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   1Fighting regressions with git bisect
   2====================================
   3:Author: Christian Couder
   4:Email: chriscool@tuxfamily.org
   5:Date: 2009/11/08
   6
   7Abstract
   8--------
   9
  10"git bisect" enables software users and developers to easily find the
  11commit that introduced a regression. We show why it is important to
  12have good tools to fight regressions. We describe how "git bisect"
  13works from the outside and the algorithms it uses inside. Then we
  14explain how to take advantage of "git bisect" to improve current
  15practices. And we discuss how "git bisect" could improve in the
  16future.
  17
  18
  19Introduction to "git bisect"
  20----------------------------
  21
  22Git is a Distributed Version Control system (DVCS) created by Linus
  23Torvalds and maintained by Junio Hamano.
  24
  25In Git like in many other Version Control Systems (VCS), the different
  26states of the data that is managed by the system are called
  27commits. And, as VCS are mostly used to manage software source code,
  28sometimes "interesting" changes of behavior in the software are
  29introduced in some commits.
  30
  31In fact people are specially interested in commits that introduce a
  32"bad" behavior, called a bug or a regression. They are interested in
  33these commits because a commit (hopefully) contains a very small set
  34of source code changes. And it's much easier to understand and
  35properly fix a problem when you only need to check a very small set of
  36changes, than when you don't know where look in the first place.
  37
  38So to help people find commits that introduce a "bad" behavior, the
  39"git bisect" set of commands was invented. And it follows of course
  40that in "git bisect" parlance, commits where the "interesting
  41behavior" is present are called "bad" commits, while other commits are
  42called "good" commits. And a commit that introduce the behavior we are
  43interested in is called a "first bad commit". Note that there could be
  44more than one "first bad commit" in the commit space we are searching.
  45
  46So "git bisect" is designed to help find a "first bad commit". And to
  47be as efficient as possible, it tries to perform a binary search.
  48
  49
  50Fighting regressions overview
  51-----------------------------
  52
  53Regressions: a big problem
  54~~~~~~~~~~~~~~~~~~~~~~~~~~
  55
  56Regressions are a big problem in the software industry. But it's
  57difficult to put some real numbers behind that claim.
  58
  59There are some numbers about bugs in general, like a NIST study in
  602002 <<1>> that said:
  61
  62_____________
  63Software bugs, or errors, are so prevalent and so detrimental that
  64they cost the U.S. economy an estimated $59.5 billion annually, or
  65about 0.6 percent of the gross domestic product, according to a newly
  66released study commissioned by the Department of Commerce's National
  67Institute of Standards and Technology (NIST). At the national level,
  68over half of the costs are borne by software users and the remainder
  69by software developers/vendors.  The study also found that, although
  70all errors cannot be removed, more than a third of these costs, or an
  71estimated $22.2 billion, could be eliminated by an improved testing
  72infrastructure that enables earlier and more effective identification
  73and removal of software defects. These are the savings associated with
  74finding an increased percentage (but not 100 percent) of errors closer
  75to the development stages in which they are introduced. Currently,
  76over half of all errors are not found until "downstream" in the
  77development process or during post-sale software use.
  78_____________
  79
  80And then:
  81
  82_____________
  83Software developers already spend approximately 80 percent of
  84development costs on identifying and correcting defects, and yet few
  85products of any type other than software are shipped with such high
  86levels of errors.
  87_____________
  88
  89Eventually the conclusion started with:
  90
  91_____________
  92The path to higher software quality is significantly improved software
  93testing.
  94_____________
  95
  96There are other estimates saying that 80% of the cost related to
  97software is about maintenance <<2>>.
  98
  99Though, according to Wikipedia <<3>>:
 100
 101_____________
 102A common perception of maintenance is that it is merely fixing
 103bugs. However, studies and surveys over the years have indicated that
 104the majority, over 80%, of the maintenance effort is used for
 105non-corrective actions (Pigosky 1997). This perception is perpetuated
 106by users submitting problem reports that in reality are functionality
 107enhancements to the system.
 108_____________
 109
 110But we can guess that improving on existing software is very costly
 111because you have to watch out for regressions. At least this would
 112make the above studies consistent among themselves.
 113
 114Of course some kind of software is developed, then used during some
 115time without being improved on much, and then finally thrown away. In
 116this case, of course, regressions may not be a big problem. But on the
 117other hand, there is a lot of big software that is continually
 118developed and maintained during years or even tens of years by a lot
 119of people. And as there are often many people who depend (sometimes
 120critically) on such software, regressions are a really big problem.
 121
 122One such software is the Linux kernel. And if we look at the Linux
 123kernel, we can see that a lot of time and effort is spent to fight
 124regressions. The release cycle start with a 2 weeks long merge
 125window. Then the first release candidate (rc) version is tagged. And
 126after that about 7 or 8 more rc versions will appear with around one
 127week between each of them, before the final release.
 128
 129The time between the first rc release and the final release is
 130supposed to be used to test rc versions and fight bugs and especially
 131regressions. And this time is more than 80% of the release cycle
 132time. But this is not the end of the fight yet, as of course it
 133continues after the release.
 134
 135And then this is what Ingo Molnar (a well known Linux kernel
 136developer) says about his use of git bisect:
 137
 138_____________
 139I most actively use it during the merge window (when a lot of trees
 140get merged upstream and when the influx of bugs is the highest) - and
 141yes, there have been cases that i used it multiple times a day. My
 142average is roughly once a day.
 143_____________
 144
 145So regressions are fought all the time by developers, and indeed it is
 146well known that bugs should be fixed as soon as possible, so as soon
 147as they are found. That's why it is interesting to have good tools for
 148this purpose.
 149
 150Other tools to fight regressions
 151~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 152
 153So what are the tools used to fight regressions? They are nearly the
 154same as those used to fight regular bugs. The only specific tools are
 155test suites and tools similar as "git bisect".
 156
 157Test suites are very nice. But when they are used alone, they are
 158supposed to be used so that all the tests are checked after each
 159commit. This means that they are not very efficient, because many
 160tests are run for no interesting result, and they suffer from
 161combinational explosion.
 162
 163In fact the problem is that big software often has many different
 164configuration options and that each test case should pass for each
 165configuration after each commit. So if you have for each release: N
 166configurations, M commits and T test cases, you should perform:
 167
 168-------------
 169N * M * T tests
 170-------------
 171
 172where N, M and T are all growing with the size your software.
 173
 174So very soon it will not be possible to completely test everything.
 175
 176And if some bugs slip through your test suite, then you can add a test
 177to your test suite. But if you want to use your new improved test
 178suite to find where the bug slipped in, then you will either have to
 179emulate a bisection process or you will perhaps bluntly test each
 180commit backward starting from the "bad" commit you have which may be
 181very wasteful.
 182
 183"git bisect" overview
 184---------------------
 185
 186Starting a bisection
 187~~~~~~~~~~~~~~~~~~~~
 188
 189The first "git bisect" subcommand to use is "git bisect start" to
 190start the search. Then bounds must be set to limit the commit
 191space. This is done usually by giving one "bad" and at least one
 192"good" commit. They can be passed in the initial call to "git bisect
 193start" like this:
 194
 195-------------
 196$ git bisect start [BAD [GOOD...]]
 197-------------
 198
 199or they can be set using:
 200
 201-------------
 202$ git bisect bad [COMMIT]
 203-------------
 204
 205and:
 206
 207-------------
 208$ git bisect good [COMMIT...]
 209-------------
 210
 211where BAD, GOOD and COMMIT are all names that can be resolved to a
 212commit.
 213
 214Then "git bisect" will checkout a commit of its choosing and ask the
 215user to test it, like this:
 216
 217-------------
 218$ git bisect start v2.6.27 v2.6.25
 219Bisecting: 10928 revisions left to test after this (roughly 14 steps)
 220[2ec65f8b89ea003c27ff7723525a2ee335a2b393] x86: clean up using max_low_pfn on 32-bit
 221-------------
 222
 223Note that the example that we will use is really a toy example, we
 224will be looking for the first commit that has a version like
 225"2.6.26-something", that is the commit that has a "SUBLEVEL = 26" line
 226in the top level Makefile. This is a toy example because there are
 227better ways to find this commit with Git than using "git bisect" (for
 228example "git blame" or "git log -S<string>").
 229
 230Driving a bisection manually
 231~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 232
 233At this point there are basically 2 ways to drive the search. It can
 234be driven manually by the user or it can be driven automatically by a
 235script or a command.
 236
 237If the user is driving it, then at each step of the search, the user
 238will have to test the current commit and say if it is "good" or "bad"
 239using the "git bisect good" or "git bisect bad" commands respectively
 240that have been described above. For example:
 241
 242-------------
 243$ git bisect bad
 244Bisecting: 5480 revisions left to test after this (roughly 13 steps)
 245[66c0b394f08fd89236515c1c84485ea712a157be] KVM: kill file->f_count abuse in kvm
 246-------------
 247
 248And after a few more steps like that, "git bisect" will eventually
 249find a first bad commit:
 250
 251-------------
 252$ git bisect bad
 2532ddcca36c8bcfa251724fe342c8327451988be0d is the first bad commit
 254commit 2ddcca36c8bcfa251724fe342c8327451988be0d
 255Author: Linus Torvalds <torvalds@linux-foundation.org>
 256Date:   Sat May 3 11:59:44 2008 -0700
 257
 258    Linux 2.6.26-rc1
 259
 260:100644 100644 5cf82581... 4492984e... M      Makefile
 261-------------
 262
 263At this point we can see what the commit does, check it out (if it's
 264not already checked out) or tinker with it, for example:
 265
 266-------------
 267$ git show HEAD
 268commit 2ddcca36c8bcfa251724fe342c8327451988be0d
 269Author: Linus Torvalds <torvalds@linux-foundation.org>
 270Date:   Sat May 3 11:59:44 2008 -0700
 271
 272    Linux 2.6.26-rc1
 273
 274diff --git a/Makefile b/Makefile
 275index 5cf8258..4492984 100644
 276--- a/Makefile
 277+++ b/Makefile
 278@@ -1,7 +1,7 @@
 279 VERSION = 2
 280 PATCHLEVEL = 6
 281-SUBLEVEL = 25
 282-EXTRAVERSION =
 283+SUBLEVEL = 26
 284+EXTRAVERSION = -rc1
 285 NAME = Funky Weasel is Jiggy wit it
 286
 287 # *DOCUMENTATION*
 288-------------
 289
 290And when we are finished we can use "git bisect reset" to go back to
 291the branch we were in before we started bisecting:
 292
 293-------------
 294$ git bisect reset
 295Checking out files: 100% (21549/21549), done.
 296Previous HEAD position was 2ddcca3... Linux 2.6.26-rc1
 297Switched to branch 'master'
 298-------------
 299
 300Driving a bisection automatically
 301~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 302
 303The other way to drive the bisection process is to tell "git bisect"
 304to launch a script or command at each bisection step to know if the
 305current commit is "good" or "bad". To do that, we use the "git bisect
 306run" command. For example:
 307
 308-------------
 309$ git bisect start v2.6.27 v2.6.25
 310Bisecting: 10928 revisions left to test after this (roughly 14 steps)
 311[2ec65f8b89ea003c27ff7723525a2ee335a2b393] x86: clean up using max_low_pfn on 32-bit
 312$
 313$ git bisect run grep '^SUBLEVEL = 25' Makefile
 314running grep ^SUBLEVEL = 25 Makefile
 315Bisecting: 5480 revisions left to test after this (roughly 13 steps)
 316[66c0b394f08fd89236515c1c84485ea712a157be] KVM: kill file->f_count abuse in kvm
 317running grep ^SUBLEVEL = 25 Makefile
 318SUBLEVEL = 25
 319Bisecting: 2740 revisions left to test after this (roughly 12 steps)
 320[671294719628f1671faefd4882764886f8ad08cb] V4L/DVB(7879): Adding cx18 Support for mxl5005s
 321...
 322...
 323running grep ^SUBLEVEL = 25 Makefile
 324Bisecting: 0 revisions left to test after this (roughly 0 steps)
 325[2ddcca36c8bcfa251724fe342c8327451988be0d] Linux 2.6.26-rc1
 326running grep ^SUBLEVEL = 25 Makefile
 3272ddcca36c8bcfa251724fe342c8327451988be0d is the first bad commit
 328commit 2ddcca36c8bcfa251724fe342c8327451988be0d
 329Author: Linus Torvalds <torvalds@linux-foundation.org>
 330Date:   Sat May 3 11:59:44 2008 -0700
 331
 332    Linux 2.6.26-rc1
 333
 334:100644 100644 5cf82581... 4492984e... M      Makefile
 335bisect run success
 336-------------
 337
 338In this example, we passed "grep '^SUBLEVEL = 25' Makefile" as
 339parameter to "git bisect run". This means that at each step, the grep
 340command we passed will be launched. And if it exits with code 0 (that
 341means success) then git bisect will mark the current state as
 342"good". If it exits with code 1 (or any code between 1 and 127
 343included, except the special code 125), then the current state will be
 344marked as "bad".
 345
 346Exit code between 128 and 255 are special to "git bisect run". They
 347make it stop immediately the bisection process. This is useful for
 348example if the command passed takes too long to complete, because you
 349can kill it with a signal and it will stop the bisection process.
 350
 351It can also be useful in scripts passed to "git bisect run" to "exit
 352255" if some very abnormal situation is detected.
 353
 354Avoiding untestable commits
 355~~~~~~~~~~~~~~~~~~~~~~~~~~~
 356
 357Sometimes it happens that the current state cannot be tested, for
 358example if it does not compile because there was a bug preventing it
 359at that time. This is what the special exit code 125 is for. It tells
 360"git bisect run" that the current commit should be marked as
 361untestable and that another one should be chosen and checked out.
 362
 363If the bisection process is driven manually, you can use "git bisect
 364skip" to do the same thing. (In fact the special exit code 125 makes
 365"git bisect run" use "git bisect skip" in the background.)
 366
 367Or if you want more control, you can inspect the current state using
 368for example "git bisect visualize". It will launch gitk (or "git log"
 369if the DISPLAY environment variable is not set) to help you find a
 370better bisection point.
 371
 372Either way, if you have a string of untestable commits, it might
 373happen that the regression you are looking for has been introduced by
 374one of these untestable commits. In this case it's not possible to
 375tell for sure which commit introduced the regression.
 376
 377So if you used "git bisect skip" (or the run script exited with
 378special code 125) you could get a result like this:
 379
 380-------------
 381There are only 'skip'ped commits left to test.
 382The first bad commit could be any of:
 38315722f2fa328eaba97022898a305ffc8172db6b1
 38478e86cf3e850bd755bb71831f42e200626fbd1e0
 385e15b73ad3db9b48d7d1ade32f8cd23a751fe0ace
 386070eab2303024706f2924822bfec8b9847e4ac1b
 387We cannot bisect more!
 388-------------
 389
 390Saving a log and replaying it
 391~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 392
 393If you want to show other people your bisection process, you can get a
 394log using for example:
 395
 396-------------
 397$ git bisect log > bisect_log.txt
 398-------------
 399
 400And it is possible to replay it using:
 401
 402-------------
 403$ git bisect replay bisect_log.txt
 404-------------
 405
 406
 407"git bisect" details
 408--------------------
 409
 410Bisection algorithm
 411~~~~~~~~~~~~~~~~~~~
 412
 413As the Git commits form a directed acyclic graph (DAG), finding the
 414best bisection commit to test at each step is not so simple. Anyway
 415Linus found and implemented a "truly stupid" algorithm, later improved
 416by Junio Hamano, that works quite well.
 417
 418So the algorithm used by "git bisect" to find the best bisection
 419commit when there are no skipped commits is the following:
 420
 4211) keep only the commits that:
 422
 423a) are ancestor of the "bad" commit (including the "bad" commit itself),
 424b) are not ancestor of a "good" commit (excluding the "good" commits).
 425
 426This means that we get rid of the uninteresting commits in the DAG.
 427
 428For example if we start with a graph like this:
 429
 430-------------
 431G-Y-G-W-W-W-X-X-X-X
 432           \ /
 433            W-W-B
 434           /
 435Y---G-W---W
 436 \ /   \
 437Y-Y     X-X-X-X
 438
 439-> time goes this way ->
 440-------------
 441
 442where B is the "bad" commit, "G" are "good" commits and W, X, and Y
 443are other commits, we will get the following graph after this first
 444step:
 445
 446-------------
 447W-W-W
 448     \
 449      W-W-B
 450     /
 451W---W
 452-------------
 453
 454So only the W and B commits will be kept. Because commits X and Y will
 455have been removed by rules a) and b) respectively, and because commits
 456G are removed by rule b) too.
 457
 458Note for Git users, that it is equivalent as keeping only the commit
 459given by:
 460
 461-------------
 462git rev-list BAD --not GOOD1 GOOD2...
 463-------------
 464
 465Also note that we don't require the commits that are kept to be
 466descendants of a "good" commit. So in the following example, commits W
 467and Z will be kept:
 468
 469-------------
 470G-W-W-W-B
 471   /
 472Z-Z
 473-------------
 474
 4752) starting from the "good" ends of the graph, associate to each
 476commit the number of ancestors it has plus one
 477
 478For example with the following graph where H is the "bad" commit and A
 479and D are some parents of some "good" commits:
 480
 481-------------
 482A-B-C
 483     \
 484      F-G-H
 485     /
 486D---E
 487-------------
 488
 489this will give:
 490
 491-------------
 4921 2 3
 493A-B-C
 494     \6 7 8
 495      F-G-H
 4961   2/
 497D---E
 498-------------
 499
 5003) associate to each commit: min(X, N - X)
 501
 502where X is the value associated to the commit in step 2) and N is the
 503total number of commits in the graph.
 504
 505In the above example we have N = 8, so this will give:
 506
 507-------------
 5081 2 3
 509A-B-C
 510     \2 1 0
 511      F-G-H
 5121   2/
 513D---E
 514-------------
 515
 5164) the best bisection point is the commit with the highest associated
 517number
 518
 519So in the above example the best bisection point is commit C.
 520
 5215) note that some shortcuts are implemented to speed up the algorithm
 522
 523As we know N from the beginning, we know that min(X, N - X) can't be
 524greater than N/2. So during steps 2) and 3), if we would associate N/2
 525to a commit, then we know this is the best bisection point. So in this
 526case we can just stop processing any other commit and return the
 527current commit.
 528
 529Bisection algorithm debugging
 530~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 531
 532For any commit graph, you can see the number associated with each
 533commit using "git rev-list --bisect-all".
 534
 535For example, for the above graph, a command like:
 536
 537-------------
 538$ git rev-list --bisect-all BAD --not GOOD1 GOOD2
 539-------------
 540
 541would output something like:
 542
 543-------------
 544e15b73ad3db9b48d7d1ade32f8cd23a751fe0ace (dist=3)
 54515722f2fa328eaba97022898a305ffc8172db6b1 (dist=2)
 54678e86cf3e850bd755bb71831f42e200626fbd1e0 (dist=2)
 547a1939d9a142de972094af4dde9a544e577ddef0e (dist=2)
 548070eab2303024706f2924822bfec8b9847e4ac1b (dist=1)
 549a3864d4f32a3bf5ed177ddef598490a08760b70d (dist=1)
 550a41baa717dd74f1180abf55e9341bc7a0bb9d556 (dist=1)
 5519e622a6dad403b71c40979743bb9d5be17b16bd6 (dist=0)
 552-------------
 553
 554Bisection algorithm discussed
 555~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 556
 557First let's define "best bisection point". We will say that a commit X
 558is a best bisection point or a best bisection commit if knowing its
 559state ("good" or "bad") gives as much information as possible whether
 560the state of the commit happens to be "good" or "bad".
 561
 562This means that the best bisection commits are the commits where the
 563following function is maximum:
 564
 565-------------
 566f(X) = min(information_if_good(X), information_if_bad(X))
 567-------------
 568
 569where information_if_good(X) is the information we get if X is good
 570and information_if_bad(X) is the information we get if X is bad.
 571
 572Now we will suppose that there is only one "first bad commit". This
 573means that all its descendants are "bad" and all the other commits are
 574"good". And we will suppose that all commits have an equal probability
 575of being good or bad, or of being the first bad commit, so knowing the
 576state of c commits gives always the same amount of information
 577wherever these c commits are on the graph and whatever c is. (So we
 578suppose that these commits being for example on a branch or near a
 579good or a bad commit does not give more or less information).
 580
 581Let's also suppose that we have a cleaned up graph like one after step
 5821) in the bisection algorithm above. This means that we can measure
 583the information we get in terms of number of commit we can remove from
 584the graph..
 585
 586And let's take a commit X in the graph.
 587
 588If X is found to be "good", then we know that its ancestors are all
 589"good", so we want to say that:
 590
 591-------------
 592information_if_good(X) = number_of_ancestors(X)  (TRUE)
 593-------------
 594
 595And this is true because at step 1) b) we remove the ancestors of the
 596"good" commits.
 597
 598If X is found to be "bad", then we know that its descendants are all
 599"bad", so we want to say that:
 600
 601-------------
 602information_if_bad(X) = number_of_descendants(X)  (WRONG)
 603-------------
 604
 605But this is wrong because at step 1) a) we keep only the ancestors of
 606the bad commit. So we get more information when a commit is marked as
 607"bad", because we also know that the ancestors of the previous "bad"
 608commit that are not ancestors of the new "bad" commit are not the
 609first bad commit. We don't know if they are good or bad, but we know
 610that they are not the first bad commit because they are not ancestor
 611of the new "bad" commit.
 612
 613So when a commit is marked as "bad" we know we can remove all the
 614commits in the graph except those that are ancestors of the new "bad"
 615commit. This means that:
 616
 617-------------
 618information_if_bad(X) = N - number_of_ancestors(X)  (TRUE)
 619-------------
 620
 621where N is the number of commits in the (cleaned up) graph.
 622
 623So in the end this means that to find the best bisection commits we
 624should maximize the function:
 625
 626-------------
 627f(X) = min(number_of_ancestors(X), N - number_of_ancestors(X))
 628-------------
 629
 630And this is nice because at step 2) we compute number_of_ancestors(X)
 631and so at step 3) we compute f(X).
 632
 633Let's take the following graph as an example:
 634
 635-------------
 636            G-H-I-J
 637           /       \
 638A-B-C-D-E-F         O
 639           \       /
 640            K-L-M-N
 641-------------
 642
 643If we compute the following non optimal function on it:
 644
 645-------------
 646g(X) = min(number_of_ancestors(X), number_of_descendants(X))
 647-------------
 648
 649we get:
 650
 651-------------
 652            4 3 2 1
 653            G-H-I-J
 6541 2 3 4 5 6/       \0
 655A-B-C-D-E-F         O
 656           \       /
 657            K-L-M-N
 658            4 3 2 1
 659-------------
 660
 661but with the algorithm used by git bisect we get:
 662
 663-------------
 664            7 7 6 5
 665            G-H-I-J
 6661 2 3 4 5 6/       \0
 667A-B-C-D-E-F         O
 668           \       /
 669            K-L-M-N
 670            7 7 6 5
 671-------------
 672
 673So we chose G, H, K or L as the best bisection point, which is better
 674than F. Because if for example L is bad, then we will know not only
 675that L, M and N are bad but also that G, H, I and J are not the first
 676bad commit (since we suppose that there is only one first bad commit
 677and it must be an ancestor of L).
 678
 679So the current algorithm seems to be the best possible given what we
 680initially supposed.
 681
 682Skip algorithm
 683~~~~~~~~~~~~~~
 684
 685When some commits have been skipped (using "git bisect skip"), then
 686the bisection algorithm is the same for step 1) to 3). But then we use
 687roughly the following steps:
 688
 6896) sort the commit by decreasing associated value
 690
 6917) if the first commit has not been skipped, we can return it and stop
 692here
 693
 6948) otherwise filter out all the skipped commits in the sorted list
 695
 6969) use a pseudo random number generator (PRNG) to generate a random
 697number between 0 and 1
 698
 69910) multiply this random number with its square root to bias it toward
 7000
 701
 70211) multiply the result by the number of commits in the filtered list
 703to get an index into this list
 704
 70512) return the commit at the computed index
 706
 707Skip algorithm discussed
 708~~~~~~~~~~~~~~~~~~~~~~~~
 709
 710After step 7) (in the skip algorithm), we could check if the second
 711commit has been skipped and return it if it is not the case. And in
 712fact that was the algorithm we used from when "git bisect skip" was
 713developed in Git version 1.5.4 (released on February 1st 2008) until
 714Git version 1.6.4 (released July 29th 2009).
 715
 716But Ingo Molnar and H. Peter Anvin (another well known linux kernel
 717developer) both complained that sometimes the best bisection points
 718all happened to be in an area where all the commits are
 719untestable. And in this case the user was asked to test many
 720untestable commits, which could be very inefficient.
 721
 722Indeed untestable commits are often untestable because a breakage was
 723introduced at one time, and that breakage was fixed only after many
 724other commits were introduced.
 725
 726This breakage is of course most of the time unrelated to the breakage
 727we are trying to locate in the commit graph. But it prevents us to
 728know if the interesting "bad behavior" is present or not.
 729
 730So it is a fact that commits near an untestable commit have a high
 731probability of being untestable themselves. And the best bisection
 732commits are often found together too (due to the bisection algorithm).
 733
 734This is why it is a bad idea to just chose the next best unskipped
 735bisection commit when the first one has been skipped.
 736
 737We found that most commits on the graph may give quite a lot of
 738information when they are tested. And the commits that will not on
 739average give a lot of information are the one near the good and bad
 740commits.
 741
 742So using a PRNG with a bias to favor commits away from the good and
 743bad commits looked like a good choice.
 744
 745One obvious improvement to this algorithm would be to look for a
 746commit that has an associated value near the one of the best bisection
 747commit, and that is on another branch, before using the PRNG. Because
 748if such a commit exists, then it is not very likely to be untestable
 749too, so it will probably give more information than a nearly randomly
 750chosen one.
 751
 752Checking merge bases
 753~~~~~~~~~~~~~~~~~~~~
 754
 755There is another tweak in the bisection algorithm that has not been
 756described in the "bisection algorithm" above.
 757
 758We supposed in the previous examples that the "good" commits were
 759ancestors of the "bad" commit. But this is not a requirement of "git
 760bisect".
 761
 762Of course the "bad" commit cannot be an ancestor of a "good" commit,
 763because the ancestors of the good commits are supposed to be
 764"good". And all the "good" commits must be related to the bad commit.
 765They cannot be on a branch that has no link with the branch of the
 766"bad" commit. But it is possible for a good commit to be related to a
 767bad commit and yet not be neither one of its ancestor nor one of its
 768descendants.
 769
 770For example, there can be a "main" branch, and a "dev" branch that was
 771forked of the main branch at a commit named "D" like this:
 772
 773-------------
 774A-B-C-D-E-F-G  <--main
 775       \
 776        H-I-J  <--dev
 777-------------
 778
 779The commit "D" is called a "merge base" for branch "main" and "dev"
 780because it's the best common ancestor for these branches for a merge.
 781
 782Now let's suppose that commit J is bad and commit G is good and that
 783we apply the bisection algorithm like it has been previously
 784described.
 785
 786As described in step 1) b) of the bisection algorithm, we remove all
 787the ancestors of the good commits because they are supposed to be good
 788too.
 789
 790So we would be left with only:
 791
 792-------------
 793H-I-J
 794-------------
 795
 796But what happens if the first bad commit is "B" and if it has been
 797fixed in the "main" branch by commit "F"?
 798
 799The result of such a bisection would be that we would find that H is
 800the first bad commit, when in fact it's B. So that would be wrong!
 801
 802And yes it can happen in practice that people working on one branch
 803are not aware that people working on another branch fixed a bug! It
 804could also happen that F fixed more than one bug or that it is a
 805revert of some big development effort that was not ready to be
 806released.
 807
 808In fact development teams often maintain both a development branch and
 809a maintenance branch, and it would be quite easy for them if "git
 810bisect" just worked when they want to bisect a regression on the
 811development branch that is not on the maintenance branch. They should
 812be able to start bisecting using:
 813
 814-------------
 815$ git bisect start dev main
 816-------------
 817
 818To enable that additional nice feature, when a bisection is started
 819and when some good commits are not ancestors of the bad commit, we
 820first compute the merge bases between the bad and the good commits and
 821we chose these merge bases as the first commits that will be checked
 822out and tested.
 823
 824If it happens that one merge base is bad, then the bisection process
 825is stopped with a message like:
 826
 827-------------
 828The merge base BBBBBB is bad.
 829This means the bug has been fixed between BBBBBB and [GGGGGG,...].
 830-------------
 831
 832where BBBBBB is the sha1 hash of the bad merge base and [GGGGGG,...]
 833is a comma separated list of the sha1 of the good commits.
 834
 835If some of the merge bases are skipped, then the bisection process
 836continues, but the following message is printed for each skipped merge
 837base:
 838
 839-------------
 840Warning: the merge base between BBBBBB and [GGGGGG,...] must be skipped.
 841So we cannot be sure the first bad commit is between MMMMMM and BBBBBB.
 842We continue anyway.
 843-------------
 844
 845where BBBBBB is the sha1 hash of the bad commit, MMMMMM is the sha1
 846hash of the merge base that is skipped and [GGGGGG,...]  is a comma
 847separated list of the sha1 of the good commits.
 848
 849So if there is no bad merge base, the bisection process continues as
 850usual after this step.
 851
 852Best bisecting practices
 853------------------------
 854
 855Using test suites and git bisect together
 856~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 857
 858If you both have a test suite and use git bisect, then it becomes less
 859important to check that all tests pass after each commit. Though of
 860course it is probably a good idea to have some checks to avoid
 861breaking too many things because it could make bisecting other bugs
 862more difficult.
 863
 864You can focus your efforts to check at a few points (for example rc
 865and beta releases) that all the T test cases pass for all the N
 866configurations. And when some tests don't pass you can use "git
 867bisect" (or better "git bisect run"). So you should perform roughly:
 868
 869-------------
 870c * N * T + b * M * log2(M) tests
 871-------------
 872
 873where c is the number of rounds of test (so a small constant) and b is
 874the ratio of bug per commit (hopefully a small constant too).
 875
 876So of course it's much better as it's O(N * T) vs O(N * T * M) if
 877you would test everything after each commit.
 878
 879This means that test suites are good to prevent some bugs from being
 880committed and they are also quite good to tell you that you have some
 881bugs. But they are not so good to tell you where some bugs have been
 882introduced. To tell you that efficiently, git bisect is needed.
 883
 884The other nice thing with test suites, is that when you have one, you
 885already know how to test for bad behavior. So you can use this
 886knowledge to create a new test case for "git bisect" when it appears
 887that there is a regression. So it will be easier to bisect the bug and
 888fix it. And then you can add the test case you just created to your
 889test suite.
 890
 891So if you know how to create test cases and how to bisect, you will be
 892subject to a virtuous circle:
 893
 894more tests => easier to create tests => easier to bisect => more tests
 895
 896So test suites and "git bisect" are complementary tools that are very
 897powerful and efficient when used together.
 898
 899Bisecting build failures
 900~~~~~~~~~~~~~~~~~~~~~~~~
 901
 902You can very easily automatically bisect broken builds using something
 903like:
 904
 905-------------
 906$ git bisect start BAD GOOD
 907$ git bisect run make
 908-------------
 909
 910Passing sh -c "some commands" to "git bisect run"
 911~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 912
 913For example:
 914
 915-------------
 916$ git bisect run sh -c "make || exit 125; ./my_app | grep 'good output'"
 917-------------
 918
 919On the other hand if you do this often, then it can be worth having
 920scripts to avoid too much typing.
 921
 922Finding performance regressions
 923~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 924
 925Here is an example script that comes slightly modified from a real
 926world script used by Junio Hamano <<4>>.
 927
 928This script can be passed to "git bisect run" to find the commit that
 929introduced a performance regression:
 930
 931-------------
 932#!/bin/sh
 933
 934# Build errors are not what I am interested in.
 935make my_app || exit 255
 936
 937# We are checking if it stops in a reasonable amount of time, so
 938# let it run in the background...
 939
 940./my_app >log 2>&1 &
 941
 942# ... and grab its process ID.
 943pid=$!
 944
 945# ... and then wait for sufficiently long.
 946sleep $NORMAL_TIME
 947
 948# ... and then see if the process is still there.
 949if kill -0 $pid
 950then
 951        # It is still running -- that is bad.
 952        kill $pid; sleep 1; kill $pid;
 953        exit 1
 954else
 955        # It has already finished (the $pid process was no more),
 956        # and we are happy.
 957        exit 0
 958fi
 959-------------
 960
 961Following general best practices
 962~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
 963
 964It is obviously a good idea not to have commits with changes that
 965knowingly break things, even if some other commits later fix the
 966breakage.
 967
 968It is also a good idea when using any VCS to have only one small
 969logical change in each commit.
 970
 971The smaller the changes in your commit, the most effective "git
 972bisect" will be. And you will probably need "git bisect" less in the
 973first place, as small changes are easier to review even if they are
 974only reviewed by the committer.
 975
 976Another good idea is to have good commit messages. They can be very
 977helpful to understand why some changes were made.
 978
 979These general best practices are very helpful if you bisect often.
 980
 981Avoiding bug prone merges
 982~~~~~~~~~~~~~~~~~~~~~~~~~
 983
 984First merges by themselves can introduce some regressions even when
 985the merge needs no source code conflict resolution. This is because a
 986semantic change can happen in one branch while the other branch is not
 987aware of it.
 988
 989For example one branch can change the semantic of a function while the
 990other branch add more calls to the same function.
 991
 992This is made much worse if many files have to be fixed to resolve
 993conflicts. That's why such merges are called "evil merges". They can
 994make regressions very difficult to track down. It can even be
 995misleading to know the first bad commit if it happens to be such a
 996merge, because people might think that the bug comes from bad conflict
 997resolution when it comes from a semantic change in one branch.
 998
 999Anyway "git rebase" can be used to linearize history. This can be used
1000either to avoid merging in the first place. Or it can be used to
1001bisect on a linear history instead of the non linear one, as this
1002should give more information in case of a semantic change in one
1003branch.
1004
1005Merges can be also made simpler by using smaller branches or by using
1006many topic branches instead of only long version related branches.
1007
1008And testing can be done more often in special integration branches
1009like linux-next for the linux kernel.
1010
1011Adapting your work-flow
1012~~~~~~~~~~~~~~~~~~~~~~~
1013
1014A special work-flow to process regressions can give great results.
1015
1016Here is an example of a work-flow used by Andreas Ericsson:
1017
1018* write, in the test suite, a test script that exposes the regression
1019* use "git bisect run" to find the commit that introduced it
1020* fix the bug that is often made obvious by the previous step
1021* commit both the fix and the test script (and if needed more tests)
1022
1023And here is what Andreas said about this work-flow <<5>>:
1024
1025_____________
1026To give some hard figures, we used to have an average report-to-fix
1027cycle of 142.6 hours (according to our somewhat weird bug-tracker
1028which just measures wall-clock time). Since we moved to Git, we've
1029lowered that to 16.2 hours. Primarily because we can stay on top of
1030the bug fixing now, and because everyone's jockeying to get to fix
1031bugs (we're quite proud of how lazy we are to let Git find the bugs
1032for us). Each new release results in ~40% fewer bugs (almost certainly
1033due to how we now feel about writing tests).
1034_____________
1035
1036Clearly this work-flow uses the virtuous circle between test suites
1037and "git bisect". In fact it makes it the standard procedure to deal
1038with regression.
1039
1040In other messages Andreas says that they also use the "best practices"
1041described above: small logical commits, topic branches, no evil
1042merge,... These practices all improve the bisectability of the commit
1043graph, by making it easier and more useful to bisect.
1044
1045So a good work-flow should be designed around the above points. That
1046is making bisecting easier, more useful and standard.
1047
1048Involving QA people and if possible end users
1049~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1050
1051One nice about "git bisect" is that it is not only a developer
1052tool. It can effectively be used by QA people or even end users (if
1053they have access to the source code or if they can get access to all
1054the builds).
1055
1056There was a discussion at one point on the linux kernel mailing list
1057of whether it was ok to always ask end user to bisect, and very good
1058points were made to support the point of view that it is ok.
1059
1060For example David Miller wrote <<6>>:
1061
1062_____________
1063What people don't get is that this is a situation where the "end node
1064principle" applies. When you have limited resources (here: developers)
1065you don't push the bulk of the burden upon them. Instead you push
1066things out to the resource you have a lot of, the end nodes (here:
1067users), so that the situation actually scales.
1068_____________
1069
1070This means that it is often "cheaper" if QA people or end users can do
1071it.
1072
1073What is interesting too is that end users that are reporting bugs (or
1074QA people that reproduced a bug) have access to the environment where
1075the bug happens. So they can often more easily reproduce a
1076regression. And if they can bisect, then more information will be
1077extracted from the environment where the bug happens, which means that
1078it will be easier to understand and then fix the bug.
1079
1080For open source projects it can be a good way to get more useful
1081contributions from end users, and to introduce them to QA and
1082development activities.
1083
1084Using complex scripts
1085~~~~~~~~~~~~~~~~~~~~~
1086
1087In some cases like for kernel development it can be worth developing
1088complex scripts to be able to fully automate bisecting.
1089
1090Here is what Ingo Molnar says about that <<7>>:
1091
1092_____________
1093i have a fully automated bootup-hang bisection script. It is based on
1094"git-bisect run". I run the script, it builds and boots kernels fully
1095automatically, and when the bootup fails (the script notices that via
1096the serial log, which it continuously watches - or via a timeout, if
1097the system does not come up within 10 minutes it's a "bad" kernel),
1098the script raises my attention via a beep and i power cycle the test
1099box. (yeah, i should make use of a managed power outlet to 100%
1100automate it)
1101_____________
1102
1103Combining test suites, git bisect and other systems together
1104~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
1105
1106We have seen that test suites an git bisect are very powerful when
1107used together. It can be even more powerful if you can combine them
1108with other systems.
1109
1110For example some test suites could be run automatically at night with
1111some unusual (or even random) configurations. And if a regression is
1112found by a test suite, then "git bisect" can be automatically
1113launched, and its result can be emailed to the author of the first bad
1114commit found by "git bisect", and perhaps other people too. And a new
1115entry in the bug tracking system could be automatically created too.
1116
1117
1118The future of bisecting
1119-----------------------
1120
1121"git replace"
1122~~~~~~~~~~~~~
1123
1124We saw earlier that "git bisect skip" is now using a PRNG to try to
1125avoid areas in the commit graph where commits are untestable. The
1126problem is that sometimes the first bad commit will be in an
1127untestable area.
1128
1129To simplify the discussion we will suppose that the untestable area is
1130a simple string of commits and that it was created by a breakage
1131introduced by one commit (let's call it BBC for bisect breaking
1132commit) and later fixed by another one (let's call it BFC for bisect
1133fixing commit).
1134
1135For example:
1136
1137-------------
1138...-Y-BBC-X1-X2-X3-X4-X5-X6-BFC-Z-...
1139-------------
1140
1141where we know that Y is good and BFC is bad, and where BBC and X1 to
1142X6 are untestable.
1143
1144In this case if you are bisecting manually, what you can do is create
1145a special branch that starts just before the BBC. The first commit in
1146this branch should be the BBC with the BFC squashed into it. And the
1147other commits in the branch should be the commits between BBC and BFC
1148rebased on the first commit of the branch and then the commit after
1149BFC also rebased on.
1150
1151For example:
1152
1153-------------
1154      (BBC+BFC)-X1'-X2'-X3'-X4'-X5'-X6'-Z'
1155     /
1156...-Y-BBC-X1-X2-X3-X4-X5-X6-BFC-Z-...
1157-------------
1158
1159where commits quoted with ' have been rebased.
1160
1161You can easily create such a branch with Git using interactive rebase.
1162
1163For example using:
1164
1165-------------
1166$ git rebase -i Y Z
1167-------------
1168
1169and then moving BFC after BBC and squashing it.
1170
1171After that you can start bisecting as usual in the new branch and you
1172should eventually find the first bad commit.
1173
1174For example:
1175
1176-------------
1177$ git bisect start Z' Y
1178-------------
1179
1180If you are using "git bisect run", you can use the same manual fix up
1181as above, and then start another "git bisect run" in the special
1182branch. Or as the "git bisect" man page says, the script passed to
1183"git bisect run" can apply a patch before it compiles and test the
1184software <<8>>. The patch should turn a current untestable commits
1185into a testable one. So the testing will result in "good" or "bad" and
1186"git bisect" will be able to find the first bad commit. And the script
1187should not forget to remove the patch once the testing is done before
1188exiting from the script.
1189
1190(Note that instead of a patch you can use "git cherry-pick BFC" to
1191apply the fix, and in this case you should use "git reset --hard
1192HEAD^" to revert the cherry-pick after testing and before returning
1193from the script.)
1194
1195But the above ways to work around untestable areas are a little bit
1196clunky. Using special branches is nice because these branches can be
1197shared by developers like usual branches, but the risk is that people
1198will get many such branches. And it disrupts the normal "git bisect"
1199work-flow. So, if you want to use "git bisect run" completely
1200automatically, you have to add special code in your script to restart
1201bisection in the special branches.
1202
1203Anyway one can notice in the above special branch example that the Z'
1204and Z commits should point to the same source code state (the same
1205"tree" in git parlance). That's because Z' result from applying the
1206same changes as Z just in a slightly different order.
1207
1208So if we could just "replace" Z by Z' when we bisect, then we would
1209not need to add anything to a script. It would just work for anyone in
1210the project sharing the special branches and the replacements.
1211
1212With the example above that would give:
1213
1214-------------
1215      (BBC+BFC)-X1'-X2'-X3'-X4'-X5'-X6'-Z'-...
1216     /
1217...-Y-BBC-X1-X2-X3-X4-X5-X6-BFC-Z
1218-------------
1219
1220That's why the "git replace" command was created. Technically it
1221stores replacements "refs" in the "refs/replace/" hierarchy. These
1222"refs" are like branches (that are stored in "refs/heads/") or tags
1223(that are stored in "refs/tags"), and that means that they can
1224automatically be shared like branches or tags among developers.
1225
1226"git replace" is a very powerful mechanism. It can be used to fix
1227commits in already released history, for example to change the commit
1228message or the author. And it can also be used instead of git "grafts"
1229to link a repository with another old repository.
1230
1231In fact it's this last feature that "sold" it to the Git community, so
1232it is now in the "master" branch of Git's Git repository and it should
1233be released in Git 1.6.5 in October or November 2009.
1234
1235One problem with "git replace" is that currently it stores all the
1236replacements refs in "refs/replace/", but it would be perhaps better
1237if the replacement refs that are useful only for bisecting would be in
1238"refs/replace/bisect/". This way the replacement refs could be used
1239only for bisecting, while other refs directly in "refs/replace/" would
1240be used nearly all the time.
1241
1242Bisecting sporadic bugs
1243~~~~~~~~~~~~~~~~~~~~~~~
1244
1245Another possible improvement to "git bisect" would be to optionally
1246add some redundancy to the tests performed so that it would be more
1247reliable when tracking sporadic bugs.
1248
1249This has been requested by some kernel developers because some bugs
1250called sporadic bugs do not appear in all the kernel builds because
1251they are very dependent on the compiler output.
1252
1253The idea is that every 3 test for example, "git bisect" could ask the
1254user to test a commit that has already been found to be "good" or
1255"bad" (because one of its descendants or one of its ancestors has been
1256found to be "good" or "bad" respectively). If it happens that a commit
1257has been previously incorrectly classified then the bisection can be
1258aborted early, hopefully before too many mistakes have been made. Then
1259the user will have to look at what happened and then restart the
1260bisection using a fixed bisect log.
1261
1262There is already a project called BBChop created by Ealdwulf Wuffinga
1263on Github that does something like that using Bayesian Search Theory
1264<<9>>:
1265
1266_____________
1267BBChop is like 'git bisect' (or equivalent), but works when your bug
1268is intermittent. That is, it works in the presence of false negatives
1269(when a version happens to work this time even though it contains the
1270bug). It assumes that there are no false positives (in principle, the
1271same approach would work, but adding it may be non-trivial).
1272_____________
1273
1274But BBChop is independent of any VCS and it would be easier for Git
1275users to have something integrated in Git.
1276
1277Conclusion
1278----------
1279
1280We have seen that regressions are an important problem, and that "git
1281bisect" has nice features that complement very well practices and
1282other tools, especially test suites, that are generally used to fight
1283regressions. But it might be needed to change some work-flows and
1284(bad) habits to get the most out of it.
1285
1286Some improvements to the algorithms inside "git bisect" are possible
1287and some new features could help in some cases, but overall "git
1288bisect" works already very well, is used a lot, and is already very
1289useful. To back up that last claim, let's give the final word to Ingo
1290Molnar when he was asked by the author how much time does he think
1291"git bisect" saves him when he uses it:
1292
1293_____________
1294a _lot_.
1295
1296About ten years ago did i do my first 'bisection' of a Linux patch
1297queue. That was prior the Git (and even prior the BitKeeper) days. I
1298literally days spent sorting out patches, creating what in essence
1299were standalone commits that i guessed to be related to that bug.
1300
1301It was a tool of absolute last resort. I'd rather spend days looking
1302at printk output than do a manual 'patch bisection'.
1303
1304With Git bisect it's a breeze: in the best case i can get a ~15 step
1305kernel bisection done in 20-30 minutes, in an automated way. Even with
1306manual help or when bisecting multiple, overlapping bugs, it's rarely
1307more than an hour.
1308
1309In fact it's invaluable because there are bugs i would never even
1310_try_ to debug if it wasn't for git bisect. In the past there were bug
1311patterns that were immediately hopeless for me to debug - at best i
1312could send the crash/bug signature to lkml and hope that someone else
1313can think of something.
1314
1315And even if a bisection fails today it tells us something valuable
1316about the bug: that it's non-deterministic - timing or kernel image
1317layout dependent.
1318
1319So git bisect is unconditional goodness - and feel free to quote that
1320;-)
1321_____________
1322
1323Acknowledgments
1324----------------
1325
1326Many thanks to Junio Hamano for his help in reviewing this paper, for
1327reviewing the patches I sent to the Git mailing list, for discussing
1328some ideas and helping me improve them, for improving "git bisect" a
1329lot and for his awesome work in maintaining and developing Git.
1330
1331Many thanks to Ingo Molnar for giving me very useful information that
1332appears in this paper, for commenting on this paper, for his
1333suggestions to improve "git bisect" and for evangelizing "git bisect"
1334on the linux kernel mailing lists.
1335
1336Many thanks to Linus Torvalds for inventing, developing and
1337evangelizing "git bisect", Git and Linux.
1338
1339Many thanks to the many other great people who helped one way or
1340another when I worked on Git, especially to Andreas Ericsson, Johannes
1341Schindelin, H. Peter Anvin, Daniel Barkalow, Bill Lear, John Hawley,
1342Shawn O. Pierce, Jeff King, Sam Vilain, Jon Seymour.
1343
1344Many thanks to the Linux-Kongress program committee for choosing the
1345author to given a talk and for publishing this paper.
1346
1347References
1348----------
1349
1350- [[[1]]] http://www.nist.gov/public_affairs/releases/n02-10.htm['Software Errors Cost U.S. Economy $59.5 Billion Annually'. Nist News Release.]
1351- [[[2]]] http://java.sun.com/docs/codeconv/html/CodeConventions.doc.html#16712['Code Conventions for the Java Programming Language'. Sun Microsystems.]
1352- [[[3]]] http://en.wikipedia.org/wiki/Software_maintenance['Software maintenance'. Wikipedia.]
1353- [[[4]]] http://article.gmane.org/gmane.comp.version-control.git/45195/[Junio C Hamano. 'Automated bisect success story'. Gmane.]
1354- [[[5]]] http://lwn.net/Articles/317154/[Christian Couder. 'Fully automated bisecting with "git bisect run"'. LWN.net.]
1355- [[[6]]] http://lwn.net/Articles/277872/[Jonathan Corbet. 'Bisection divides users and developers'. LWN.net.]
1356- [[[7]]] http://article.gmane.org/gmane.linux.scsi/36652/[Ingo Molnar. 'Re: BUG 2.6.23-rc3 can't see sd partitions on Alpha'. Gmane.]
1357- [[[8]]] http://www.kernel.org/pub/software/scm/git/docs/git-bisect.html[Junio C Hamano and the git-list. 'git-bisect(1) Manual Page'. Linux Kernel Archives.]
1358- [[[9]]] http://github.com/Ealdwulf/bbchop[Ealdwulf. 'bbchop'. GitHub.]