import math


def normal_dist_cdf(x, mu=0.0, sigma=1.0):
    return 0.5 * (1.0 + math.erf((x - mu) / (sigma * (2 ** 0.5))))


def normal_dist_inv_cdf(p, mu=0.0, sigma=1.0):
    # There is no closed-form solution to the inverse CDF for the normal
    # distribution, so we use a rational approximation instead:
    # Wichura, M.J. (1988). "Algorithm AS241: The Percentage Points of the
    # Normal Distribution".  Applied Statistics. Blackwell Publishing. 37
    # (3): 477–484. doi:10.2307/2347330. JSTOR 2347330.
    q = p - 0.5
    if math.fabs(q) <= 0.425:
        r = 0.180625 - q * q
        # Hash sum: 55.88319_28806_14901_4439
        num = (((((((2.5090809287301226727e+3 * r +
                     3.3430575583588128105e+4) * r +
                    6.7265770927008700853e+4) * r +
                   4.5921953931549871457e+4) * r +
                  1.3731693765509461125e+4) * r +
                 1.9715909503065514427e+3) * r +
                1.3314166789178437745e+2) * r +
               3.3871328727963666080e+0) * q
        den = (((((((5.2264952788528545610e+3 * r +
                     2.8729085735721942674e+4) * r +
                    3.9307895800092710610e+4) * r +
                   2.1213794301586595867e+4) * r +
                  5.3941960214247511077e+3) * r +
                 6.8718700749205790830e+2) * r +
                4.2313330701600911252e+1) * r +
               1.0)
        x = num / den
        return mu + (x * sigma)
    r = p if q <= 0.0 else 1.0 - p
    r = math.sqrt(-math.log(r))
    if r <= 5.0:
        r = r - 1.6
        # Hash sum: 49.33206_50330_16102_89036
        num = (((((((7.74545014278341407640e-4 * r +
                     2.27238449892691845833e-2) * r +
                    2.41780725177450611770e-1) * r +
                   1.27045825245236838258e+0) * r +
                  3.64784832476320460504e+0) * r +
                 5.76949722146069140550e+0) * r +
                4.63033784615654529590e+0) * r +
               1.42343711074968357734e+0)
        den = (((((((1.05075007164441684324e-9 * r +
                     5.47593808499534494600e-4) * r +
                    1.51986665636164571966e-2) * r +
                   1.48103976427480074590e-1) * r +
                  6.89767334985100004550e-1) * r +
                 1.67638483018380384940e+0) * r +
                2.05319162663775882187e+0) * r +
               1.0)
    else:
        r = r - 5.0
        # Hash sum: 47.52583317549289671629
        num = (((((((2.01033439929228813265e-7 * r +
                     2.71155556874348757815e-5) * r +
                    1.24266094738807843860e-3) * r +
                   2.65321895265761230930e-2) * r +
                  2.96560571828504891230e-1) * r +
                 1.78482653991729133580e+0) * r +
                5.46378491116411436990e+0) * r +
               6.65790464350110377720e+0)
        den = (((((((2.04426310338993978564e-15 * r +
                     1.42151175831644588870e-7) * r +
                    1.84631831751005468180e-5) * r +
                   7.86869131145613259100e-4) * r +
                  1.48753612908506148525e-2) * r +
                 1.36929880922735805310e-1) * r +
                5.99832206555887937690e-1) * r +
               1.0)
    x = num / den
    if q < 0.0:
        x = -x
    return mu + (x * sigma)