(11C) Purchase or Lease

[Dieter]

(09-26-2018 12:53 PM)Albert Chan Wrote:  HP-12C, Horner's for NPV (cash-flow input in reverse)
Multiply flipped rate => 907.76891 00
Divide discount rate => 907.76893 00

Hey, that's a difference of 0,002 Cents!!!!1111eleven!!!

(09-26-2018 12:53 PM)Albert Chan Wrote:  HP-12C, Horner's for FV, then back to NPV => 907.76892 51

HP-12C build-in CFj/Nj code => 907.76893 37 ... Amazingly Good

How does HP-12C code get it so close ?

Simple. Since 1976 most HPs use three additional guard digits for their internal functions. So the 10-digit 12C internally works with 13 digits. Only this way the usual 10-digit accuracy for scientific or financial functions can be achieved. Well, mostly, but not always. Try the infamous 3²⁰¹ example on a regular 10-digit HP. ;-)

(09-26-2018 12:53 PM)Albert Chan Wrote:  My guess is calculations all done in 13 internal digits, only 1 rounding for the NPV result ...

That's the main reason. But there is more. As far as I remember the financial calculators have special internal functions that evaluate (1+i)ⁿ and (1+i)ⁿ–1 with higher accuracy. I assume this means they have something like ln(1+x) and e^x–1 implemented so that these expressions can be accurately calculated even for small interest rates. There is a nice example in the HP15C Advanced Functions Handbook that shows what happens when these expressions are evaluated with standard functions.

Dieter