rnl
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.
Contents
Formulas
Nethack 3.4.3
See source code in [2]
Let R be a random integer number from 0 to x-1 (inclusively).
If Luck=0, R is returned.
Otherwise, with 1/(50-Luck) chance (since Luck is from -13 to 13, the chance should be between 1/37 and 1/63), R is returned.
Otherwise, if x<=15 and Luck>=-5 (that is, x is not too big and the user is not too unlucky), Luck/3 (truncated towards zero) is subtracted from R, than R is forced into the [0,x-1] interval.
Otherwise, Luck is subtracted from R, after which R is forced into the [0,x-1] interval and returned.
Note this means that for Luck in [-2,2] interval and x<=15, rnl just returns a random integer in the [0,x-1] interval. For Luck of -3 or worse, however, rnl have very tiny chance to return 0 (between 1/(53*x) and 1/(63*x), depending on the Luck). For Luck of 3 or better, the chance to return 0 increases significantly. For example, if x=10, the chance will be 10% for Luck between -2 and 2, about 20% for Luck of 3 to 5, and about 49% for Luck of 12 to 13.
Nethack 3.6.0
See source code in [3]
Let R be a random integer number from 0 to x-1 (inclusively).
If Luck=0, R is returned.
Otherwise, let A be
- Luck/3 rounded to the closest integer, if x<=15 (note that in this case A is from -4 to 4)
- Luck, otherwise
If A=0, R is returned.
Otherwise, with chance 1/(37+|A|), R is still returned (note that the chance is always between 1/38 and 1/50).
Otherwise, R-A is forced into [0,x-1] interval and returned.
Note this means that for Luck in [-1,1] interval and x<=15, rnl just returns a random integer in the [0,x-1] interval (because A=0). For Luck of -2 or worse, however, rnl have very tiny chance to return 0 (between 1/(38*x) and 1/(50*x), depending on the Luck). For Luck of 2 or better, the chance to return 0 increases significantly. For example, if x=10, the chance will be 10% for Luck between -1 and 1, about 20% for Luck of 2 to 4, and about 49% for Luck of 11 to 13.
How it's used
rnl is used for the following purposes:
| Situation | Code | Luck | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| -11 | -8 | -5 | -2 | 0 | +2 | +5 | +8 | +11 | ||
| Whether your god will help if you pray with exactly 0 alignment | rnl(2)==0[4] | 01% | 01% | 01% | 01% | 50% | 99% | 99% | 99% | 99% |
| Chance of writing a scroll you have not seen | rnl(15)==0[5] | .2% | .2% | .2% | .2% | 6.7% | 13.2% | 19.7% | 26.2% | 32.7% |
| Chance of writing a scroll you have not seen, as a Wizard | rnl(3)==0[6] | .8% | .8% | .9% | .9% | 33.3% | 65.8% | 98.3% | 98.3% | 98.3% |
| Chance to fail to set a landmine (then it blows up) or bear trap (nothing happens) from a horse when riding while unskilled | rnl(10)>5[7] | 79% | 69% | 59% | 50% | 40% | 30% | 21% | 11% | 1% |
| Chance to fail to set a landmine (then it blows up) or bear trap (nothing happens) from a horse when riding while unskilled, while fumbling or with a cursed trap | rnl(10)>3[8] | 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 fumbled | rnl(10)>5[9] | 79% | 69% | 59% | 50% | 40% | 30% | 21% | 11% | 1% |
| Chance that an otherwise pickup-able object does not slip free when you use a whip to pick it up | rnl(6)==0[10] | 0.4% | 0.4% | 0.4% | 0.4% | 16.7% | 32.9% | 49.1% | 65.4% | 81.8% |
| Chance of finding secret corridors and doors on each surrounding square when searching | rnl(7)==0[11] | 0.3% | 0.4% | 0.4% | 0.4% | 14.3% | 28.2% | 42.1% | 56.1% | 70.0% |
| Chance of finding secret corridors and doors on each surrounding square when searching, while wearing lenses | rnl(5)==0[12] | 0.5% | 0.5% | 0.5% | 0.5% | 20.0% | 39.5% | 59.0% | 78.5% | 98.0% |
| Chance of finding secret corridors 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[13] | 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[14] | 0.3% | 0.3% | 0.3% | 0.3% | 12.5% | 24.7% | 36.9% | 49.1% | 61.3% |
| Chance that a blessed projectile is not broken when thrown | rnl(4)==0[15] | 0.6% | 0.6% | 0.6% | 0.7% | 25% | 49.3% | 73.7% | 98.1% | 98.2% |
| Chance that you are petrified if your saddle slips off[16] or is detroyed(?), or you are polymorphed and lose stone resistance[17], and are riding a steed that petrifies on touch (likely due to your horse having just polymorphed into a footrice) | 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[18] | 99.5% | 99.5% | 99.5% | 99.5% | 80.0% | 60.5% | 41.0% | 21.5% | 2.0% |
| Chance that blessed equipment is not eroded by rust[19] or water[20] damage, or blessed flammable items are not burned by fire damage[21] | rnl(4)==0 | 0.6% | 0.6% | 0.6% | 0.7% | 25% | 49.3% | 73.7% | 98.1% | 98.2% |
| Chance that a monster becomes angry when it steps in a trap set by you[22], or that you trigger a trap when failing to disarm it[23] | rnl(5)>0 | 99.5% | 99.5% | 99.5% | 99.5% | 80.0% | 60.5% | 41.0% | 21.5% | 2.0% |
| Chance that a hostile, awake, unfrozen, mindful, nonhuman monster becomes peaceful when you free it from a trap not set by you | rnl(10)<8[24] | 40.1% | 50.8% | 60.5% | 70.3% | 80.0% | 89.7% | 99.5% | 99.5% | 99.5% |
| Chance that a lawful player gets +1 alignment for the above | rnl(8)==0 times 1/3[25] | 0.1% | 0.1% | 0.1% | 0.1% | 4.2% | 8.2% | 12.3% | 16.4% | 20.4% |
| Chance that you get +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[26] | 0.7% | 0.8% | 10.5% | 20.2% | 30% | 39.7% | 49.5% | 59.3% | 69.0% |
| Chance that a boomerang will break when used as a melee weapon | rnl(4)==3[27] | 98.2% | 98.1% | 73.7% | 49.3% | 25% | 0.7% | 0.6% | 0.6% | 0.6% |
| Chance that a successful joust, against a solid monster, breaks your non-artifact lance (very smooth distribution, 4.7% at Luck -13, .01% at Luck 1) | rnl(50)==49, 1/5 of the time[28] | 4.71% | 3.53% | 2.35% | 1.18% | 0.4% | 0.01% | 0.01% | 0.01% | 0.01% |
| Probability based on non-Luck stats | ||||||||||
| Chance that you can kick down a door (somewhat smooth distribution) | rnl(35)<X where X = avg( str, dex, con) + (dex if Samurai or Monk)[29] | Approx (Luck+x)/35 chance[30] | ||||||||
| Chance that you can kick down a door, avg(str,dex,con)=10 | rnl(35)<10 | 0.5% | 6.2% | 14.6% | 23.0% | 28.6% | 34.1% | 42.5% | 50.9% | 59.3% |
| Chance that you can kick down a door, avg(str,dex,con)=15 | rnl(35)<15 | 12.1% | 20.5% | 28.9% | 37.3% | 42.9% | 48.4% | 56.8% | 65.2% | 73.6% |
| Chance that you can kick down a door, Samurai or Monk and avg(str,con)=15, dex=15 | rnl(35)<30 | 54.9% | 63.4% | 71.8% | 80.1% | 85.7% | 91.3% | 99.7% | 99.7% | 99.7% |
| Chance that you can open a door (smoother distribution) | rnl(20)<X where X = avg( str, dex, con)[31] | Approx (Luck+x)/20 chance[32] | ||||||||
| Chance that you can open a door, avg(str,dex,con)=10 | rnl(20)<10 | 1% | 11% | 26% | 40% | 50% | 60% | 74% | 89% | 99% |
| Chance that you can open a door, avg(str,dex,con)=15 | rnl(20)<15 | 21% | 36% | 51% | 65% | 75% | 85% | 99% | 99% | 99% |
| One criterion (the other being alignment-based) for an altar conversion[33] (or attempt[34]) producing a summoned minion. | rnl(your level)>6 | Impossible before level 8, more likely with low luck, more likely at 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 (smoother distribution) | 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 (smoother distribution) | 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[35] | 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 (smoother distribution) | 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 (smoother distribution) | rnl(32)>0 | 99.9% | 99.9% | 99.9% | 99.9% | 96.9% | 90.8% | 81.6% | 72.4% | 63.2% |
| Numerical Distributions | ||||||||||
| Average number of fruit that fall from a tree if you kick it | 8-rnl(7)[36] | 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[37] | 5 | 5 | 4.7 | 4.2 | 3.5 | 2.8 | 2.3 | 2 | 2 |
| Amount of gold left from a gold golem's death (average, smooth distribution) | 200-rnl(101)[38] | 140 | 143 | 145 | 148 | 150 | 152 | 155 | 157 | 160 |
| Number of darts/arrows/etc left behind when disarming a trap (average, smooth distribution) | 50-rnl(50)[39] | 16 | 18 | 21 | 24 | 25.5 | 27 | 30 | 33 | 35 |
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".
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 as the luck number 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 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 00% are 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 | 00% | 00% | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 49% |
| -12 | 00% | 00% | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 49% |
| -11 | 00% | 00% | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 49% |
| -10 | 00% | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 10% | 39% |
| -9 | 00% | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 10% | 39% |
| -8 | 00% | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 10% | 39% |
| -7 | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 29% |
| -6 | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 29% |
| -5 | 00% | 00% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 29% |
| -4 | 00% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 20% |
| -3 | 00% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 20% |
| -2 | 00% | 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% | 00% |
| 3 | 20% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 00% |
| 4 | 20% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 00% |
| 5 | 29% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 00% | 00% |
| 6 | 29% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 00% | 00% |
| 7 | 29% | 10% | 10% | 10% | 10% | 10% | 10% | 10% | 00% | 00% |
| 8 | 39% | 10% | 10% | 10% | 10% | 10% | 10% | 00% | 00% | 00% |
| 9 | 39% | 10% | 10% | 10% | 10% | 10% | 10% | 00% | 00% | 00% |
| 10 | 39% | 10% | 10% | 10% | 10% | 10% | 10% | 00% | 00% | 00% |
| 11 | 49% | 10% | 10% | 10% | 10% | 10% | 00% | 00% | 00% | 00% |
| 12 | 49% | 10% | 10% | 10% | 10% | 10% | 00% | 00% | 00% | 00% |
| 13 | 49% | 10% | 10% | 10% | 10% | 10% | 00% | 00% | 00% | 00% |
References
- ↑ rnl in rnd.c
- ↑ rnl code for Nethack 3.4.3
- ↑ rnl code for Nethack 3.6.0
- ↑ pray.c in NetHack 3.4.3, line 828
- ↑ write.c in NetHack 3.4.3, line 201
- ↑ write.c in NetHack 3.4.3, line 201
- ↑ apply.c in NetHack 3.4.3, line 2034
- ↑ apply.c in NetHack 3.4.3, line 2034
- ↑ apply.c in NetHack 3.4.3, line 2097
- ↑ apply.c in NetHack 3.4.3, line 2174
- ↑ detect.c in NetHack 3.4.3, line 1191
- ↑ detect.c in NetHack 3.4.3, line 1183
- ↑ detect.c in NetHack 3.4.3, line 1183
- ↑ detect.c in NetHack 3.4.3, line 1248
- ↑ dothrow.c in NetHack 3.4.3, line 1332
- ↑ worn.c in NetHack 3.4.3, line 741
- ↑ polyself.c in NetHack 3.4.3, line 485
- ↑ teleport.c in NetHack 3.4.3, line 606
- ↑ trap.c in NetHack 3.4.3, line 162
- ↑ trap.c in NetHack 3.4.3, line 2735
- ↑ trap.c in NetHack 3.4.3, line 2666
- ↑ trap.c in NetHack 3.4.3, line 1716
- ↑ trap.c in NetHack 3.4.3, line 3122
- ↑ trap.c in NetHack 3.4.3, line 3166
- ↑ trap.c in NetHack 3.4.3, line 3176
- ↑ trap.c in NetHack 3.4.3, line 3294
- ↑ uhitm.c in NetHack 3.4.3, line 587
- ↑ uhitm.c in NetHack 3.4.3, line 1153
- ↑ dokick.c in NetHack 3.4.3, line 1046
- ↑ computed numbers slightly incorrect
- ↑ lock.c in NetHack 3.4.3, line 562
- ↑ computed numbers slightly incorrect
- ↑ pray.c in NetHack 3.4.3, line 1396
- ↑ pray.c in NetHack 3.4.3, line 1408
- ↑ pray.c in NetHack 3.4.3, line 1618
- ↑ dokick.c in NetHack 3.4.3, line 900
- ↑ dokick.c in NetHack 3.4.3, line 900
- ↑ mon.c in NetHack 3.4.3, line 278
- ↑ trap.c in NetHack 3.4.3, line 3274
