One wants to determine $1/(2+\sqrt 3)^4$ having access to an approximate value of $\sqrt 3$.

Question

Can someone help me with this question please:

One wants to determine $1/(2+\sqrt 3)^4$ having access to an approximate value of $\sqrt 3$. Compare the relative errors on direct computation and on using the equivalent expression $97 - 56\sqrt 3$.

---|---|--|--|-|-|-|-|||||--|--|||||-|-|-|-|--|--|--|---------------
-4    -3    -2      -1      0      1       2     3     4

Tick marks indicates all 25 numbers in floating-point system having $\beta=2, p=3, L = -1$, and $U = 1$

OFL = $(1.11)$₂ = $2^1 = (3.5)$₁₀

UFL = $(1.00)$₂ = ${2^-}^1 = (0.5)$₁₀

Answer 1

You should as a first step compute the actual values for some approximations. Then try theoretical reasons for the difference against them

ar3         1/(2+ar3)**4    97-56*ar3     1/(97+56*ar3)
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1.73      0.00516612217572  0.12         0.00515782958531
1.732     0.00515505685776  0.008        0.00515485174646
1.7321    0.00515450436992  0.0024       0.00515470294478
1.73205   0.00515478060459  0.0052       0.00515477734455