717.sagews

this is a very fast approximation of a trigonometric function like cos using polynomials

cosplot = plot(cos(x), (x,-pi/2,pi), color="red")

approx(x,k) = 1-(x/(pi/2))^2 + k *(x^2*(x - (pi/2)))

l_sum=[(k, sum([abs(cos(_) - approx(_,k).n()) for _ in srange(0,pi/2,0.05,include_endpoint=True)])) for k in sxrange(0.093000,0.095000,.000001,include_endpoint=True)]
k_best_sum = sorted(l_sum, key=lambda x:x[1])[0][0]
print k_best_sum

0.0937900000000008

l_max=[(k, max([abs(cos(_) - approx(_,k).n()) for _ in srange(0,pi/2,0.05,include_endpoint=True)])) for k in sxrange(0.093000,0.0950000,.000001,include_endpoint=True)]
k_best_max = sorted(l_max, key=lambda x:x[1])[0][0]
print k_best_max

0.0945440000000015

p1 = plot(cos(x) - approx(x,k_best_sum), (x,-0.1, pi/2+.1), color="green")
p2 = plot(cos(x) - approx(x,k_best_max), (x,-0.1, pi/2+.1), color="orange")
print 'error plot: green=best-sum, orange=best-max'
show(p1 + p2)

error plot: green=best-sum, orange=best-max

show(cosplot + plot(approx(x,k_best_max), (x,0,pi/2)), ymin=-1, ymax=1)