Merge branch 'master' into grids-4

.github/workflows/diffcalc.yml

@@ -104,6 +104,25 @@ env:
   EXECUTION_ID: execution-${{ github.run_id }}-${{ github.run_number }}-${{ github.run_attempt }}
 
 jobs:
+  master-environment:
+    name: Save master environment
+    runs-on: ubuntu-latest
+    outputs:
+      HEAD: ${{ steps.get-head.outputs.HEAD }}
+    steps:
+      - name: Checkout osu
+        uses: actions/checkout@v4
+        with:
+          ref: master
+          sparse-checkout: |
+            README.md
+
+      - name: Get HEAD ref
+        id: get-head
+        run: |
+          ref=$(git log -1 --format='%H')
+          echo "HEAD=https://github.com/${{ github.repository }}/commit/${ref}" >> "${GITHUB_OUTPUT}"
+
   check-permissions:
     name: Check permissions
     runs-on: ubuntu-latest
@@ -121,7 +140,7 @@ jobs:
 
   create-comment:
     name: Create PR comment
-    needs: check-permissions
+    needs: [ master-environment, check-permissions ]
     runs-on: ubuntu-latest
     if: ${{ github.event_name == 'issue_comment' && github.event.issue.pull_request }}
     steps:
@@ -158,7 +177,7 @@ jobs:
 
   environment:
     name: Setup environment
-    needs: directory
+    needs: [ master-environment, directory ]
     runs-on: self-hosted
     env:
       VARS_JSON: ${{ toJSON(vars) }}
@@ -182,6 +201,10 @@ jobs:
               fi
           done
 
+      - name: Add master environment
+        run: |
+          sed -i "s;^OSU_A=.*$;OSU_A=${{ needs.master-environment.outputs.HEAD }};" "${{ needs.directory.outputs.GENERATOR_ENV }}"
+
       - name: Add pull-request environment
         if: ${{ github.event_name == 'issue_comment' && github.event.issue.pull_request }}
         run: |

osu.Android/GameplayScreenRotationLocker.cs

@@ -6,28 +6,29 @@ using osu.Framework.Allocation;
 using osu.Framework.Bindables;
 using osu.Framework.Graphics;
 using osu.Game;
+using osu.Game.Screens.Play;
 
 namespace osu.Android
 {
     public partial class GameplayScreenRotationLocker : Component
     {
-        private Bindable<bool> localUserPlaying = null!;
+        private IBindable<LocalUserPlayingState> localUserPlaying = null!;
 
         [Resolved]
         private OsuGameActivity gameActivity { get; set; } = null!;
 
         [BackgroundDependencyLoader]
-        private void load(OsuGame game)
+        private void load(ILocalUserPlayInfo localUserPlayInfo)
         {
-            localUserPlaying = game.LocalUserPlaying.GetBoundCopy();
+            localUserPlaying = localUserPlayInfo.PlayingState.GetBoundCopy();
             localUserPlaying.BindValueChanged(updateLock, true);
         }
 
-        private void updateLock(ValueChangedEvent<bool> userPlaying)
+        private void updateLock(ValueChangedEvent<LocalUserPlayingState> userPlaying)
         {
             gameActivity.RunOnUiThread(() =>
             {
-                gameActivity.RequestedOrientation = userPlaying.NewValue ? ScreenOrientation.Locked : gameActivity.DefaultOrientation;
+                gameActivity.RequestedOrientation = userPlaying.NewValue != LocalUserPlayingState.NotPlaying ? ScreenOrientation.Locked : gameActivity.DefaultOrientation;
             });
         }
     }

osu.Desktop/Updater/VelopackUpdateManager.cs

@@ -25,6 +25,8 @@ namespace osu.Desktop.Updater
         [Resolved]
         private ILocalUserPlayInfo? localUserInfo { get; set; }
 
+        private bool isInGameplay => localUserInfo?.PlayingState.Value != LocalUserPlayingState.NotPlaying;
+
         private UpdateInfo? pendingUpdate;
 
         public VelopackUpdateManager()
@@ -43,7 +45,7 @@ namespace osu.Desktop.Updater
 
         protected override async Task<bool> PerformUpdateCheck() => await checkForUpdateAsync().ConfigureAwait(false);
 
-        private async Task<bool> checkForUpdateAsync(UpdateProgressNotification? notification = null)
+        private async Task<bool> checkForUpdateAsync()
         {
             // whether to check again in 30 minutes. generally only if there's an error or no update was found (yet).
             bool scheduleRecheck = false;
@@ -51,10 +53,10 @@ namespace osu.Desktop.Updater
             try
             {
                 // Avoid any kind of update checking while gameplay is running.
-                if (localUserInfo?.IsPlaying.Value == true)
+                if (isInGameplay)
                 {
                     scheduleRecheck = true;
-                    return false;
+                    return true;
                 }
 
                 // TODO: we should probably be checking if there's a more recent update, rather than shortcutting here.
@@ -84,27 +86,22 @@ namespace osu.Desktop.Updater
                 }
 
                 // An update is found, let's notify the user and start downloading it.
-                if (notification == null)
+                UpdateProgressNotification notification = new UpdateProgressNotification
                 {
-                    notification = new UpdateProgressNotification
+                    CompletionClickAction = () =>
                     {
-                        CompletionClickAction = () =>
-                        {
-                            Task.Run(restartToApplyUpdate);
-                            return true;
-                        },
-                    };
-
-                    Schedule(() => notificationOverlay.Post(notification));
-                }
+                        Task.Run(restartToApplyUpdate);
+                        return true;
+                    },
+                };
 
+                runOutsideOfGameplay(() => notificationOverlay.Post(notification));
                 notification.StartDownload();
 
                 try
                 {
                     await updateManager.DownloadUpdatesAsync(pendingUpdate, p => notification.Progress = p / 100f).ConfigureAwait(false);
-
-                    notification.State = ProgressNotificationState.Completed;
+                    runOutsideOfGameplay(() => notification.State = ProgressNotificationState.Completed);
                 }
                 catch (Exception e)
                 {
@@ -131,6 +128,17 @@ namespace osu.Desktop.Updater
             return true;
         }
 
+        private void runOutsideOfGameplay(Action action)
+        {
+            if (isInGameplay)
+            {
+                Scheduler.AddDelayed(() => runOutsideOfGameplay(action), 1000);
+                return;
+            }
+
+            action();
+        }
+
         private async Task restartToApplyUpdate()
         {
             await updateManager.WaitExitThenApplyUpdatesAsync(pendingUpdate?.TargetFullRelease).ConfigureAwait(false);

osu.Desktop/Windows/GameplayWinKeyBlocker.cs

@@ -13,7 +13,7 @@ namespace osu.Desktop.Windows
     public partial class GameplayWinKeyBlocker : Component
     {
         private Bindable<bool> disableWinKey = null!;
-        private IBindable<bool> localUserPlaying = null!;
+        private IBindable<LocalUserPlayingState> localUserPlaying = null!;
         private IBindable<bool> isActive = null!;
 
         [Resolved]
@@ -22,7 +22,7 @@ namespace osu.Desktop.Windows
         [BackgroundDependencyLoader]
         private void load(ILocalUserPlayInfo localUserInfo, OsuConfigManager config)
         {
-            localUserPlaying = localUserInfo.IsPlaying.GetBoundCopy();
+            localUserPlaying = localUserInfo.PlayingState.GetBoundCopy();
             localUserPlaying.BindValueChanged(_ => updateBlocking());
 
             isActive = host.IsActive.GetBoundCopy();
@@ -34,7 +34,7 @@ namespace osu.Desktop.Windows
 
         private void updateBlocking()
         {
-            bool shouldDisable = isActive.Value && disableWinKey.Value && localUserPlaying.Value;
+            bool shouldDisable = isActive.Value && disableWinKey.Value && localUserPlaying.Value == LocalUserPlayingState.Playing;
 
             if (shouldDisable)
                 host.InputThread.Scheduler.Add(WindowsKey.Disable);

osu.Game.Rulesets.Mania/Difficulty/ManiaDifficultyCalculator.cs

@@ -29,7 +29,7 @@ namespace osu.Game.Rulesets.Mania.Difficulty
         private readonly bool isForCurrentRuleset;
         private readonly double originalOverallDifficulty;
 
-        public override int Version => 20230817;
+        public override int Version => 20241007;
 
         public ManiaDifficultyCalculator(IRulesetInfo ruleset, IWorkingBeatmap beatmap)
             : base(ruleset, beatmap)

osu.Game.Rulesets.Osu/Difficulty/OsuDifficultyAttributes.cs

@@ -46,6 +46,12 @@ namespace osu.Game.Rulesets.Osu.Difficulty
         [JsonProperty("slider_factor")]
         public double SliderFactor { get; set; }
 
+        [JsonProperty("aim_difficult_strain_count")]
+        public double AimDifficultStrainCount { get; set; }
+
+        [JsonProperty("speed_difficult_strain_count")]
+        public double SpeedDifficultStrainCount { get; set; }
+
         /// <summary>
         /// The perceived approach rate inclusive of rate-adjusting mods (DT/HT/etc).
         /// </summary>
@@ -99,6 +105,9 @@ namespace osu.Game.Rulesets.Osu.Difficulty
                 yield return (ATTRIB_ID_FLASHLIGHT, FlashlightDifficulty);
 
             yield return (ATTRIB_ID_SLIDER_FACTOR, SliderFactor);
+
+            yield return (ATTRIB_ID_AIM_DIFFICULT_STRAIN_COUNT, AimDifficultStrainCount);
+            yield return (ATTRIB_ID_SPEED_DIFFICULT_STRAIN_COUNT, SpeedDifficultStrainCount);
             yield return (ATTRIB_ID_SPEED_NOTE_COUNT, SpeedNoteCount);
         }
 
@@ -113,8 +122,9 @@ namespace osu.Game.Rulesets.Osu.Difficulty
             StarRating = values[ATTRIB_ID_DIFFICULTY];
             FlashlightDifficulty = values.GetValueOrDefault(ATTRIB_ID_FLASHLIGHT);
             SliderFactor = values[ATTRIB_ID_SLIDER_FACTOR];
+            AimDifficultStrainCount = values[ATTRIB_ID_AIM_DIFFICULT_STRAIN_COUNT];
+            SpeedDifficultStrainCount = values[ATTRIB_ID_SPEED_DIFFICULT_STRAIN_COUNT];
             SpeedNoteCount = values[ATTRIB_ID_SPEED_NOTE_COUNT];
-
             DrainRate = onlineInfo.DrainRate;
             HitCircleCount = onlineInfo.CircleCount;
             SliderCount = onlineInfo.SliderCount;

osu.Game.Rulesets.Osu/Difficulty/OsuDifficultyCalculator.cs

@@ -24,7 +24,7 @@ namespace osu.Game.Rulesets.Osu.Difficulty
     {
         private const double difficulty_multiplier = 0.0675;
 
-        public override int Version => 20220902;
+        public override int Version => 20241007;
 
         public OsuDifficultyCalculator(IRulesetInfo ruleset, IWorkingBeatmap beatmap)
             : base(ruleset, beatmap)
@@ -48,6 +48,9 @@ namespace osu.Game.Rulesets.Osu.Difficulty
 
             double sliderFactor = aimRating > 0 ? aimRatingNoSliders / aimRating : 1;
 
+            double aimDifficultyStrainCount = ((OsuStrainSkill)skills[0]).CountDifficultStrains();
+            double speedDifficultyStrainCount = ((OsuStrainSkill)skills[2]).CountDifficultStrains();
+
             if (mods.Any(m => m is OsuModTouchDevice))
             {
                 aimRating = Math.Pow(aimRating, 0.8);
@@ -100,6 +103,8 @@ namespace osu.Game.Rulesets.Osu.Difficulty
                 SpeedNoteCount = speedNotes,
                 FlashlightDifficulty = flashlightRating,
                 SliderFactor = sliderFactor,
+                AimDifficultStrainCount = aimDifficultyStrainCount,
+                SpeedDifficultStrainCount = speedDifficultyStrainCount,
                 ApproachRate = preempt > 1200 ? (1800 - preempt) / 120 : (1200 - preempt) / 150 + 5,
                 OverallDifficulty = (80 - hitWindowGreat) / 6,
                 DrainRate = drainRate,

osu.Game.Rulesets.Osu/Difficulty/OsuPerformanceCalculator.cs

@@ -97,11 +97,8 @@ namespace osu.Game.Rulesets.Osu.Difficulty
                                  (totalHits > 2000 ? Math.Log10(totalHits / 2000.0) * 0.5 : 0.0);
             aimValue *= lengthBonus;
 
-            // Penalize misses by assessing # of misses relative to the total # of objects. Default a 3% reduction for any # of misses.
             if (effectiveMissCount > 0)
-                aimValue *= 0.97 * Math.Pow(1 - Math.Pow(effectiveMissCount / totalHits, 0.775), effectiveMissCount);
-
-            aimValue *= getComboScalingFactor(attributes);
+                aimValue *= calculateMissPenalty(effectiveMissCount, attributes.AimDifficultStrainCount);
 
             double approachRateFactor = 0.0;
             if (attributes.ApproachRate > 10.33)
@@ -150,11 +147,8 @@ namespace osu.Game.Rulesets.Osu.Difficulty
                                  (totalHits > 2000 ? Math.Log10(totalHits / 2000.0) * 0.5 : 0.0);
             speedValue *= lengthBonus;
 
-            // Penalize misses by assessing # of misses relative to the total # of objects. Default a 3% reduction for any # of misses.
             if (effectiveMissCount > 0)
-                speedValue *= 0.97 * Math.Pow(1 - Math.Pow(effectiveMissCount / totalHits, 0.775), Math.Pow(effectiveMissCount, .875));
-
-            speedValue *= getComboScalingFactor(attributes);
+                speedValue *= calculateMissPenalty(effectiveMissCount, attributes.SpeedDifficultStrainCount);
 
             double approachRateFactor = 0.0;
             if (attributes.ApproachRate > 10.33)
@@ -271,6 +265,10 @@ namespace osu.Game.Rulesets.Osu.Difficulty
             return Math.Max(countMiss, comboBasedMissCount);
         }
 
+        // Miss penalty assumes that a player will miss on the hardest parts of a map,
+        // so we use the amount of relatively difficult sections to adjust miss penalty
+        // to make it more punishing on maps with lower amount of hard sections.
+        private double calculateMissPenalty(double missCount, double difficultStrainCount) => 0.96 / ((missCount / (4 * Math.Pow(Math.Log(difficultStrainCount), 0.94))) + 1);
         private double getComboScalingFactor(OsuDifficultyAttributes attributes) => attributes.MaxCombo <= 0 ? 1.0 : Math.Min(Math.Pow(scoreMaxCombo, 0.8) / Math.Pow(attributes.MaxCombo, 0.8), 1.0);
         private int totalHits => countGreat + countOk + countMeh + countMiss;
     }

osu.Game.Rulesets.Osu/Difficulty/Skills/Aim.cs

@@ -34,6 +34,7 @@ namespace osu.Game.Rulesets.Osu.Difficulty.Skills
         {
             currentStrain *= strainDecay(current.DeltaTime);
             currentStrain += AimEvaluator.EvaluateDifficultyOf(current, withSliders) * skillMultiplier;
+            ObjectStrains.Add(currentStrain);
 
             return currentStrain;
         }

osu.Game.Rulesets.Osu/Difficulty/Skills/OsuStrainSkill.cs

@@ -23,6 +23,9 @@ namespace osu.Game.Rulesets.Osu.Difficulty.Skills
         /// </summary>
         protected virtual double ReducedStrainBaseline => 0.75;
 
+        protected List<double> ObjectStrains = new List<double>();
+        protected double Difficulty;
+
         protected OsuStrainSkill(Mod[] mods)
             : base(mods)
         {
@@ -30,7 +33,7 @@ namespace osu.Game.Rulesets.Osu.Difficulty.Skills
 
         public override double DifficultyValue()
         {
-            double difficulty = 0;
+            Difficulty = 0;
             double weight = 1;
 
             // Sections with 0 strain are excluded to avoid worst-case time complexity of the following sort (e.g. /b/2351871).
@@ -50,11 +53,25 @@ namespace osu.Game.Rulesets.Osu.Difficulty.Skills
             // We're sorting from highest to lowest strain.
             foreach (double strain in strains.OrderDescending())
             {
-                difficulty += strain * weight;
+                Difficulty += strain * weight;
                 weight *= DecayWeight;
             }
 
-            return difficulty;
+            return Difficulty;
+        }
+
+        /// <summary>
+        /// Returns the number of strains weighted against the top strain.
+        /// The result is scaled by clock rate as it affects the total number of strains.
+        /// </summary>
+        public double CountDifficultStrains()
+        {
+            if (Difficulty == 0)
+                return 0.0;
+
+            double consistentTopStrain = Difficulty / 10; // What would the top strain be if all strain values were identical
+            // Use a weighted sum of all strains. Constants are arbitrary and give nice values
+            return ObjectStrains.Sum(s => 1.1 / (1 + Math.Exp(-10 * (s / consistentTopStrain - 0.88))));
         }
 
         public static double DifficultyToPerformance(double difficulty) => Math.Pow(5.0 * Math.Max(1.0, difficulty / 0.0675) - 4.0, 3.0) / 100000.0;

osu.Game.Rulesets.Osu/Difficulty/Skills/Speed.cs

@@ -6,7 +6,6 @@ using osu.Game.Rulesets.Difficulty.Preprocessing;
 using osu.Game.Rulesets.Mods;
 using osu.Game.Rulesets.Osu.Difficulty.Evaluators;
 using osu.Game.Rulesets.Osu.Difficulty.Preprocessing;
-using System.Collections.Generic;
 using System.Linq;
 
 namespace osu.Game.Rulesets.Osu.Difficulty.Skills
@@ -24,8 +23,6 @@ namespace osu.Game.Rulesets.Osu.Difficulty.Skills
 
         protected override int ReducedSectionCount => 5;
 
-        private readonly List<double> objectStrains = new List<double>();
-
         public Speed(Mod[] mods)
             : base(mods)
         {
@@ -43,23 +40,21 @@ namespace osu.Game.Rulesets.Osu.Difficulty.Skills
             currentRhythm = RhythmEvaluator.EvaluateDifficultyOf(current);
 
             double totalStrain = currentStrain * currentRhythm;
-
-            objectStrains.Add(totalStrain);
+            ObjectStrains.Add(totalStrain);
 
             return totalStrain;
         }
 
         public double RelevantNoteCount()
         {
-            if (objectStrains.Count == 0)
+            if (ObjectStrains.Count == 0)
                 return 0;
 
-            double maxStrain = objectStrains.Max();
-
+            double maxStrain = ObjectStrains.Max();
             if (maxStrain == 0)
                 return 0;
 
-            return objectStrains.Sum(strain => 1.0 / (1.0 + Math.Exp(-(strain / maxStrain * 12.0 - 6.0))));
+            return ObjectStrains.Sum(strain => 1.0 / (1.0 + Math.Exp(-(strain / maxStrain * 12.0 - 6.0))));
         }
     }
 }

osu.Game.Rulesets.Taiko/Difficulty/TaikoDifficultyAttributes.cs

@@ -43,6 +43,15 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
         [JsonProperty("great_hit_window")]
         public double GreatHitWindow { get; set; }
 
+        /// <summary>
+        /// The perceived hit window for an OK hit inclusive of rate-adjusting mods (DT/HT/etc).
+        /// </summary>
+        /// <remarks>
+        /// Rate-adjusting mods don't directly affect the hit window, but have a perceived effect as a result of adjusting audio timing.
+        /// </remarks>
+        [JsonProperty("ok_hit_window")]
+        public double OkHitWindow { get; set; }
+
         public override IEnumerable<(int attributeId, object value)> ToDatabaseAttributes()
         {
             foreach (var v in base.ToDatabaseAttributes())
@@ -50,6 +59,7 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
 
             yield return (ATTRIB_ID_DIFFICULTY, StarRating);
             yield return (ATTRIB_ID_GREAT_HIT_WINDOW, GreatHitWindow);
+            yield return (ATTRIB_ID_OK_HIT_WINDOW, OkHitWindow);
         }
 
         public override void FromDatabaseAttributes(IReadOnlyDictionary<int, double> values, IBeatmapOnlineInfo onlineInfo)
@@ -58,6 +68,7 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
 
             StarRating = values[ATTRIB_ID_DIFFICULTY];
             GreatHitWindow = values[ATTRIB_ID_GREAT_HIT_WINDOW];
+            OkHitWindow = values[ATTRIB_ID_OK_HIT_WINDOW];
         }
     }
 }

osu.Game.Rulesets.Taiko/Difficulty/TaikoDifficultyCalculator.cs

@@ -25,7 +25,7 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
         private const double colour_skill_multiplier = 0.375 * difficulty_multiplier;
         private const double stamina_skill_multiplier = 0.375 * difficulty_multiplier;
 
-        public override int Version => 20221107;
+        public override int Version => 20241007;
 
         public TaikoDifficultyCalculator(IRulesetInfo ruleset, IWorkingBeatmap beatmap)
             : base(ruleset, beatmap)
@@ -99,6 +99,7 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
                 ColourDifficulty = colourRating,
                 PeakDifficulty = combinedRating,
                 GreatHitWindow = hitWindows.WindowFor(HitResult.Great) / clockRate,
+                OkHitWindow = hitWindows.WindowFor(HitResult.Ok) / clockRate,
                 MaxCombo = beatmap.GetMaxCombo(),
             };
 

osu.Game.Rulesets.Taiko/Difficulty/TaikoPerformanceAttributes.cs

@@ -18,6 +18,9 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
         [JsonProperty("effective_miss_count")]
         public double EffectiveMissCount { get; set; }
 
+        [JsonProperty("estimated_unstable_rate")]
+        public double? EstimatedUnstableRate { get; set; }
+
         public override IEnumerable<PerformanceDisplayAttribute> GetAttributesForDisplay()
         {
             foreach (var attribute in base.GetAttributesForDisplay())

osu.Game.Rulesets.Taiko/Difficulty/TaikoPerformanceCalculator.cs

@@ -9,6 +9,7 @@ using osu.Game.Rulesets.Mods;
 using osu.Game.Rulesets.Scoring;
 using osu.Game.Rulesets.Taiko.Objects;
 using osu.Game.Scoring;
+using osu.Game.Utils;
 
 namespace osu.Game.Rulesets.Taiko.Difficulty
 {
@@ -18,7 +19,7 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
         private int countOk;
         private int countMeh;
         private int countMiss;
-        private double accuracy;
+        private double? estimatedUnstableRate;
 
         private double effectiveMissCount;
 
@@ -35,7 +36,7 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
             countOk = score.Statistics.GetValueOrDefault(HitResult.Ok);
             countMeh = score.Statistics.GetValueOrDefault(HitResult.Meh);
             countMiss = score.Statistics.GetValueOrDefault(HitResult.Miss);
-            accuracy = customAccuracy;
+            estimatedUnstableRate = computeDeviationUpperBound(taikoAttributes) * 10;
 
             // The effectiveMissCount is calculated by gaining a ratio for totalSuccessfulHits and increasing the miss penalty for shorter object counts lower than 1000.
             if (totalSuccessfulHits > 0)
@@ -65,6 +66,7 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
                 Difficulty = difficultyValue,
                 Accuracy = accuracyValue,
                 EffectiveMissCount = effectiveMissCount,
+                EstimatedUnstableRate = estimatedUnstableRate,
                 Total = totalValue
             };
         }
@@ -85,35 +87,94 @@ namespace osu.Game.Rulesets.Taiko.Difficulty
                 difficultyValue *= 1.025;
 
             if (score.Mods.Any(m => m is ModHardRock))
-                difficultyValue *= 1.050;
+                difficultyValue *= 1.10;
 
             if (score.Mods.Any(m => m is ModFlashlight<TaikoHitObject>))
                 difficultyValue *= 1.050 * lengthBonus;
 
-            return difficultyValue * Math.Pow(accuracy, 2.0);
+            if (estimatedUnstableRate == null)
+                return 0;
+
+            return difficultyValue * Math.Pow(SpecialFunctions.Erf(400 / (Math.Sqrt(2) * estimatedUnstableRate.Value)), 2.0);
         }
 
         private double computeAccuracyValue(ScoreInfo score, TaikoDifficultyAttributes attributes, bool isConvert)
         {
-            if (attributes.GreatHitWindow <= 0)
+            if (attributes.GreatHitWindow <= 0 || estimatedUnstableRate == null)
                 return 0;
 
-            double accuracyValue = Math.Pow(60.0 / attributes.GreatHitWindow, 1.1) * Math.Pow(accuracy, 8.0) * Math.Pow(attributes.StarRating, 0.4) * 27.0;
+            double accuracyValue = Math.Pow(70 / estimatedUnstableRate.Value, 1.1) * Math.Pow(attributes.StarRating, 0.4) * 100.0;
 
             double lengthBonus = Math.Min(1.15, Math.Pow(totalHits / 1500.0, 0.3));
-            accuracyValue *= lengthBonus;
 
             // Slight HDFL Bonus for accuracy. A clamp is used to prevent against negative values.
             if (score.Mods.Any(m => m is ModFlashlight<TaikoHitObject>) && score.Mods.Any(m => m is ModHidden) && !isConvert)
-                accuracyValue *= Math.Max(1.0, 1.1 * lengthBonus);
+                accuracyValue *= Math.Max(1.0, 1.05 * lengthBonus);
 
             return accuracyValue;
         }
 
+        /// <summary>
+        /// Computes an upper bound on the player's tap deviation based on the OD, number of circles and sliders,
+        /// and the hit judgements, assuming the player's mean hit error is 0. The estimation is consistent in that
+        /// two SS scores on the same map with the same settings will always return the same deviation.
+        /// </summary>
+        private double? computeDeviationUpperBound(TaikoDifficultyAttributes attributes)
+        {
+            if (totalSuccessfulHits == 0 || attributes.GreatHitWindow <= 0)
+                return null;
+
+            double h300 = attributes.GreatHitWindow;
+            double h100 = attributes.OkHitWindow;
+
+            const double z = 2.32634787404; // 99% critical value for the normal distribution (one-tailed).
+
+            // The upper bound on deviation, calculated with the ratio of 300s to objects, and the great hit window.
+            double? calcDeviationGreatWindow()
+            {
+                if (countGreat == 0) return null;
+
+                double n = totalHits;
+
+                // Proportion of greats hit.
+                double p = countGreat / n;
+
+                // We can be 99% confident that p is at least this value.
+                double pLowerBound = (n * p + z * z / 2) / (n + z * z) - z / (n + z * z) * Math.Sqrt(n * p * (1 - p) + z * z / 4);
+
+                // We can be 99% confident that the deviation is not higher than:
+                return h300 / (Math.Sqrt(2) * SpecialFunctions.ErfInv(pLowerBound));
+            }
+
+            // The upper bound on deviation, calculated with the ratio of 300s + 100s to objects, and the good hit window.
+            // This will return a lower value than the first method when the number of 100s is high, but the miss count is low.
+            double? calcDeviationGoodWindow()
+            {
+                if (totalSuccessfulHits == 0) return null;
+
+                double n = totalHits;
+
+                // Proportion of greats + goods hit.
+                double p = totalSuccessfulHits / n;
+
+                // We can be 99% confident that p is at least this value.
+                double pLowerBound = (n * p + z * z / 2) / (n + z * z) - z / (n + z * z) * Math.Sqrt(n * p * (1 - p) + z * z / 4);
+
+                // We can be 99% confident that the deviation is not higher than:
+                return h100 / (Math.Sqrt(2) * SpecialFunctions.ErfInv(pLowerBound));
+            }
+
+            double? deviationGreatWindow = calcDeviationGreatWindow();
+            double? deviationGoodWindow = calcDeviationGoodWindow();
+
+            if (deviationGreatWindow is null)
+                return deviationGoodWindow;
+
+            return Math.Min(deviationGreatWindow.Value, deviationGoodWindow!.Value);
+        }
+
         private int totalHits => countGreat + countOk + countMeh + countMiss;
 
         private int totalSuccessfulHits => countGreat + countOk + countMeh;
-
-        private double customAccuracy => totalHits > 0 ? (countGreat * 300 + countOk * 150) / (totalHits * 300.0) : 0;
     }
 }

osu.Game.Tests/Database/BackgroundDataStoreProcessorTests.cs

@@ -22,9 +22,9 @@ namespace osu.Game.Tests.Database
     [HeadlessTest]
     public partial class BackgroundDataStoreProcessorTests : OsuTestScene, ILocalUserPlayInfo
     {
-        public IBindable<bool> IsPlaying => isPlaying;
+        public IBindable<LocalUserPlayingState> PlayingState => isPlaying;
 
-        private readonly Bindable<bool> isPlaying = new Bindable<bool>();
+        private readonly Bindable<LocalUserPlayingState> isPlaying = new Bindable<LocalUserPlayingState>();
 
         private BeatmapSetInfo importedSet = null!;
 
@@ -37,7 +37,7 @@ namespace osu.Game.Tests.Database
         [SetUpSteps]
         public void SetUpSteps()
         {
-            AddStep("Set not playing", () => isPlaying.Value = false);
+            AddStep("Set not playing", () => isPlaying.Value = LocalUserPlayingState.NotPlaying);
         }
 
         [Test]
@@ -89,7 +89,7 @@ namespace osu.Game.Tests.Database
                 });
             });
 
-            AddStep("Set playing", () => isPlaying.Value = true);
+            AddStep("Set playing", () => isPlaying.Value = LocalUserPlayingState.Playing);
 
             AddStep("Reset difficulty", () =>
             {
@@ -117,7 +117,7 @@ namespace osu.Game.Tests.Database
                 });
             });
 
-            AddStep("Set not playing", () => isPlaying.Value = false);
+            AddStep("Set not playing", () => isPlaying.Value = LocalUserPlayingState.NotPlaying);
 
             AddUntilStep("wait for difficulties repopulated", () =>
             {

osu.Game.Tests/Input/ConfineMouseTrackerTest.cs

@@ -3,11 +3,13 @@
 
 using NUnit.Framework;
 using osu.Framework.Allocation;
+using osu.Framework.Bindables;
 using osu.Framework.Configuration;
 using osu.Framework.Input;
 using osu.Framework.Testing;
 using osu.Game.Configuration;
 using osu.Game.Input;
+using osu.Game.Screens.Play;
 using osu.Game.Tests.Visual;
 
 namespace osu.Game.Tests.Input
@@ -15,9 +17,20 @@ namespace osu.Game.Tests.Input
     [HeadlessTest]
     public partial class ConfineMouseTrackerTest : OsuGameTestScene
     {
+        private readonly Bindable<LocalUserPlayingState> playingState = new Bindable<LocalUserPlayingState>();
+
         [Resolved]
         private FrameworkConfigManager frameworkConfigManager { get; set; } = null!;
 
+        [SetUpSteps]
+        public override void SetUpSteps()
+        {
+            base.SetUpSteps();
+
+            // a bit dodgy.
+            AddStep("bind playing state", () => ((IBindable<LocalUserPlayingState>)playingState).BindTo(((ILocalUserPlayInfo)Game).PlayingState));
+        }
+
         [TestCase(WindowMode.Windowed)]
         [TestCase(WindowMode.Borderless)]
         public void TestDisableConfining(WindowMode windowMode)
@@ -88,7 +101,7 @@ namespace osu.Game.Tests.Input
             => AddStep($"set {mode} game-side", () => Game.LocalConfig.SetValue(OsuSetting.ConfineMouseMode, mode));
 
         private void setLocalUserPlayingTo(bool playing)
-            => AddStep($"local user {(playing ? "playing" : "not playing")}", () => Game.LocalUserPlaying.Value = playing);
+            => AddStep($"local user {(playing ? "playing" : "not playing")}", () => playingState.Value = playing ? LocalUserPlayingState.Playing : LocalUserPlayingState.NotPlaying);
 
         private void gameSideModeIs(OsuConfineMouseMode mode)
             => AddAssert($"mode is {mode} game-side", () => Game.LocalConfig.Get<OsuConfineMouseMode>(OsuSetting.ConfineMouseMode) == mode);

osu.Game.Tests/Visual/Gameplay/TestSceneOverlayActivation.cs

@@ -1,8 +1,6 @@
 // Copyright (c) ppy Pty Ltd <contact@ppy.sh>. Licensed under the MIT Licence.
 // See the LICENCE file in the repository root for full licence text.
 
-#nullable disable
-
 using System.Linq;
 using NUnit.Framework;
 using osu.Game.Overlays;
@@ -12,7 +10,7 @@ namespace osu.Game.Tests.Visual.Gameplay
 {
     public partial class TestSceneOverlayActivation : OsuPlayerTestScene
     {
-        protected new OverlayTestPlayer Player => base.Player as OverlayTestPlayer;
+        protected new OverlayTestPlayer Player => (OverlayTestPlayer)base.Player;
 
         public override void SetUpSteps()
         {

osu.Game.Tests/Visual/Multiplayer/TestSceneGameplayChatDisplay.cs

@@ -25,14 +25,14 @@ namespace osu.Game.Tests.Visual.Multiplayer
         [Cached(typeof(ILocalUserPlayInfo))]
         private ILocalUserPlayInfo localUserInfo;
 
-        private readonly Bindable<bool> localUserPlaying = new Bindable<bool>();
+        private readonly Bindable<LocalUserPlayingState> playingState = new Bindable<LocalUserPlayingState>();
 
         private TextBox textBox => chatDisplay.ChildrenOfType<TextBox>().First();
 
         public TestSceneGameplayChatDisplay()
         {
             var mockLocalUserInfo = new Mock<ILocalUserPlayInfo>();
-            mockLocalUserInfo.SetupGet(i => i.IsPlaying).Returns(localUserPlaying);
+            mockLocalUserInfo.SetupGet(i => i.PlayingState).Returns(playingState);
 
             localUserInfo = mockLocalUserInfo.Object;
         }
@@ -124,6 +124,6 @@ namespace osu.Game.Tests.Visual.Multiplayer
             AddAssert($"chat {(isFocused ? "focused" : "not focused")}", () => textBox.HasFocus == isFocused);
 
         private void setLocalUserPlaying(bool playing) =>
-            AddStep($"local user {(playing ? "playing" : "not playing")}", () => localUserPlaying.Value = playing);
+            AddStep($"local user {(playing ? "playing" : "not playing")}", () => playingState.Value = playing ? LocalUserPlayingState.Playing : LocalUserPlayingState.NotPlaying);
     }
 }

osu.Game/Database/BackgroundDataStoreProcessor.cs

@@ -606,7 +606,7 @@ namespace osu.Game.Database
         {
             // Importantly, also sleep if high performance session is active.
             // If we don't do this, memory usage can become runaway due to GC running in a more lenient mode.
-            while (localUserPlayInfo?.IsPlaying.Value == true || highPerformanceSessionManager?.IsSessionActive == true)
+            while (localUserPlayInfo?.PlayingState.Value != LocalUserPlayingState.NotPlaying || highPerformanceSessionManager?.IsSessionActive == true)
             {
                 Logger.Log("Background processing sleeping due to active gameplay...");
                 Thread.Sleep(TimeToSleepDuringGameplay);

osu.Game/Input/ConfineMouseTracker.cs

@@ -15,7 +15,7 @@ namespace osu.Game.Input
 {
     /// <summary>
     /// Connects <see cref="OsuSetting.ConfineMouseMode"/> with <see cref="FrameworkSetting.ConfineMouseMode"/>.
-    /// If <see cref="OsuGame.LocalUserPlaying"/> is true, we should also confine the mouse cursor if it has been
+    /// If <see cref="ILocalUserPlayInfo.PlayingState"/> is playing, we should also confine the mouse cursor if it has been
     /// requested with <see cref="OsuConfineMouseMode.DuringGameplay"/>.
     /// </summary>
     public partial class ConfineMouseTracker : Component
@@ -25,7 +25,7 @@ namespace osu.Game.Input
         private Bindable<bool> frameworkMinimiseOnFocusLossInFullscreen;
 
         private Bindable<OsuConfineMouseMode> osuConfineMode;
-        private IBindable<bool> localUserPlaying;
+        private IBindable<LocalUserPlayingState> localUserPlaying;
 
         [BackgroundDependencyLoader]
         private void load(ILocalUserPlayInfo localUserInfo, FrameworkConfigManager frameworkConfigManager, OsuConfigManager osuConfigManager)
@@ -37,7 +37,7 @@ namespace osu.Game.Input
             frameworkMinimiseOnFocusLossInFullscreen.BindValueChanged(_ => updateConfineMode());
 
             osuConfineMode = osuConfigManager.GetBindable<OsuConfineMouseMode>(OsuSetting.ConfineMouseMode);
-            localUserPlaying = localUserInfo.IsPlaying.GetBoundCopy();
+            localUserPlaying = localUserInfo.PlayingState.GetBoundCopy();
 
             osuConfineMode.ValueChanged += _ => updateConfineMode();
             localUserPlaying.BindValueChanged(_ => updateConfineMode(), true);
@@ -63,7 +63,7 @@ namespace osu.Game.Input
                     break;
 
                 case OsuConfineMouseMode.DuringGameplay:
-                    frameworkConfineMode.Value = localUserPlaying.Value ? ConfineMouseMode.Always : ConfineMouseMode.Never;
+                    frameworkConfineMode.Value = localUserPlaying.Value == LocalUserPlayingState.Playing ? ConfineMouseMode.Always : ConfineMouseMode.Never;
                     break;
 
                 case OsuConfineMouseMode.Always:

osu.Game/Input/OsuUserInputManager.cs

@@ -3,15 +3,16 @@
 
 using osu.Framework.Bindables;
 using osu.Framework.Input;
+using osu.Game.Screens.Play;
 using osuTK.Input;
 
 namespace osu.Game.Input
 {
     public partial class OsuUserInputManager : UserInputManager
     {
-        protected override bool AllowRightClickFromLongTouch => !LocalUserPlaying.Value;
+        protected override bool AllowRightClickFromLongTouch => PlayingState.Value != LocalUserPlayingState.Playing;
 
-        public readonly BindableBool LocalUserPlaying = new BindableBool();
+        public readonly IBindable<LocalUserPlayingState> PlayingState = new Bindable<LocalUserPlayingState>();
 
         internal OsuUserInputManager()
         {

osu.Game/OsuGame.cs

@@ -175,14 +175,9 @@ namespace osu.Game
         /// </summary>
         public readonly IBindable<OverlayActivation> OverlayActivationMode = new Bindable<OverlayActivation>();
 
-        /// <summary>
-        /// Whether the local user is currently interacting with the game in a way that should not be interrupted.
-        /// </summary>
-        /// <remarks>
-        /// This is exclusively managed by <see cref="Player"/>. If other components are mutating this state, a more
-        /// resilient method should be used to ensure correct state.
-        /// </remarks>
-        public Bindable<bool> LocalUserPlaying = new BindableBool();
+        IBindable<LocalUserPlayingState> ILocalUserPlayInfo.PlayingState => playingState;
+
+        private readonly Bindable<LocalUserPlayingState> playingState = new Bindable<LocalUserPlayingState>();
 
         protected OsuScreenStack ScreenStack;
 
@@ -302,7 +297,7 @@ namespace osu.Game
         protected override UserInputManager CreateUserInputManager()
         {
             var userInputManager = base.CreateUserInputManager();
-            (userInputManager as OsuUserInputManager)?.LocalUserPlaying.BindTo(LocalUserPlaying);
+            (userInputManager as OsuUserInputManager)?.PlayingState.BindTo(playingState);
             return userInputManager;
         }
 
@@ -391,11 +386,11 @@ namespace osu.Game
             // Transfer any runtime changes back to configuration file.
             SkinManager.CurrentSkinInfo.ValueChanged += skin => configSkin.Value = skin.NewValue.ID.ToString();
 
-            LocalUserPlaying.BindValueChanged(p =>
+            playingState.BindValueChanged(p =>
             {
-                BeatmapManager.PauseImports = p.NewValue;
-                SkinManager.PauseImports = p.NewValue;
-                ScoreManager.PauseImports = p.NewValue;
+                BeatmapManager.PauseImports = p.NewValue != LocalUserPlayingState.NotPlaying;
+                SkinManager.PauseImports = p.NewValue != LocalUserPlayingState.NotPlaying;
+                ScoreManager.PauseImports = p.NewValue != LocalUserPlayingState.NotPlaying;
             }, true);
 
             IsActive.BindValueChanged(active => updateActiveState(active.NewValue), true);
@@ -1553,6 +1548,16 @@ namespace osu.Game
                 scope.SetTag(@"screen", newScreen?.GetType().ReadableName() ?? @"none");
             });
 
+            switch (current)
+            {
+                case Player player:
+                    player.PlayingState.UnbindFrom(playingState);
+
+                    // reset for sanity.
+                    playingState.Value = LocalUserPlayingState.NotPlaying;
+                    break;
+            }
+
             switch (newScreen)
             {
                 case IntroScreen intro:
@@ -1565,14 +1570,15 @@ namespace osu.Game
                     versionManager?.Show();
                     break;
 
+                case Player player:
+                    player.PlayingState.BindTo(playingState);
+                    break;
+
                 default:
                     versionManager?.Hide();
                     break;
             }
 
-            // reset on screen change for sanity.
-            LocalUserPlaying.Value = false;
-
             if (current is IOsuScreen currentOsuScreen)
             {
                 OverlayActivationMode.UnbindFrom(currentOsuScreen.OverlayActivationMode);
@@ -1621,7 +1627,5 @@ namespace osu.Game
             if (newScreen == null)
                 Exit();
         }
-
-        IBindable<bool> ILocalUserPlayInfo.IsPlaying => LocalUserPlaying;
     }
 }

osu.Game/Overlays/BeatmapListingOverlay.cs

@@ -98,7 +98,7 @@ namespace osu.Game.Overlays
             apiUser.BindValueChanged(_ => Schedule(() =>
             {
                 if (api.IsLoggedIn)
-                    replaceResultsAreaContent(Drawable.Empty());
+                    replaceResultsAreaContent(Empty());
             }));
         }
 

osu.Game/Overlays/Chat/DaySeparator.cs

@@ -75,7 +75,7 @@ namespace osu.Game.Overlays.Chat
                                         Height = LineHeight,
                                         Colour = colourProvider?.Background5 ?? Colour4.White,
                                     },
-                                    Drawable.Empty(),
+                                    Empty(),
                                     new OsuSpriteText
                                     {
                                         Anchor = Anchor.CentreRight,
@@ -87,7 +87,7 @@ namespace osu.Game.Overlays.Chat
                                 }
                             },
                         },
-                        Drawable.Empty(),
+                        Empty(),
                         new Circle
                         {
                             Anchor = Anchor.Centre,

osu.Game/Overlays/Settings/SettingsSidebar.cs

@@ -66,7 +66,7 @@ namespace osu.Game.Overlays.Settings
             [BackgroundDependencyLoader]
             private void load()
             {
-                Size = new Vector2(SettingsSidebar.EXPANDED_WIDTH);
+                Size = new Vector2(EXPANDED_WIDTH);
 
                 Padding = new MarginPadding(40);
 

osu.Game/Rulesets/Difficulty/DifficultyAttributes.cs

@@ -26,6 +26,9 @@ namespace osu.Game.Rulesets.Difficulty
         protected const int ATTRIB_ID_FLASHLIGHT = 17;
         protected const int ATTRIB_ID_SLIDER_FACTOR = 19;
         protected const int ATTRIB_ID_SPEED_NOTE_COUNT = 21;
+        protected const int ATTRIB_ID_SPEED_DIFFICULT_STRAIN_COUNT = 23;
+        protected const int ATTRIB_ID_AIM_DIFFICULT_STRAIN_COUNT = 25;
+        protected const int ATTRIB_ID_OK_HIT_WINDOW = 27;
 
         /// <summary>
         /// The mods which were applied to the beatmap.

osu.Game/Screens/OnlinePlay/Multiplayer/GameplayChatDisplay.cs

@@ -23,9 +23,9 @@ namespace osu.Game.Screens.OnlinePlay.Multiplayer
         [CanBeNull]
         private ILocalUserPlayInfo localUserInfo { get; set; }
 
-        private readonly IBindable<bool> localUserPlaying = new Bindable<bool>();
+        private readonly IBindable<LocalUserPlayingState> localUserPlaying = new Bindable<LocalUserPlayingState>();
 
-        public override bool PropagatePositionalInputSubTree => !localUserPlaying.Value;
+        public override bool PropagatePositionalInputSubTree => localUserPlaying.Value != LocalUserPlayingState.Playing;
 
         public Bindable<bool> Expanded = new Bindable<bool>();
 
@@ -58,7 +58,7 @@ namespace osu.Game.Screens.OnlinePlay.Multiplayer
             base.LoadComplete();
 
             if (localUserInfo != null)
-                localUserPlaying.BindTo(localUserInfo.IsPlaying);
+                localUserPlaying.BindTo(localUserInfo.PlayingState);
 
             localUserPlaying.BindValueChanged(playing =>
             {
@@ -67,7 +67,7 @@ namespace osu.Game.Screens.OnlinePlay.Multiplayer
                 TextBox.HoldFocus = false;
 
                 // only hold focus (after sending a message) during breaks
-                TextBox.ReleaseFocusOnCommit = playing.NewValue;
+                TextBox.ReleaseFocusOnCommit = playing.NewValue == LocalUserPlayingState.Playing;
             }, true);
 
             Expanded.BindValueChanged(_ => updateExpandedState(), true);

osu.Game/Screens/Play/ILocalUserPlayInfo.cs

@@ -10,8 +10,8 @@ namespace osu.Game.Screens.Play
     public interface ILocalUserPlayInfo
     {
         /// <summary>
-        /// Whether the local user is currently playing.
+        /// Whether the local user is currently interacting (playing) with the game in a way that should not be interrupted.
         /// </summary>
-        IBindable<bool> IsPlaying { get; }
+        IBindable<LocalUserPlayingState> PlayingState { get; }
     }
 }

osu.Game/Screens/Play/LocalUserPlayingState.cs

@@ -0,0 +1,23 @@
+// Copyright (c) ppy Pty Ltd <contact@ppy.sh>. Licensed under the MIT Licence.
+// See the LICENCE file in the repository root for full licence text.
+
+namespace osu.Game.Screens.Play
+{
+    public enum LocalUserPlayingState
+    {
+        /// <summary>
+        /// The local player is not current in gameplay. If watching a replay, gameplay always remains in this state.
+        /// </summary>
+        NotPlaying,
+
+        /// <summary>
+        /// The local player is in a break, paused, or failed but still at the gameplay screen.
+        /// </summary>
+        Break,
+
+        /// <summary>
+        /// The local user is in active gameplay.
+        /// </summary>
+        Playing,
+    }
+}

osu.Game/Screens/Play/Player.cs

@@ -94,6 +94,7 @@ namespace osu.Game.Screens.Play
         public IBindable<bool> LocalUserPlaying => localUserPlaying;
 
         private readonly Bindable<bool> localUserPlaying = new Bindable<bool>();
+        private readonly Bindable<LocalUserPlayingState> playingState = new Bindable<LocalUserPlayingState>();
 
         public int RestartCount;
 
@@ -231,9 +232,6 @@ namespace osu.Game.Screens.Play
             if (game != null)
                 gameActive.BindTo(game.IsActive);
 
-            if (game is OsuGame osuGame)
-                LocalUserPlaying.BindTo(osuGame.LocalUserPlaying);
-
             DrawableRuleset = ruleset.CreateDrawableRulesetWith(playableBeatmap, gameplayMods);
             dependencies.CacheAs(DrawableRuleset);
 
@@ -510,9 +508,16 @@ namespace osu.Game.Screens.Play
 
         private void updateGameplayState()
         {
-            bool inGameplay = !DrawableRuleset.HasReplayLoaded.Value && !DrawableRuleset.IsPaused.Value && !breakTracker.IsBreakTime.Value && !GameplayState.HasFailed;
-            OverlayActivationMode.Value = inGameplay ? OverlayActivation.Disabled : OverlayActivation.UserTriggered;
-            localUserPlaying.Value = inGameplay;
+            bool inGameplay = !DrawableRuleset.HasReplayLoaded.Value && !GameplayState.HasPassed && !GameplayState.HasFailed;
+            bool inBreak = breakTracker.IsBreakTime.Value || DrawableRuleset.IsPaused.Value;
+
+            if (inGameplay)
+                playingState.Value = inBreak ? LocalUserPlayingState.Break : LocalUserPlayingState.Playing;
+            else
+                playingState.Value = LocalUserPlayingState.NotPlaying;
+
+            localUserPlaying.Value = playingState.Value == LocalUserPlayingState.Playing;
+            OverlayActivationMode.Value = playingState.Value == LocalUserPlayingState.Playing ? OverlayActivation.Disabled : OverlayActivation.UserTriggered;
         }
 
         private void updateSampleDisabledState()
@@ -1279,6 +1284,6 @@ namespace osu.Game.Screens.Play
 
         IBindable<bool> ISamplePlaybackDisabler.SamplePlaybackDisabled => samplePlaybackDisabled;
 
-        IBindable<bool> ILocalUserPlayInfo.IsPlaying => LocalUserPlaying;
+        public IBindable<LocalUserPlayingState> PlayingState => playingState;
     }
 }

osu.Game/Utils/SpecialFunctions.cs

@@ -0,0 +1,694 @@
+// Copyright (c) ppy Pty Ltd <contact@ppy.sh>. Licensed under the MIT Licence.
+// See the LICENCE file in the repository root for full licence text.
+
+// All code is referenced from the following:
+// https://github.com/mathnet/mathnet-numerics/blob/master/src/Numerics/SpecialFunctions/Erf.cs
+// https://github.com/mathnet/mathnet-numerics/blob/master/src/Numerics/Optimization/NelderMeadSimplex.cs
+
+/*
+ Copyright (c) 2002-2022 Math.NET
+Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
+The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
+ */
+
+using System;
+
+namespace osu.Game.Utils
+{
+    public class SpecialFunctions
+    {
+        private const double sqrt2_pi = 2.5066282746310005024157652848110452530069867406099d;
+
+        /// <summary>
+        /// **************************************
+        /// COEFFICIENTS FOR METHOD ErfImp       *
+        /// **************************************
+        /// </summary>
+        /// <summary> Polynomial coefficients for a numerator of ErfImp
+        /// calculation for Erf(x) in the interval [1e-10, 0.5].
+        /// </summary>
+        private static readonly double[] erf_imp_an = { 0.00337916709551257388990745, -0.00073695653048167948530905, -0.374732337392919607868241, 0.0817442448733587196071743, -0.0421089319936548595203468, 0.0070165709512095756344528, -0.00495091255982435110337458, 0.000871646599037922480317225 };
+
+        /// <summary> Polynomial coefficients for  a denominator of ErfImp
+        /// calculation for Erf(x) in the interval [1e-10, 0.5].
+        /// </summary>
+        private static readonly double[] erf_imp_ad = { 1, -0.218088218087924645390535, 0.412542972725442099083918, -0.0841891147873106755410271, 0.0655338856400241519690695, -0.0120019604454941768171266, 0.00408165558926174048329689, -0.000615900721557769691924509 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [0.5, 0.75].
+        /// </summary>
+        private static readonly double[] erf_imp_bn = { -0.0361790390718262471360258, 0.292251883444882683221149, 0.281447041797604512774415, 0.125610208862766947294894, 0.0274135028268930549240776, 0.00250839672168065762786937 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [0.5, 0.75].
+        /// </summary>
+        private static readonly double[] erf_imp_bd = { 1, 1.8545005897903486499845, 1.43575803037831418074962, 0.582827658753036572454135, 0.124810476932949746447682, 0.0113724176546353285778481 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [0.75, 1.25].
+        /// </summary>
+        private static readonly double[] erf_imp_cn = { -0.0397876892611136856954425, 0.153165212467878293257683, 0.191260295600936245503129, 0.10276327061989304213645, 0.029637090615738836726027, 0.0046093486780275489468812, 0.000307607820348680180548455 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [0.75, 1.25].
+        /// </summary>
+        private static readonly double[] erf_imp_cd = { 1, 1.95520072987627704987886, 1.64762317199384860109595, 0.768238607022126250082483, 0.209793185936509782784315, 0.0319569316899913392596356, 0.00213363160895785378615014 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [1.25, 2.25].
+        /// </summary>
+        private static readonly double[] erf_imp_dn = { -0.0300838560557949717328341, 0.0538578829844454508530552, 0.0726211541651914182692959, 0.0367628469888049348429018, 0.00964629015572527529605267, 0.00133453480075291076745275, 0.778087599782504251917881e-4 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [1.25, 2.25].
+        /// </summary>
+        private static readonly double[] erf_imp_dd = { 1, 1.75967098147167528287343, 1.32883571437961120556307, 0.552528596508757581287907, 0.133793056941332861912279, 0.0179509645176280768640766, 0.00104712440019937356634038, -0.106640381820357337177643e-7 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [2.25, 3.5].
+        /// </summary>
+        private static readonly double[] erf_imp_en = { -0.0117907570137227847827732, 0.014262132090538809896674, 0.0202234435902960820020765, 0.00930668299990432009042239, 0.00213357802422065994322516, 0.00025022987386460102395382, 0.120534912219588189822126e-4 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [2.25, 3.5].
+        /// </summary>
+        private static readonly double[] erf_imp_ed = { 1, 1.50376225203620482047419, 0.965397786204462896346934, 0.339265230476796681555511, 0.0689740649541569716897427, 0.00771060262491768307365526, 0.000371421101531069302990367 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [3.5, 5.25].
+        /// </summary>
+        private static readonly double[] erf_imp_fn = { -0.00546954795538729307482955, 0.00404190278731707110245394, 0.0054963369553161170521356, 0.00212616472603945399437862, 0.000394984014495083900689956, 0.365565477064442377259271e-4, 0.135485897109932323253786e-5 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [3.5, 5.25].
+        /// </summary>
+        private static readonly double[] erf_imp_fd = { 1, 1.21019697773630784832251, 0.620914668221143886601045, 0.173038430661142762569515, 0.0276550813773432047594539, 0.00240625974424309709745382, 0.891811817251336577241006e-4, -0.465528836283382684461025e-11 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [5.25, 8].
+        /// </summary>
+        private static readonly double[] erf_imp_gn = { -0.00270722535905778347999196, 0.0013187563425029400461378, 0.00119925933261002333923989, 0.00027849619811344664248235, 0.267822988218331849989363e-4, 0.923043672315028197865066e-6 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [5.25, 8].
+        /// </summary>
+        private static readonly double[] erf_imp_gd = { 1, 0.814632808543141591118279, 0.268901665856299542168425, 0.0449877216103041118694989, 0.00381759663320248459168994, 0.000131571897888596914350697, 0.404815359675764138445257e-11 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [8, 11.5].
+        /// </summary>
+        private static readonly double[] erf_imp_hn = { -0.00109946720691742196814323, 0.000406425442750422675169153, 0.000274499489416900707787024, 0.465293770646659383436343e-4, 0.320955425395767463401993e-5, 0.778286018145020892261936e-7 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [8, 11.5].
+        /// </summary>
+        private static readonly double[] erf_imp_hd = { 1, 0.588173710611846046373373, 0.139363331289409746077541, 0.0166329340417083678763028, 0.00100023921310234908642639, 0.24254837521587225125068e-4 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [11.5, 17].
+        /// </summary>
+        private static readonly double[] erf_imp_in = { -0.00056907993601094962855594, 0.000169498540373762264416984, 0.518472354581100890120501e-4, 0.382819312231928859704678e-5, 0.824989931281894431781794e-7 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [11.5, 17].
+        /// </summary>
+        private static readonly double[] erf_imp_id = { 1, 0.339637250051139347430323, 0.043472647870310663055044, 0.00248549335224637114641629, 0.535633305337152900549536e-4, -0.117490944405459578783846e-12 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [17, 24].
+        /// </summary>
+        private static readonly double[] erf_imp_jn = { -0.000241313599483991337479091, 0.574224975202501512365975e-4, 0.115998962927383778460557e-4, 0.581762134402593739370875e-6, 0.853971555085673614607418e-8 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [17, 24].
+        /// </summary>
+        private static readonly double[] erf_imp_jd = { 1, 0.233044138299687841018015, 0.0204186940546440312625597, 0.000797185647564398289151125, 0.117019281670172327758019e-4 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [24, 38].
+        /// </summary>
+        private static readonly double[] erf_imp_kn = { -0.000146674699277760365803642, 0.162666552112280519955647e-4, 0.269116248509165239294897e-5, 0.979584479468091935086972e-7, 0.101994647625723465722285e-8 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [24, 38].
+        /// </summary>
+        private static readonly double[] erf_imp_kd = { 1, 0.165907812944847226546036, 0.0103361716191505884359634, 0.000286593026373868366935721, 0.298401570840900340874568e-5 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [38, 60].
+        /// </summary>
+        private static readonly double[] erf_imp_ln = { -0.583905797629771786720406e-4, 0.412510325105496173512992e-5, 0.431790922420250949096906e-6, 0.993365155590013193345569e-8, 0.653480510020104699270084e-10 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [38, 60].
+        /// </summary>
+        private static readonly double[] erf_imp_ld = { 1, 0.105077086072039915406159, 0.00414278428675475620830226, 0.726338754644523769144108e-4, 0.477818471047398785369849e-6 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [60, 85].
+        /// </summary>
+        private static readonly double[] erf_imp_mn = { -0.196457797609229579459841e-4, 0.157243887666800692441195e-5, 0.543902511192700878690335e-7, 0.317472492369117710852685e-9 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [60, 85].
+        /// </summary>
+        private static readonly double[] erf_imp_md = { 1, 0.052803989240957632204885, 0.000926876069151753290378112, 0.541011723226630257077328e-5, 0.535093845803642394908747e-15 };
+
+        /// <summary> Polynomial coefficients for a numerator in ErfImp
+        /// calculation for Erfc(x) in the interval [85, 110].
+        /// </summary>
+        private static readonly double[] erf_imp_nn = { -0.789224703978722689089794e-5, 0.622088451660986955124162e-6, 0.145728445676882396797184e-7, 0.603715505542715364529243e-10 };
+
+        /// <summary> Polynomial coefficients for a denominator in ErfImp
+        /// calculation for Erfc(x) in the interval [85, 110].
+        /// </summary>
+        private static readonly double[] erf_imp_nd = { 1, 0.0375328846356293715248719, 0.000467919535974625308126054, 0.193847039275845656900547e-5 };
+
+        /// <summary>
+        /// **************************************
+        /// COEFFICIENTS FOR METHOD ErfInvImp    *
+        /// **************************************
+        /// </summary>
+        /// <summary> Polynomial coefficients for a numerator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0, 0.5].
+        /// </summary>
+        private static readonly double[] erv_inv_imp_an = { -0.000508781949658280665617, -0.00836874819741736770379, 0.0334806625409744615033, -0.0126926147662974029034, -0.0365637971411762664006, 0.0219878681111168899165, 0.00822687874676915743155, -0.00538772965071242932965 };
+
+        /// <summary> Polynomial coefficients for a denominator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0, 0.5].
+        /// </summary>
+        private static readonly double[] erv_inv_imp_ad = { 1, -0.970005043303290640362, -1.56574558234175846809, 1.56221558398423026363, 0.662328840472002992063, -0.71228902341542847553, -0.0527396382340099713954, 0.0795283687341571680018, -0.00233393759374190016776, 0.000886216390456424707504 };
+
+        /// <summary> Polynomial coefficients for a numerator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.5, 0.75].
+        /// </summary>
+        private static readonly double[] erv_inv_imp_bn = { -0.202433508355938759655, 0.105264680699391713268, 8.37050328343119927838, 17.6447298408374015486, -18.8510648058714251895, -44.6382324441786960818, 17.445385985570866523, 21.1294655448340526258, -3.67192254707729348546 };
+
+        /// <summary> Polynomial coefficients for a denominator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.5, 0.75].
+        /// </summary>
+        private static readonly double[] erv_inv_imp_bd = { 1, 6.24264124854247537712, 3.9713437953343869095, -28.6608180499800029974, -20.1432634680485188801, 48.5609213108739935468, 10.8268667355460159008, -22.6436933413139721736, 1.72114765761200282724 };
+
+        /// <summary> Polynomial coefficients for a numerator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x less than 3.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_cn = { -0.131102781679951906451, -0.163794047193317060787, 0.117030156341995252019, 0.387079738972604337464, 0.337785538912035898924, 0.142869534408157156766, 0.0290157910005329060432, 0.00214558995388805277169, -0.679465575181126350155e-6, 0.285225331782217055858e-7, -0.681149956853776992068e-9 };
+
+        /// <summary> Polynomial coefficients for a denominator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x less than 3.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_cd = { 1, 3.46625407242567245975, 5.38168345707006855425, 4.77846592945843778382, 2.59301921623620271374, 0.848854343457902036425, 0.152264338295331783612, 0.01105924229346489121 };
+
+        /// <summary> Polynomial coefficients for a numerator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x between 3 and 6.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_dn = { -0.0350353787183177984712, -0.00222426529213447927281, 0.0185573306514231072324, 0.00950804701325919603619, 0.00187123492819559223345, 0.000157544617424960554631, 0.460469890584317994083e-5, -0.230404776911882601748e-9, 0.266339227425782031962e-11 };
+
+        /// <summary> Polynomial coefficients for a denominator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x between 3 and 6.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_dd = { 1, 1.3653349817554063097, 0.762059164553623404043, 0.220091105764131249824, 0.0341589143670947727934, 0.00263861676657015992959, 0.764675292302794483503e-4 };
+
+        /// <summary> Polynomial coefficients for a numerator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x between 6 and 18.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_en = { -0.0167431005076633737133, -0.00112951438745580278863, 0.00105628862152492910091, 0.000209386317487588078668, 0.149624783758342370182e-4, 0.449696789927706453732e-6, 0.462596163522878599135e-8, -0.281128735628831791805e-13, 0.99055709973310326855e-16 };
+
+        /// <summary> Polynomial coefficients for a denominator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x between 6 and 18.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_ed = { 1, 0.591429344886417493481, 0.138151865749083321638, 0.0160746087093676504695, 0.000964011807005165528527, 0.275335474764726041141e-4, 0.282243172016108031869e-6 };
+
+        /// <summary> Polynomial coefficients for a numerator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x between 18 and 44.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_fn = { -0.0024978212791898131227, -0.779190719229053954292e-5, 0.254723037413027451751e-4, 0.162397777342510920873e-5, 0.396341011304801168516e-7, 0.411632831190944208473e-9, 0.145596286718675035587e-11, -0.116765012397184275695e-17 };
+
+        /// <summary> Polynomial coefficients for a denominator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x between 18 and 44.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_fd = { 1, 0.207123112214422517181, 0.0169410838120975906478, 0.000690538265622684595676, 0.145007359818232637924e-4, 0.144437756628144157666e-6, 0.509761276599778486139e-9 };
+
+        /// <summary> Polynomial coefficients for a numerator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x greater than 44.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_gn = { -0.000539042911019078575891, -0.28398759004727721098e-6, 0.899465114892291446442e-6, 0.229345859265920864296e-7, 0.225561444863500149219e-9, 0.947846627503022684216e-12, 0.135880130108924861008e-14, -0.348890393399948882918e-21 };
+
+        /// <summary> Polynomial coefficients for a denominator of ErfInvImp
+        /// calculation for Erf^-1(z) in the interval [0.75, 1] with x greater than 44.
+        /// </summary>
+        private static readonly double[] erv_inv_imp_gd = { 1, 0.0845746234001899436914, 0.00282092984726264681981, 0.468292921940894236786e-4, 0.399968812193862100054e-6, 0.161809290887904476097e-8, 0.231558608310259605225e-11 };
+
+        /// <summary>Calculates the error function.</summary>
+        /// <param name="x">The value to evaluate.</param>
+        /// <returns>the error function evaluated at given value.</returns>
+        /// <remarks>
+        ///     <list type="bullet">
+        ///         <item>returns 1 if <c>x == double.PositiveInfinity</c>.</item>
+        ///         <item>returns -1 if <c>x == double.NegativeInfinity</c>.</item>
+        ///     </list>
+        /// </remarks>
+        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;
+            }
+
+            return erfImp(x, false);
+        }
+
+        /// <summary>Calculates the complementary error function.</summary>
+        /// <param name="x">The value to evaluate.</param>
+        /// <returns>the complementary error function evaluated at given value.</returns>
+        /// <remarks>
+        ///     <list type="bullet">
+        ///         <item>returns 0 if <c>x == double.PositiveInfinity</c>.</item>
+        ///         <item>returns 2 if <c>x == double.NegativeInfinity</c>.</item>
+        ///     </list>
+        /// </remarks>
+        public static double Erfc(double x)
+        {
+            if (x == 0)
+            {
+                return 1;
+            }
+
+            if (double.IsPositiveInfinity(x))
+            {
+                return 0;
+            }
+
+            if (double.IsNegativeInfinity(x))
+            {
+                return 2;
+            }
+
+            if (double.IsNaN(x))
+            {
+                return double.NaN;
+            }
+
+            return erfImp(x, true);
+        }
+
+        /// <summary>Calculates the inverse error function evaluated at z.</summary>
+        /// <returns>The inverse error function evaluated at given value.</returns>
+        /// <remarks>
+        ///   <list type="bullet">
+        ///     <item>returns double.PositiveInfinity if <c>z &gt;= 1.0</c>.</item>
+        ///     <item>returns double.NegativeInfinity if <c>z &lt;= -1.0</c>.</item>
+        ///   </list>
+        /// </remarks>
+        /// <summary>Calculates the inverse error function evaluated at z.</summary>
+        /// <param name="z">value to evaluate.</param>
+        /// <returns>the inverse error function evaluated at Z.</returns>
+        public static double ErfInv(double z)
+        {
+            if (z == 0.0)
+            {
+                return 0.0;
+            }
+
+            if (z >= 1.0)
+            {
+                return double.PositiveInfinity;
+            }
+
+            if (z <= -1.0)
+            {
+                return double.NegativeInfinity;
+            }
+
+            double p, q, s;
+
+            if (z < 0)
+            {
+                p = -z;
+                q = 1 - p;
+                s = -1;
+            }
+            else
+            {
+                p = z;
+                q = 1 - z;
+                s = 1;
+            }
+
+            return erfInvImpl(p, q, s);
+        }
+
+        /// <summary>
+        /// Implementation of the error function.
+        /// </summary>
+        /// <param name="z">Where to evaluate the error function.</param>
+        /// <param name="invert">Whether to compute 1 - the error function.</param>
+        /// <returns>the error function.</returns>
+        private static double erfImp(double z, bool invert)
+        {
+            if (z < 0)
+            {
+                if (!invert)
+                {
+                    return -erfImp(-z, false);
+                }
+
+                if (z < -0.5)
+                {
+                    return 2 - erfImp(-z, true);
+                }
+
+                return 1 + erfImp(-z, false);
+            }
+
+            double result;
+
+            // Big bunch of selection statements now to pick which
+            // implementation to use, try to put most likely options
+            // first:
+            if (z < 0.5)
+            {
+                // We're going to calculate erf:
+                if (z < 1e-10)
+                {
+                    result = (z * 1.125) + (z * 0.003379167095512573896158903121545171688);
+                }
+                else
+                {
+                    // Worst case absolute error found: 6.688618532e-21
+                    result = (z * 1.125) + (z * evaluatePolynomial(z, erf_imp_an) / evaluatePolynomial(z, erf_imp_ad));
+                }
+            }
+            else if (z < 110)
+            {
+                // We'll be calculating erfc:
+                invert = !invert;
+                double r, b;
+
+                if (z < 0.75)
+                {
+                    // Worst case absolute error found: 5.582813374e-21
+                    r = evaluatePolynomial(z - 0.5, erf_imp_bn) / evaluatePolynomial(z - 0.5, erf_imp_bd);
+                    b = 0.3440242112F;
+                }
+                else if (z < 1.25)
+                {
+                    // Worst case absolute error found: 4.01854729e-21
+                    r = evaluatePolynomial(z - 0.75, erf_imp_cn) / evaluatePolynomial(z - 0.75, erf_imp_cd);
+                    b = 0.419990927F;
+                }
+                else if (z < 2.25)
+                {
+                    // Worst case absolute error found: 2.866005373e-21
+                    r = evaluatePolynomial(z - 1.25, erf_imp_dn) / evaluatePolynomial(z - 1.25, erf_imp_dd);
+                    b = 0.4898625016F;
+                }
+                else if (z < 3.5)
+                {
+                    // Worst case absolute error found: 1.045355789e-21
+                    r = evaluatePolynomial(z - 2.25, erf_imp_en) / evaluatePolynomial(z - 2.25, erf_imp_ed);
+                    b = 0.5317370892F;
+                }
+                else if (z < 5.25)
+                {
+                    // Worst case absolute error found: 8.300028706e-22
+                    r = evaluatePolynomial(z - 3.5, erf_imp_fn) / evaluatePolynomial(z - 3.5, erf_imp_fd);
+                    b = 0.5489973426F;
+                }
+                else if (z < 8)
+                {
+                    // Worst case absolute error found: 1.700157534e-21
+                    r = evaluatePolynomial(z - 5.25, erf_imp_gn) / evaluatePolynomial(z - 5.25, erf_imp_gd);
+                    b = 0.5571740866F;
+                }
+                else if (z < 11.5)
+                {
+                    // Worst case absolute error found: 3.002278011e-22
+                    r = evaluatePolynomial(z - 8, erf_imp_hn) / evaluatePolynomial(z - 8, erf_imp_hd);
+                    b = 0.5609807968F;
+                }
+                else if (z < 17)
+                {
+                    // Worst case absolute error found: 6.741114695e-21
+                    r = evaluatePolynomial(z - 11.5, erf_imp_in) / evaluatePolynomial(z - 11.5, erf_imp_id);
+                    b = 0.5626493692F;
+                }
+                else if (z < 24)
+                {
+                    // Worst case absolute error found: 7.802346984e-22
+                    r = evaluatePolynomial(z - 17, erf_imp_jn) / evaluatePolynomial(z - 17, erf_imp_jd);
+                    b = 0.5634598136F;
+                }
+                else if (z < 38)
+                {
+                    // Worst case absolute error found: 2.414228989e-22
+                    r = evaluatePolynomial(z - 24, erf_imp_kn) / evaluatePolynomial(z - 24, erf_imp_kd);
+                    b = 0.5638477802F;
+                }
+                else if (z < 60)
+                {
+                    // Worst case absolute error found: 5.896543869e-24
+                    r = evaluatePolynomial(z - 38, erf_imp_ln) / evaluatePolynomial(z - 38, erf_imp_ld);
+                    b = 0.5640528202F;
+                }
+                else if (z < 85)
+                {
+                    // Worst case absolute error found: 3.080612264e-21
+                    r = evaluatePolynomial(z - 60, erf_imp_mn) / evaluatePolynomial(z - 60, erf_imp_md);
+                    b = 0.5641309023F;
+                }
+                else
+                {
+                    // Worst case absolute error found: 8.094633491e-22
+                    r = evaluatePolynomial(z - 85, erf_imp_nn) / evaluatePolynomial(z - 85, erf_imp_nd);
+                    b = 0.5641584396F;
+                }
+
+                double g = Math.Exp(-z * z) / z;
+                result = (g * b) + (g * r);
+            }
+            else
+            {
+                // Any value of z larger than 28 will underflow to zero:
+                result = 0;
+                invert = !invert;
+            }
+
+            if (invert)
+            {
+                result = 1 - result;
+            }
+
+            return result;
+        }
+
+        /// <summary>Calculates the complementary inverse error function evaluated at z.</summary>
+        /// <returns>The complementary inverse error function evaluated at given value.</returns>
+        /// <remarks> We have tested this implementation against the arbitrary precision mpmath library
+        /// and found cases where we can only guarantee 9 significant figures correct.
+        ///     <list type="bullet">
+        ///         <item>returns double.PositiveInfinity if <c>z &lt;= 0.0</c>.</item>
+        ///         <item>returns double.NegativeInfinity if <c>z &gt;= 2.0</c>.</item>
+        ///     </list>
+        /// </remarks>
+        /// <summary>calculates the complementary inverse error function evaluated at z.</summary>
+        /// <param name="z">value to evaluate.</param>
+        /// <returns>the complementary inverse error function evaluated at Z.</returns>
+        public static double ErfcInv(double z)
+        {
+            if (z <= 0.0)
+            {
+                return double.PositiveInfinity;
+            }
+
+            if (z >= 2.0)
+            {
+                return double.NegativeInfinity;
+            }
+
+            double p, q, s;
+
+            if (z > 1)
+            {
+                q = 2 - z;
+                p = 1 - q;
+                s = -1;
+            }
+            else
+            {
+                p = 1 - z;
+                q = z;
+                s = 1;
+            }
+
+            return erfInvImpl(p, q, s);
+        }
+
+        /// <summary>
+        /// The implementation of the inverse error function.
+        /// </summary>
+        /// <param name="p">First intermediate parameter.</param>
+        /// <param name="q">Second intermediate parameter.</param>
+        /// <param name="s">Third intermediate parameter.</param>
+        /// <returns>the inverse error function.</returns>
+        private static double erfInvImpl(double p, double q, double s)
+        {
+            double result;
+
+            if (p <= 0.5)
+            {
+                // Evaluate inverse erf using the rational approximation:
+                //
+                // x = p(p+10)(Y+R(p))
+                //
+                // Where Y is a constant, and R(p) is optimized for a low
+                // absolute error compared to |Y|.
+                //
+                // double: Max error found: 2.001849e-18
+                // long double: Max error found: 1.017064e-20
+                // Maximum Deviation Found (actual error term at infinite precision) 8.030e-21
+                const float y = 0.0891314744949340820313f;
+                double g = p * (p + 10);
+                double r = evaluatePolynomial(p, erv_inv_imp_an) / evaluatePolynomial(p, erv_inv_imp_ad);
+                result = (g * y) + (g * r);
+            }
+            else if (q >= 0.25)
+            {
+                // Rational approximation for 0.5 > q >= 0.25
+                //
+                // x = sqrt(-2*log(q)) / (Y + R(q))
+                //
+                // Where Y is a constant, and R(q) is optimized for a low
+                // absolute error compared to Y.
+                //
+                // double : Max error found: 7.403372e-17
+                // long double : Max error found: 6.084616e-20
+                // Maximum Deviation Found (error term) 4.811e-20
+                const float y = 2.249481201171875f;
+                double g = Math.Sqrt(-2 * Math.Log(q));
+                double xs = q - 0.25;
+                double r = evaluatePolynomial(xs, erv_inv_imp_bn) / evaluatePolynomial(xs, erv_inv_imp_bd);
+                result = g / (y + r);
+            }
+            else
+            {
+                // For q < 0.25 we have a series of rational approximations all
+                // of the general form:
+                //
+                // let: x = sqrt(-log(q))
+                //
+                // Then the result is given by:
+                //
+                // x(Y+R(x-B))
+                //
+                // where Y is a constant, B is the lowest value of x for which
+                // the approximation is valid, and R(x-B) is optimized for a low
+                // absolute error compared to Y.
+                //
+                // Note that almost all code will really go through the first
+                // or maybe second approximation.  After than we're dealing with very
+                // small input values indeed: 80 and 128 bit long double's go all the
+                // way down to ~ 1e-5000 so the "tail" is rather long...
+                double x = Math.Sqrt(-Math.Log(q));
+
+                if (x < 3)
+                {
+                    // Max error found: 1.089051e-20
+                    const float y = 0.807220458984375f;
+                    double xs = x - 1.125;
+                    double r = evaluatePolynomial(xs, erv_inv_imp_cn) / evaluatePolynomial(xs, erv_inv_imp_cd);
+                    result = (y * x) + (r * x);
+                }
+                else if (x < 6)
+                {
+                    // Max error found: 8.389174e-21
+                    const float y = 0.93995571136474609375f;
+                    double xs = x - 3;
+                    double r = evaluatePolynomial(xs, erv_inv_imp_dn) / evaluatePolynomial(xs, erv_inv_imp_dd);
+                    result = (y * x) + (r * x);
+                }
+                else if (x < 18)
+                {
+                    // Max error found: 1.481312e-19
+                    const float y = 0.98362827301025390625f;
+                    double xs = x - 6;
+                    double r = evaluatePolynomial(xs, erv_inv_imp_en) / evaluatePolynomial(xs, erv_inv_imp_ed);
+                    result = (y * x) + (r * x);
+                }
+                else if (x < 44)
+                {
+                    // Max error found: 5.697761e-20
+                    const float y = 0.99714565277099609375f;
+                    double xs = x - 18;
+                    double r = evaluatePolynomial(xs, erv_inv_imp_fn) / evaluatePolynomial(xs, erv_inv_imp_fd);
+                    result = (y * x) + (r * x);
+                }
+                else
+                {
+                    // Max error found: 1.279746e-20
+                    const float y = 0.99941349029541015625f;
+                    double xs = x - 44;
+                    double r = evaluatePolynomial(xs, erv_inv_imp_gn) / evaluatePolynomial(xs, erv_inv_imp_gd);
+                    result = (y * x) + (r * x);
+                }
+            }
+
+            return s * result;
+        }
+
+        /// <summary>
+        /// Evaluate a polynomial at point x.
+        /// Coefficients are ordered ascending by power with power k at index k.
+        /// Example: coefficients [3,-1,2] represent y=2x^2-x+3.
+        /// </summary>
+        /// <param name="z">The location where to evaluate the polynomial at.</param>
+        /// <param name="coefficients">The coefficients of the polynomial, coefficient for power k at index k.</param>
+        /// <exception cref="ArgumentNullException">
+        /// <paramref name="coefficients"/> is a null reference.
+        /// </exception>
+        private static double evaluatePolynomial(double z, params double[] coefficients)
+        {
+            // 2020-10-07 jbialogrodzki #730 Since this is public API we should probably
+            // handle null arguments? It doesn't seem to have been done consistently in this class though.
+            if (coefficients == null)
+            {
+                throw new ArgumentNullException(nameof(coefficients));
+            }
+
+            // 2020-10-07 jbialogrodzki #730 Zero polynomials need explicit handling.
+            // Without this check, we attempted to peek coefficients at negative indices!
+            int n = coefficients.Length;
+
+            if (n == 0)
+            {
+                return 0;
+            }
+
+            double sum = coefficients[n - 1];
+
+            for (int i = n - 2; i >= 0; --i)
+            {
+                sum *= z;
+                sum += coefficients[i];
+            }
+
+            return sum;
+        }
+    }
+}

osu.Game/osu.Game.csproj

@@ -36,7 +36,7 @@
     </PackageReference>
     <PackageReference Include="Realm" Version="11.5.0" />
     <PackageReference Include="ppy.osu.Framework" Version="2024.1007.0" />
-    <PackageReference Include="ppy.osu.Game.Resources" Version="2024.904.0" />
+    <PackageReference Include="ppy.osu.Game.Resources" Version="2024.1003.0" />
     <PackageReference Include="Sentry" Version="4.3.0" />
     <!-- Held back due to 0.34.0 failing AOT compilation on ZstdSharp.dll dependency. -->
     <PackageReference Include="SharpCompress" Version="0.36.0" />