# rnl

Not to be confused with rn1.

rnl(x)[1] is a pseudo-random number function used in NetHack when a result should be weighted by Luck.

rnl(x) has the same range as rn2(x) (that is, 0 ≤ rnl(x) < x), but is weighted toward 0 if you have positive Luck, and toward x − 1 if you have negative Luck.

## Formula

See source code in [1]

For rnl(x), let L equal your Luck, including the luck item bonus or penalty, whether positive, negative, or zero.

If x ≤ 15, L = L3, rounded toward the nearest integer.

If L = 0, return rn2(x).

With chance 137 + |L|, return rn2(x).

Otherwise, return rn2(x) - L, but no less than 0 and no more than x - 1.

This means that rnl(x) is almost always equivalent to rn2(x) - Luck when x > 15. The chance of Luck being ignored is less than 3% in any case, and even smaller for stronger (positive or negative) Luck. For smaller values of x, rn2(x) - Luck3 is the most likely approximation.

## How it's used

In the tables below, some very high (99.8%) and very low (0.2%) values above are rounded to 100% or 0%. Nothing is completely 100% or 0% likely, except where spelled out as "certain" or "impossible".

### Events based solely on Luck

Situation Code Luck
−11 −8 −5 −2 0 +2 +5 +8 +11
Chance of your god helping you if you pray with exactly 0 alignment `rnl(2)==0`[2] 1% 1% 1% 1% 50% 99% 99% 99% 99%
Chance of writing a scroll you have not seen `rnl(15)==0`[3] 0.2% 0.2% 0.2% 0.2% 6.7% 13.2% 19.7% 26.2% 32.7%
Chance of writing a scroll you have not seen, as a Wizard `rnl(5)==0`[4] 0.5% 0.5% 0.5% 0.5% 20.0% 39.5% 59.0% 78.5% 98.0%
Chance of failing to set a land mine (in which case it blows up) or beartrap (nothing happens) while riding and unskilled at riding `rnl(10)>5`[5] 79% 69% 59% 50% 40% 30% 21% 11% 1%
Chance of failing to set a land mine (in which case it blows up) or beartrap (nothing happens) while riding and unskilled at riding, and fumbling or using a cursed trap `rnl(10)>3`[6] 99% 89% 79% 70% 60% 50% 41% 31% 21%
Chance that a trap goes off after you set it, if it was cursed or you were fumbling `rnl(10)>5`[7] 79% 69% 59% 50% 40% 30% 21% 11% 1%
Chance of snagging an item you can pick up when using a bullwhip (instead of having it slip free) `rnl(6)==0`[8] 0.4% 0.4% 0.4% 0.4% 16.7% 32.9% 49.1% 65.4% 81.8%
Chance of finding secret passages and doors on each surrounding square when searching `rnl(7)==0`[9] 0.3% 0.4% 0.4% 0.4% 14.3% 28.2% 42.1% 56.1% 70.0%
Chance of finding secret passages and doors on each surrounding square when searching, while wearing a pair of lenses `rnl(5)==0`[10] 0.5% 0.5% 0.5% 0.5% 20.0% 39.5% 59.0% 78.5% 98.0%
Chance of finding secret passages and doors on each surrounding square when searching, while wielding a +5 Excalibur, or +3 Excalibur and lenses, or any better combination `rnl(2)==0`[11] 1.2% 1.3% 1.3% 1.3% 50% 98.7% 98.7% 98.8% 98.8%
Chance of finding a hidden trap when searching `rnl(8)==0`[12] 0.3% 0.3% 0.3% 0.3% 12.5% 24.7% 36.9% 49.1% 61.3%
Chance of a blessed projectile not breaking when thrown `rnl(4)==0`[13] 0.6% 0.6% 0.6% 0.7% 25% 49.3% 73.7% 98.1% 98.2%
Chance of being petrified if your saddle slips off[14] or is destroyed, or you are polymorphed and lose stone resistance,[15] when you are riding a steed that petrifies on touch (likely due to your horse having just polymorphed into a cockatrice) `rnl(3)>0` 99.2% 99.2% 99.1% 99.1% 66.7% 34.2% 1.7% 1.7% 1.6%
Chance of a random levelport when attempting to controlled levelport (or cancel) while confused `rnl(5)>0`[16] 99.5% 99.5% 99.5% 99.5% 80.0% 60.5% 41.0% 21.5% 2.0%
Chance of blessed equipment not being eroded by rust[17] or water[18] damage, or blessed flammable items not being burned by fire damage[19] `rnl(4)==0` 0.6% 0.6% 0.6% 0.7% 25% 49.3% 73.7% 98.1% 98.2%
Chance of a monster becoming angry when it steps in a trap set by you,[20] or of you triggering a trap when failing to disarm it[21] `rnl(5)>0` 99.5% 99.5% 99.5% 99.5% 80.0% 60.5% 41.0% 21.5% 2.0%
Chance of a hostile, awake, unfrozen, mindful, non-​@ monster becoming peaceful when you free it from a trap not set by you `rnl(10)<8`[22] 40.1% 50.8% 60.5% 70.3% 80.0% 89.7% 99.5% 99.5% 99.5%
Chance of a lawful player getting +1 alignment for the above `rnl(8)==0`, one-third of the time[23] 0.1% 0.1% 0.1% 0.1% 4.2% 8.2% 12.3% 16.4% 20.4%
Chance of getting +1 alignment for trying and failing to lift a hostile, mobile, mindful, nonhuman monster out of a pit not dug by you `rnl(10)<3`[24] 0.7% 0.8% 10.5% 20.2% 30% 39.7% 49.5% 59.3% 69.0%
Chance of having a boomerang break when used as a melee weapon `rnl(4)==3`[25] 98.2% 98.1% 73.7% 49.3% 25% 0.7% 0.6% 0.6% 0.6%
Chance of having a non-artifact lance break when successfully jousting a solid monster `rnl(50)==49`, 15 of the time[26] 4.71% 3.53% 2.35% 1.18% 0.4% 0.01% 0.01% 0.01% 0.01%

### Events based on Luck and other stats

Situation Code Luck
−11 −8 −5 −2 0 +2 +5 +8 +11
Chance of kicking down a door `rnl(35)<X`, where X = avg(str, dex, con) + (dex if Samurai or Monk)[27] Approx Luck + X35 chance
Chance of kicking down a door when avg(str, dex, con) is 10 `rnl(35)<10` 0.5% 6.2% 14.6% 23.0% 28.6% 34.1% 42.5% 50.9% 59.3%
Chance of kicking down a door when avg(str, dex, con) is 15 `rnl(35)<15` 12.1% 20.5% 28.9% 37.3% 42.9% 48.4% 56.8% 65.2% 73.6%
Chance of kicking down a door as a Samurai or Monk when avg(str, con) is 15 and dex is 15 `rnl(35)<30` 54.9% 63.4% 71.8% 80.1% 85.7% 91.3% 99.7% 99.7% 99.7%
Chance of opening a door `rnl(20)<X`, where X = avg(str, dex, con)[28] Approx Luck + X20 chance[29]
Chance of opening a door when avg(str, dex, con) is 10 `rnl(20)<10` 1% 11% 26% 40% 50% 60% 74% 89% 99%
Chance of opening a door when avg(str, dex, con) is 15 `rnl(20)<15` 21% 36% 51% 65% 75% 85% 99% 99% 99%
One criterion (the other being alignment-based) for a minion to be summoned when converting[30] (or attempting to convert[31]) an altar `rnl(your level)>6` Impossible before level 8, more likely with low luck or higher level
Chance of summoning a minion, all other criteria being met, at level 8 `rnl(8)>6` 61.3% 49.1% 36.9% 24.7% 12.5% 0.3% 0.3% 0.3% 0.3%
Chance of summoning a minion, all other criteria being met, at level 12 `rnl(12)>6` 74.2% 66.0% 58.0% 49.8% 41.7% 33.6% 25.4% 17.3% 9.1%
Chance of summoning a minion, all other criteria being met, at level 16 `rnl(16)>6` 99.1% 99.0% 86.8% 68.4% 56.3% 44.1% 25.7% 7.4% 1.2%
Chance of summoning a minion, all other criteria being met, at level 24 `rnl(24)>6` 99.4% 99.4% 91.2% 79.0% 70.8% 62.7% 50.5% 38.2% 26.0%
Chance of angering your god by praying in Gehennom `rnl(alignment)>0`[32] Certain with negative alignment, impossible at alignment 1, more likely with higher alignment, less likely with higher luck
Chance of angering your god by praying in Gehennom at alignment 2 `rnl(2)>0` 98.8% 98.8% 98.7% 98.7% 50% 1.3% 1.3% 1.3% 1.2%
Chance of angering your god by praying in Gehennom at alignment 4 `rnl(4)>0` 99.4% 99.4% 99.3% 99.3% 75% 50.7% 26.3% 1.9% 1.8%
Chance of angering your god by praying in Gehennom at alignment 8 `rnl(8)>0` 99.7% 99.7% 99.7% 99.7% 87.5% 75.3% 63.1% 50.9% 38.7%
Chance of angering your god by praying in Gehennom at alignment 16 `rnl(16)>0` 99.9% 99.9% 99.9% 99.9% 93.8% 81.6% 63.2% 44.9% 26.4%
Chance of angering your god by praying in Gehennom at alignment 32 `rnl(32)>0` 99.9% 99.9% 99.9% 99.9% 96.9% 90.8% 81.6% 72.4% 63.2%

### Numerical distributions

Value Code Luck
−11 −8 −5 −2 0 +2 +5 +8 +11
Average number of fruit that fall from a tree when you kick it `8-rnl(7)`[33] 2.5 2.9 3.5 4.2 5 5.8 6.5 7.1 7.5
Average number of bees that are summoned from a tree if you kick it `rnl(4)+2`[34] 5 5 4.7 4.2 3.5 2.8 2.3 2 2
Average number of gold pieces dropped by a gold golem on death `200-rnl(101)`[35] 140 143 145 148 150 152 155 157 160
Average number of darts or arrows left behind when disarming a dart trap or arrow trap `50-rnl(50)`[36] 16 18 21 24 25.5 27 30 33 35

## Mathematical analysis

Basically, rnl(x) generates a number between 0 and x − 1, and adjusts it in proportion to the player's Luck, with a small chance (which is smaller when Luck tends toward the extremes) of leaving it alone. The net effect is that with positive or negative Luck, the highest or lowest numbers on the list become vanishingly unlikely, and the odds of getting the best and worst result increase, while the odds of getting every result in between stays the same.

Further complicating the matter is that, when x is 15 or less, the adjustment is divided by 3, which means that for those calls there is a stepwise distribution where the odds of everything are almost completely the same within blocks of 3 numbers (−13 to −11, −10 to −8, −7 to −5, −4 to −2, −1 to 1, 2 to 4, 5 to 7, 8 to 10, 11 to 13). Larger parameters have smoother distributions.

In addition, the function often being called in unusual manners, such as comparing it to a non-zero constant, which results in a "slice" being taken from the distribution graph at a less easily explainable point. This is evidenced in the distrubtions for rnl(A)<B or rnl(A)>B above.

Since many of the comparisons are made against 0, having even −2 Luck can cause many results to swing against the player's favor with near certainty. −1 Luck is not as bad as most of these calls also have a small parameter, although it does negatively affect many things that do not call rnl (like prayer or wishing). On the flip side, positive Luck's effect tends to be much less dramatic, and +2 Luck still has a lot of room for improvement. +1 Luck changes almost nothing significantly; the only pronounced effect it has is to cause jousting to be much safer.

## Sample distribution

This table is the computed odds of rnl(10) at each possible Luck for each possible result. The "0%" values are actually around 0.24–0.26% depending on the magnitude of the Luck.

Luck Simplified results
0 1 2 3 4 5 6 7 8 9
−13 0% 0% 0% 0% 10% 10% 10% 10% 10% 49%
−12 0% 0% 0% 0% 10% 10% 10% 10% 10% 49%
−11 0% 0% 0% 0% 10% 10% 10% 10% 10% 49%
−10 0% 0% 0% 10% 10% 10% 10% 10% 10% 39%
−9 0% 0% 0% 10% 10% 10% 10% 10% 10% 39%
−8 0% 0% 0% 10% 10% 10% 10% 10% 10% 39%
−7 0% 0% 10% 10% 10% 10% 10% 10% 10% 29%
−6 0% 0% 10% 10% 10% 10% 10% 10% 10% 29%
−5 0% 0% 10% 10% 10% 10% 10% 10% 10% 29%
−4 0% 10% 10% 10% 10% 10% 10% 10% 10% 20%
−3 0% 10% 10% 10% 10% 10% 10% 10% 10% 20%
−2 0% 10% 10% 10% 10% 10% 10% 10% 10% 20%
−1 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%
0 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%
1 10% 10% 10% 10% 10% 10% 10% 10% 10% 10%
2 20% 10% 10% 10% 10% 10% 10% 10% 10% 0%
3 20% 10% 10% 10% 10% 10% 10% 10% 10% 0%
4 20% 10% 10% 10% 10% 10% 10% 10% 0% 0%
5 29% 10% 10% 10% 10% 10% 10% 10% 0% 0%
6 29% 10% 10% 10% 10% 10% 10% 10% 0% 0%
7 29% 10% 10% 10% 10% 10% 10% 10% 0% 0%
8 39% 10% 10% 10% 10% 10% 10% 0% 0% 0%
9 39% 10% 10% 10% 10% 10% 10% 0% 0% 0%
10 39% 10% 10% 10% 10% 10% 10% 0% 0% 0%
11 49% 10% 10% 10% 10% 10% 0% 0% 0% 0%
12 49% 10% 10% 10% 10% 10% 0% 0% 0% 0%
13 49% 10% 10% 10% 10% 10% 0% 0% 0% 0%

## History

In NetHack 3.4.3, the formulas were slightly different.[37]

Let R be a random integer number from 0 to x − 1 (inclusive).

If Luck = 0, R is returned.

Otherwise, with 150 − Luck chance (since Luck ranges from −13 to 13, the chance should be between 137 and 163), R is returned.

Otherwise, if x ≤  15 and Luck ≥ −5 (that is, x is not too big and the player is not too unlucky), Luck3 (truncated towards zero) is subtracted from R, then R is forced into the interval [0, x − 1] and returned.

Otherwise, Luck is subtracted from R, then R is forced into the interval [0, x − 1] and returned.