HP Prime complex division problem in CAS

[Albert Chan]

(11-27-2020 08:30 AM)lyuka Wrote:  I found relatively simple workaround for complex division in CAS,
that is to apply scaling process to Smith's method (1962).

Unlike IEEE double, HP Prime does not support gradual underflow.
It does not do flush-to-zero either.

CAS> MINREAL/2             → 2.2250738585e−308
CAS> MINREAL/2^47       → 2.2250738585e−308
CAS> MINREAL/2^48       → 0.

With this underflow behavior, it may be best to scale up, to some safe maximum value.

Smith's method, with u = IM(y)/RE(y), |u| ≤ 1 : x/y = x * (1 - u*i) / (RE(y) + u*IM(y))

As long as x, y parts at or below MAXREAL/2, RHS numerator and denominator will not overflow.

x/y = cdiv_smith(x*s, y*s), where s = (MAXREAL/2) / max(|RE(x)|,|IM(x)|,|RE(y)|,|IM(y)|)

Example, with g = MAXREAL/2:

(2g + 2g*i) / (g + 2g*i)
= (g + g*i) / (0.5*g + g*i)                                        // scaled by s = g / (2g) = 0.5
= conj((g + g*i) / (g + 0.5*g*i))                               // flip to force u=0.5, |u| ≤ 1
= conj((g + g*i) * (1 - 0.5*i)) / (g + 0.5*(0.5*g))      // smith method
= (1.5*g - 0.5*g*i) / (1.25*g)
= 1.2 - 0.4*i