154416+(.24022650695869054994+
(0.55504108675285271942e-1+(0.96181289721472527028e-2
+(0.13333568212100120656e-2+(0.15403075872034576440e-3
+(0.15265399313676342402e-4+(0.13003468358428470167e-5
+0.12113766044841794408e-6*x)*x)*x)*x)*x)*x)*x)*x)*x

By contrast, the Maclaurin polynomial of the same degree has
maximum error about $7.11 \times 10^{-12}$ on this interval.