def score_fun_first_term(vals_hist,a_mid):
sum = 0.0
lim = int(np.max(vals_hist.keys()))
for i in range(0, lim+1):
if (vals_hist[i] > 0):
inner_sum = 0.0
for j in range(0, i):
inner_sum += j/(1.0 + a_mid*j)
sum += vals_hist[i]*inner_sum
return sum
##############################
## in-line functions
##############################
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