(15C) Fibonacci Numbers

[Albert Chan]

(03-23-2017 01:32 PM)Eddie W. Shore Wrote:  n = 44
HP15C LE: 701408728.7
HP Prime: 701408733.002

n = 49
HP 15C LE: 7778741992
HP Prime: 7778742049.02

I think we may have found a bug on the HP 15C LE.

√5 ≈ 2.23606797749979

Round to 10 digits, we have huge ulp error, almost 1/2 ULP
(1+√5) relative error = 0.49979E-9 / 3.23607 = 1.54E-10

(1 + ε)^n ≈ 1 + n*ε

F(44) corrections ≈ 44 * 1.54E-10 * 701408728.7 ≈ 4.8
F(44) ≈ 701408728.7 + 4.8 = 701408733.5 (error = -0.5)

F(49) corrections ≈ 49 * 1.54E-10 * 7778741992 ≈ 59
F(49) ≈ 7778741992 + 59 = 7778742051 (error = -2)

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√20 is better, ulp error = 2 * 0.49979 - 1 = -0.00042 ULP

F(n) ≈ (2+√20)^n / (2^(2n-1)*√20)

On my HP12C:

F(44) = 701,408,733.1 (error -0.1)
F(49) = 7,778,742,050 (error -1)