HP-71B Enhanced Math LEX

[J-F Garnier]

(01-27-2020 11:35 AM)Albert Chan Wrote:  

(01-27-2020 08:05 AM)J-F Garnier Wrote:  I understand that the alternate formula (x+y)*(x-y) saves a multiplication, but does it guarantee that the result will be *always* better?

Yes, I believe (x*x-y*y) is better calculated as (x+y)*(x-y)

|x| + |y| always accurate to whatever the system precision (unless it overflows)
|x| − |y| is exact if ratio of |x| and |y| = 1/2 to 2. (unless it underflows)
(see Kahans Miscalculating Area and Angles of a Needle-like Triangle, Section 4, Why Cancellation Cannot Hurt)

I tend to agree with you. (x+y)*(x-y) is better than (x*x-y*y) especially when x and y are close.
For instance:
>X=PI @ Y=PI-1E-11
>Z=(X,Y)
>Z*Z
(6.283E-11,19.7392088021)
>(X+Y)*(X-Y)
6.28318530717E-11
The last formula gives 8 more significant digits for the real part.
Note, however, than in this particular case, the involution formula gives an even worst result:
>Z^(2,0)
(6.27015817311E-11,19.7392088021)

I checked the x^2 (or SQ in RPL) function on HP Saturn-based machines up to the 49G+ and they all seem to use the x*x-y*y formula.
So I will refrain from implementing the (x+y)*(x-y) approach in the 71 Math 2 LEX, my goal is to align its performance with other machines of the time (75C, 28S, 42S), and the Z^2 case implemented as Z*Z already provides an interesting improvement.

J-F