def F(C):
    x2 = x**2
    C5_2, C3_2, C1_2 = C
    y = (((C5_2 * x2) + C3_2) * x2 + C1_2) * x
    y *= 2        # scaling
    return y

def Hrms(C):
    return 1E+08 * ((F(C) - truth)**2).sum()

def Hmax(C):
    return 1E+06 * np.abs((F(C)-truth)).max()

import numpy as np
from scipy.special import factorial
from scipy.optimize import minimize
import matplotlib.pyplot as plt

quarterpi, halfpi, pi, twopi = [f*np.pi for f in (0.25, 0.5, 1, 2)]

C5_2 =  0.03635510    # per https://space.stackexchange.com/a/30955/12102
C3_2 = -0.32161470
C1_2 =  0.78531340

C0   = np.array([C5_2, C3_2, C1_2])

Cnom = np.array([halfpi**5 /(2.*factorial(5)),
                 halfpi**3 /(2.*factorial(3)),
                 halfpi**1 /(2.*factorial(1))]) # if expanding about 0

angle = np.linspace(0, halfpi, 1001)
truth = np.sin(angle)
x     = angle/halfpi   # scaling

Cst = np.array([0.1, 0.0, 0.5])  # arbitrary starting point

opts = {'xtol': 1E-12, 'disp': False, 'maxiter':1000, 'maxfev':2000}

resrms = minimize(Hrms, Cst, method='Nelder-Mead', options=opts)
Crms   = resrms['x']

resmax = minimize(Hmax, Cst, method='Nelder-Mead', options=opts)
Cmax   = resmax['x']

np.set_printoptions(precision=6)

print ("C0:   {}, Hmax(C0):   {:0.2f}, Hmrms(C0):   {:0.2f}".format(C0, Hmax(C0), Hrms(C0)))
print ("Cnom: {}, Hmax(Cnom): {:0.2f}, Hmrms(Cnom): {:0.2f}".format(Cnom, Hmax(Cnom), Hrms(Cnom)))
print ("Crms: {}, Hmax(Crms): {:0.2f}, Hmrms(Crms): {:0.2f}".format(Crms, Hmax(Crms), Hrms(Crms)))
print ("Cmax: {}, Hmax(Cmax): {:0.2f}, Hmrms(Cmax): {:0.2f}".format(Cmax, Hmax(Cmax), Hrms(Cmax)))