Python implementation of the sigmoid to map a point value to a rating (where points are milliseconds and the rating is the resulting score).

rating.ipynb

rating.py

%matplotlib
"""
in reference to http://programmers.stackexchange.com/questions/225487/mapping-values-over-a-curve/225503#225503
"""
import numpy as np
import matplotlib.pyplot as plt
import math

MIN_RATING=1
MAX_RATING=2000

MIN_POINTS=1
MAX_POINTS=8000

SIGMOID_RANGE=25

def sigmoid(x):
    return 1 / (1 + math.exp(-x))

def map2range(value, low, high, new_low, new_high):
    '''map a value from one range to another'''
    return value * 1.0 / (high - low + 1) * (new_high - new_low + 1)

def map2sigmoid(value, max=SIGMOID_RANGE):
    '''map a value from its rating range to the sigmoid range'''
    return map2range(value, MIN_RATING, MAX_RATING, 1.0, max) - (max / 2.0)  

def map2rating(value, max=SIGMOID_RANGE):
    '''map a points value to a rating value using the sigmoid function'''
    rating_range = map2range(value, MIN_POINTS, MAX_POINTS, MIN_RATING, MAX_RATING)
    sigmoid_range = map2sigmoid(rating_range, max=max)
    s = sigmoid(sigmoid_range) 
    if s < 0.5:
        r = math.floor(s * MAX_RATING)
    else:
        r = math.ceil(s * MAX_RATING)
    return r

ax = plt.subplot(1,1,1)
x = np.arange(MIN_RATING,MAX_RATING,1);
y = [sigmoid(map2sigmoid(y, max=10)) for y in x]
ax.plot(x, y, label="range=10")
y = [sigmoid(map2sigmoid(y, max=20)) for y in x]
ax.plot(x, y, label="range=20")
y = [sigmoid(map2sigmoid(y, max=30)) for y in x]
ax.plot(x, y, label="range=30")

handles, labels = ax.get_legend_handles_labels()
ax.legend(handles, labels, loc=2)

print "%5s %7s %7s" % ("ms", "r=6", "r=20")
for points in range(0, 8001, 500):
    print "%5d %7.0f %7.0f" % (points, map2rating(points,max=6), map2rating(points, max=20))