TVM solve for interest rate, revisited

[Albert Chan]

(06-10-2024 03:27 PM)Albert Chan Wrote:  Lets make bottom branch more accurate, with log1p_sub / expm1_sub
...
(log1p_sub / expm1_sub / fma) still does not help f with tiny rate!

I figured out why Newton on f does not work well with tiny rate!
Say, with non-zero a, we want to solve f(x) = g(x) - a

Newton's method: x ← x - f(x)/f'(x) = x - (g(x)-a) / g'(x)

With iteration zeroed-in to true root, g(x) ≈ a
f(x) will get less and less accurate, due to cancellation errors.
It does not matter if g(x) is super-accurate, even if correctly rounded!
Cancellation errors may happen for slope too, if a is also a function of x

This is normally not an issue. If f(x) only get half-precision, iterated x holds the other half.
With final f turns into garbage, more Newton's iteration may make it worse.

lua> x = 1
lua> for i=1,6 do f=x*x-2; print(x,f); x=x-f/(2*x) end
1                       -1
1.5                     0.25
1.4166666666666667      0.006944444444444642
1.4142156862745099      6.007304882871267e-06
1.4142135623746899      4.510614104447086e-12
1.4142135623730951      4.440892098500626e-16

But, that does not hold if X is very close to zero. Iterated x unable to keep its share.
This is the reason tiny rate top branch is needed, to keep cancellation errors at bay.
The other way is to throw in more-precision to the problem, but that is expensive.