The mean field theory (nMF):

$\mathbf{J}^\text{nMF}=-\mathbf{C}^{-1}, (i \neq j) $,

$h_{i}^\text{nMF}=\tanh^{-1}\left< s_{i} \right>-\sum_{j=1}^{N} J{ij}^\text{nMF}\left< s_{j} \right> $ .

The independent-pair theory (IP):

$J_{ij}^\text{pair}=0.25*\ln\left[ \frac{(1+m_{i}+m_{j}+C_{ij}^{*}) (1-m_{i}-m_{j}+C_{ij}^{*})} {(1-m_{i}+m_{j}-C_{ij}^{*})(1+m_{i}-m_{j}-C_{ij}^{*})} \right]$, with $C_{ij}^{*}=C_{ij}+m_{i}m_{j} , (i \neq j)$

$h_{i}^\text{pair}=0.5*\ln\left( \frac{1+m_{i}} {1-m_{i}} \right)-\sum_{j=1}^{N} J_{ij}^\text{pair} m_{j}$.

The Thouless-Anderson-Palmer theory (TAP):

$\mathbf{C}^{-1}+J_{ij}^\text{TAP}+2(J_{ij}^\text{TAP})^{2}m_{i}m_{j}=0 , (i \neq j)$

$h_{i}^\text{TAP}=h_{i}^\text{nMF}-m_{i}\sum_{i=1}^{N} (J_{ij}^\text{TAP})^{2} (1-m_{j})^{2}$.

The Sessak and Monasson theory (SM):

$J_{ij}^\text{SM}=J_{ij}^\text{nMF}+J_{ij}^\text{pair}-\frac{C_{ij}}{(1-m_{i}^{2})(1-m_{j}^{2})-C_{ij}^{2}} , (i \neq j)$

$h_{i}^\text{SM}=0.5*\ln\left( \frac{1+m_{i}} {1-m_{i}} \right)-\sum_{j=1}^{N} J_{ij}^\text{SM} m_{j}$.

The coefficient of $R^{2}$ is:

$R^{2}=1-\frac{\sum_{ij} \left(J_{ij}^{\text{approx}}-J_{ij}^{\text{Exact}} \right)^{2} }{\sum_{ij} \left(J_{ij}^{\text{Exact}}-J_{ij}^{\text{$\overline{Exact}$}} \right)^{2}}$ , with

The rms error is: $\sqrt{\frac{1}{N(N-1)} \sum_{i \neq j} (J_{ij}^\text{approx}-J_{ij}^\text{Exact})^{2} }$.

We set the mean equal to zero and the variance of normal distribution of h and J from $\frac{1}{\sqrt{200}}$: $\frac{1}{\sqrt{150}}$: $\frac{1}{\sqrt{100}}$:$\frac{1}{\sqrt{50}}$:$\frac{1}{\sqrt{20}} \approx 0.07 : 0.08 : 0.1 : 0.14 : 0.22 $.

We use the normal distribution of h and J to get the coefficients of $R^{2}$ and the RMSerror at T=1.

We consider the cases in the beta=1.0.

The external fields are h~n(0.1,0.1)

The coupling distributions are case 1: J~ 0.8*n(-0.2,0.1) +0.1*n(0,0.2)+0.1*n(0.5,0.5)

case2: J~ 0.5*n(-0.1,0.1) +0.3*n(0,0.3)+0.2*n(0.5,0.5)

case3: J~ 1*n(0,0.1)

case4: J~ 0.5*n(-0.2,0.1) +0.5*n(0.2,0.1)

case5: J~ 1*uniform(-0.15,0.1)