Cube root [HP-35]

[Gerson W. Barbosa]

just a proof of concept on the hp 33s to reduce the number of iterations in this variation of Juan’s algorithm. Line D0013 should be replaced by a short routine to compute ln(x) on the HP-16C, where this might make some sense.

Code:


C0001 LBL C
C0002 12
C0003 STO i
C0004 x<>y
C0005 ENTER 
C0006 ENTER
C0007 ENTER
C0008 √x
D0001 LBL D
D0002 ÷
D0003 √x
D0004 DSE i
D0005 GTO D
D0006 LASTx
D0007 R↑
D0008 x<>y
D0009 ÷
D0010 x<>y
D0011 8192
D0012 R↑
D0013 LN
D0014 ÷
D0015 5..18 ; 5/18 = 2.77777777778E-1
D0016 +
D0017 1/x
D0018 +/-
D0019 2
D0020 +
D0021 ×
D0022 LASTx
D0023 R↓
D0024 +
D0025 R↑
D0026 1
D0027 +
D0028 ÷
D0029 RTN

27 XEQ C -> 3

1E50 XEQ C -> 4.64158883316E16

1E50 ³√x -> 4.64158883316E16

Algorithm:

r₀ = √x
r₁ = √(x/r₀)
r₂ = √(x/r₁)
...
rn = √(x/rn-1)

a = rn
b = rn-1

³√x ~ {a[2 - 1/(2ⁿ⁺¹/lnx + 5/18)] + b}/[3 - 1/(2ⁿ⁺¹/lnx + 5/18)]