Path: blob/master/osu.Game/Rulesets/Difficulty/Utils/DifficultyCalculationUtils.cs
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// Copyright (c) ppy Pty Ltd <[email protected]>. Licensed under the MIT Licence. // See the LICENCE file in the repository root for full licence text. using System; using System.Linq; namespace osu.Game.Rulesets.Difficulty.Utils { public static partial class DifficultyCalculationUtils { /// <summary> /// Converts BPM value into milliseconds /// </summary> /// <param name="bpm">Beats per minute</param> /// <param name="delimiter">Which rhythm delimiter to use, default is 1/4</param> /// <returns>BPM conveted to milliseconds</returns> public static double BPMToMilliseconds(double bpm, int delimiter = 4) { return 60000.0 / delimiter / bpm; } /// <summary> /// Converts milliseconds value into a BPM value /// </summary> /// <param name="ms">Milliseconds</param> /// <param name="delimiter">Which rhythm delimiter to use, default is 1/4</param> /// <returns>Milliseconds conveted to beats per minute</returns> public static double MillisecondsToBPM(double ms, int delimiter = 4) { return 60000.0 / (ms * delimiter); } /// <summary> /// Calculates a S-shaped logistic function (https://en.wikipedia.org/wiki/Logistic_function) /// </summary> /// <param name="x">Value to calculate the function for</param> /// <param name="maxValue">Maximum value returnable by the function</param> /// <param name="multiplier">Growth rate of the function</param> /// <param name="midpointOffset">How much the function midpoint is offset from zero <paramref name="x"/></param> /// <returns>The output of logistic function of <paramref name="x"/></returns> public static double Logistic(double x, double midpointOffset, double multiplier, double maxValue = 1) => maxValue / (1 + Math.Exp(multiplier * (midpointOffset - x))); /// <summary> /// Calculates a S-shaped logistic function (https://en.wikipedia.org/wiki/Logistic_function) /// </summary> /// <param name="maxValue">Maximum value returnable by the function</param> /// <param name="exponent">Exponent</param> /// <returns>The output of logistic function</returns> public static double Logistic(double exponent, double maxValue = 1) => maxValue / (1 + Math.Exp(exponent)); /// <summary> /// Returns the <i>p</i>-norm of an <i>n</i>-dimensional vector (https://en.wikipedia.org/wiki/Norm_(mathematics)) /// </summary> /// <param name="p">The value of <i>p</i> to calculate the norm for.</param> /// <param name="values">The coefficients of the vector.</param> /// <returns>The <i>p</i>-norm of the vector.</returns> public static double Norm(double p, params double[] values) => Math.Pow(values.Sum(x => Math.Pow(x, p)), 1 / p); /// <summary> /// Calculates a Gaussian-based bell curve function (https://en.wikipedia.org/wiki/Gaussian_function) /// </summary> /// <param name="x">Value to calculate the function for</param> /// <param name="mean">The mean (center) of the bell curve</param> /// <param name="width">The width (spread) of the curve</param> /// <param name="multiplier">Multiplier to adjust the curve's height</param> /// <returns>The output of the bell curve function of <paramref name="x"/></returns> public static double BellCurve(double x, double mean, double width, double multiplier = 1.0) => multiplier * Math.Exp(Math.E * -(Math.Pow(x - mean, 2) / Math.Pow(width, 2))); /// <summary> /// Calculates a Smoothstep Bellcurve that returns returns 1 for x = mean, and smoothly reducing it's value to 0 over width /// </summary> /// <param name="x">Value to calculate the function for</param> /// <param name="mean">Value of x, for which return value will be the highest (=1)</param> /// <param name="width">Range [mean - width, mean + width] where function will change values</param> /// <returns>The output of the smoothstep bell curve function of <paramref name="x"/></returns> public static double SmoothstepBellCurve(double x, double mean = 0.5, double width = 0.5) { x -= mean; x = x > 0 ? (width - x) : (width + x); return Smoothstep(x, 0, width); } /// <summary> /// Smoothstep function (https://en.wikipedia.org/wiki/Smoothstep) /// </summary> /// <param name="x">Value to calculate the function for</param> /// <param name="start">Value at which function returns 0</param> /// <param name="end">Value at which function returns 1</param> public static double Smoothstep(double x, double start, double end) { x = Math.Clamp((x - start) / (end - start), 0.0, 1.0); return x * x * (3.0 - 2.0 * x); } /// <summary> /// Smootherstep function (https://en.wikipedia.org/wiki/Smoothstep#Variations) /// </summary> /// <param name="x">Value to calculate the function for</param> /// <param name="start">Value at which function returns 0</param> /// <param name="end">Value at which function returns 1</param> public static double Smootherstep(double x, double start, double end) { x = Math.Clamp((x - start) / (end - start), 0.0, 1.0); return x * x * x * (x * (6.0 * x - 15.0) + 10.0); } /// <summary> /// Reverse linear interpolation function (https://en.wikipedia.org/wiki/Linear_interpolation) /// </summary> /// <param name="x">Value to calculate the function for</param> /// <param name="start">Value at which function returns 0</param> /// <param name="end">Value at which function returns 1</param> public static double ReverseLerp(double x, double start, double end) { return Math.Clamp((x - start) / (end - start), 0.0, 1.0); } /// <summary> /// Error function (https://en.wikipedia.org/wiki/Error_function) /// </summary> /// <param name="x">Value to calculate the function for</param> public static double Erf(double x) { if (x == 0) return 0; if (double.IsPositiveInfinity(x)) return 1; if (double.IsNegativeInfinity(x)) return -1; if (double.IsNaN(x)) return double.NaN; // Constants for approximation (Abramowitz and Stegun formula 7.1.26) double t = 1.0 / (1.0 + 0.3275911 * Math.Abs(x)); double tau = t * (0.254829592 + t * (-0.284496736 + t * (1.421413741 + t * (-1.453152027 + t * 1.061405429)))); double erf = 1.0 - tau * Math.Exp(-x * x); return x >= 0 ? erf : -erf; } /// <summary> /// Complementary error function (https://en.wikipedia.org/wiki/Error_function) /// </summary> /// <param name="x">Value to calculate the function for</param> public static double Erfc(double x) => 1 - Erf(x); /// <summary> /// Inverse error function (https://en.wikipedia.org/wiki/Error_function) /// </summary> /// <param name="x">Value to calculate the function for</param> public static double ErfInv(double x) { if (x <= -1) return double.NegativeInfinity; if (x >= 1) return double.PositiveInfinity; if (x == 0) return 0; const double a = 0.147; double sgn = Math.Sign(x); x = Math.Abs(x); double ln = Math.Log(1 - x * x); double t1 = 2 / (Math.PI * a) + ln / 2; double t2 = ln / a; double baseApprox = Math.Sqrt(t1 * t1 - t2) - t1; // Correction reduces max error from -0.005 to -0.00045. double c = x >= 0.85 ? Math.Pow((x - 0.85) / 0.293, 8) : 0; double erfInv = sgn * (Math.Sqrt(baseApprox) + c); return erfInv; } /// <summary> /// Inverse complementary error function (https://en.wikipedia.org/wiki/Error_function) /// </summary> /// <param name="x">Value to calculate the function for</param> public static double ErfcInv(double x) => ErfInv(1 - x); } }