zenno/doc/frbm: precise example for uphill power diff

zenno/src/routes/doc/frbm/+page.svelte

@@ -5,6 +5,7 @@
 		AptitudeLevel,
 		downhillAccelEnterChance,
 		frontModeEnterChance,
+		HORSE_LENGTH,
 		moveLaneModifier,
 		paceUpEnterChance,
 		Phase,
@@ -31,6 +32,20 @@
 		return x * (1 - p) + x * bonus * p;
 	}
 
+	const uphillSlowPow = 800;
+	const uphillFastPow = 1200;
+	const uphillBaseSpeed = mean2(sectionSpeed(2500, 1200, 1200, RunningStyle.FrontRunner, Phase.MidRace));
+	const uphillSlopes = [
+		{ per: 2, len: 110 },
+		{ per: 1.5, len: 200 },
+	]
+		.map((s) => ({ ...s, slowSpeed: uphillMod(uphillSlowPow, s.per), fastSpeed: uphillMod(uphillFastPow, s.per) }))
+		.map((s) => ({ ...s, slowTime: s.len / (uphillBaseSpeed + s.slowSpeed), fastTime: s.len / (uphillBaseSpeed + s.fastSpeed) }))
+		// TODO(zeph): should include the difference with decel down to uphill speed and accel out of it,
+		// since the high power horse is favored for both
+		.map((s) => ({ ...s, dx: (s.fastSpeed - s.slowSpeed) * s.fastTime + s.slowSpeed * (s.slowTime - s.fastTime) }));
+	const uphillLengths = uphillSlopes.reduce((m, { dx }) => m + dx, 0) / HORSE_LENGTH;
+
 	const secSpeedS = 1200;
 	const secSpeedA = 1000;
 	const secSpeedExample = $derived(sectionSpeed(raceLen, secSpeedS, secSpeedS * 1.1, RunningStyle.FrontRunner, Phase.EarlyRace));
@@ -425,14 +440,15 @@
 	<Sec h="3" id="uphills">Uphills</Sec>
 	<p>
 		Running uphill carries a penalty to target speed. This penalty scales negatively with the power stat; that is, higher power
-		means faster uphill running. It scales positively with slope angle.
+		means faster uphill running. It scales positively with slope angle. There is also a flat reduction in base acceleration for
+		running uphill, which does not change with stats nor slope angle.
 	</p>
 	<div class="mb-4 h-60 w-full md:h-96">
 		<StatChart
 			class="mx-auto h-full w-full max-w-3xl"
 			stat={Stat.Power}
 			y={uphillSeries}
-			yLabel="Speed Modifier (m/s)"
+			yLabel="Uphill Speed Modifier (m/s)"
 			yRule={uphillYRule}
 			range={[-2, 0]}
 		/>
@@ -441,12 +457,9 @@
 		Note that surface aptitude <i>does not</i> affect uphill speed, nor power generally. It only affects acceleration.
 	</p>
 	<p>
-		The practical impact is that steep early- and mid-race hills filter out front runners with low power. Even with an otherwise
-		perfect build, an 800 power VBourbon is likely to be passed by a 1280 power (<Skill skill={200152} hint="firm" mention /> + <Skill
-			skill={200282}
-			hint="comp spirit"
-			mention
-		/>) Seiun Sky.
+		The practical impact is that steep early- and mid-race hills filter out front runners with low power. All else equal, a Seiun
+		Sky with {uphillFastPow} power gains an average of about {uphillLengths.toFixed(1)} lengths over a VBourbon with {uphillSlowPow}
+		power on the Arima Kinen mid-race hills.
 	</p>
 
 	<Sec h="3" id="downhills">Downhills</Sec>