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.
Calculator – How Pocket Calculators Calculate Exponents
calculatorexponential functionexponentiation
Related Solutions
This is what I would do to calculate $e^x$ with perfect 8-digit accuracy.
Take $\lfloor x\rfloor$ and $b = x - \lfloor x\rfloor$, the whole and fractional parts respectively. ($e^x = e^{\lfloor x \rfloor + (x - \lfloor x\rfloor)} = e^{\lfloor x \rfloor} e^{x - \lfloor x \rfloor}$)
Use the $(((b/4 + 1)b/3 + 1)b/2 + 1)b + 1$ pattern to calculate $e^{x - \lfloor x\rfloor}$. (If the fractional part of the exponent is .4 or less, only terms up to $b/7$ are needed for 8-digit accuracy. Worst-case scenario ($b \rightarrow 1$) terms up to $b/10$ are needed.) This method is easy to implement on a simple calculator, especially ones with a memory slot to quickly reinsert $b$.
When that is finished, multiply by $e$ ($\approx 2.71828183$ by memorization) and press the equals button $\lfloor x\rfloor$ times to repeat multiply. The result is $e^x$.
I've analyzed the number of terms required for full accuracy. Here is the chart (click to view full image):
Basically, if terms up to $b/t$ are needed to fully calculate $e^x$ up to 8-digit precision (7 digits after the decimal point), accounting for rounding (the $\frac{1}{2}$ term), the relation between $x$ and $t$ is given by $\sqrt[t]{\frac{1}{2}10^{-7}t!} = x$.
In response to edit of initial post, then answer is clearly $30$. Basic calculators are assumed to evaluate in order from left to right.
Original post I responded to
In a Mensa calander, IQ dialy challenge I got this and put a challenge up at work.
Using +,-,time and divide only once. Use the math operator only once to get the highest answer.
5 ? 4 ? 7 ? 3 ? 2 =
We all worked out
5 + 4 x 7 - 3 / 2 = 30
Except that my result answer was 31.5 and not 30, like in the answers of the MENSA calendar.
Why was I the only one that applied the rules of maths on this? ANd when I asked why nobody else applied the rule of maths, I got the weirdest looks. Nobody knew about multiplication before division, subtraction before adding? I thought that was why the question was marked as the most difficutl to test if you knew this.
Response to original post
Sadly, many people forget the basic rules of arithmetic as they (a) don't view them as affecting their lives, (b) didn't like maths, and/or (c) know technology can handle the problem for them. The issue with the last point is that different technologies handle things differently. The Google calculator (much like most graphing calculators) will handle order of operations for you correctly. The standard Windows calculator appears to be operating like an old 4 function calculator which evaluates after every operation is completed as opposed to correct order of operations. Though this can also happen when users hit enter after every operation is finished as opposed to when the whole expression is finished. (Don't have access to a Windows calculator right now so can't tell which is the reason for the wrong answer.)
Best Answer
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 $1.57 \times 10^{-17}$:
By contrast, the Maclaurin polynomial of the same degree has maximum error about $7.11 \times 10^{-12}$ on this interval.