ans = {};
x = {{0, 0}, {0, 1}, {1, 0}, {1, 1}};
t = {{0, 1, 0}, { 1, 1, 1}, {1, 1, 1}, {1, 0, 0}};
t = {{0}, {1}, {1}, {0}};
inD = Length[First[x]];
outD = Length[First[t]];
hSize = 2;
spread = .2;
v = .5 + RandomReal[{-spread, spread}, {inD + 1, hSize}];
w = .5 + RandomReal[{-spread, spread}, {hSize + 1, outD}];
slope = 1;
tf[z_] := N[1/(1 + Exp[-4*slope*z])];

xb = Map[Append[#, -1] & , x];

ans = Reap[
Do[
  (*1 based indexing of arrays. *)
  (* FORWARD *)
  h = xb . v;
  h = Map[tf, h, {2}];
  hb = Map[Append[#, -1] & , h];
  y = hb . w;
  y = Map[tf, y, {2}];
  Sow[First[Transpose[y]]];
  
  (* BACKWARD *)
  dy = (y - t) y (1 - y);
  dhb = hb (1 - hb) (dy . Transpose[w]);
  
  (* UPDATE *)
  eta = .2;
  (* now adjust w after using it *)
  
  w -= eta*(Transpose[hb] . dy);
  dh = Map[Drop[#, -1] &, dhb];
  v -= eta*(Transpose[xb] . dh);
, 2000]];
ans = first[second[ans]];

(* check final answer *)
h = xb. v;
h = Map[tf, h, {2}];
h = Map[Append[#, -1] & , h];
y = h . w;
y = Map[tf, y, {2}]
ListPlot[Transpose[ans], Frame -> True]