<pre>
  BIP: 441
  Layer: Consensus (soft fork)
  Title: Restoration of disabled script (Tapleaf 0xC2)
  Authors: Rusty Russell <rusty@rustcorp.com.au>
           Julian Moik <julianmoik@gmail.com>
  Status: Draft
  Type: Specification
  Assigned: 2026-03-25
  License: BSD-3-Clause
  Discussion: https://groups.google.com/g/bitcoindev/c/GisTcPb8Jco/m/8znWcWwKAQAJ
  Version: 0.2.2
  Requires: 440
</pre>

==Introduction==

===Abstract===

This BIP introduces a new tapleaf version (0xc2) which restores Bitcoin script
to its pre-0.3.1 capability, relying on the Varops Budget in
[[bip-0440.mediawiki|BIP440]] to prevent the excessive
computational time which caused CVE-2010-5137.

In particular, this BIP:
* Reenables disabled opcodes.
* Increases the maximum stack object size from 520 bytes to 4,000,000 bytes.
* Introduces a total stack byte limit of 8,000,000 bytes.
* Increases the maximum total number of stack objects from 1,000 to 32,768.
* Removes the 32-bit size restriction on numerical values.
* Treats all numerical values as unsigned.

All opcodes are described in exact (painstaking) byte-by-byte operations, so
that their varops budget can be easily derived.  Note that this level of
detail is unnecessary to users of script, only being of interest to
implementers.

===Copyright===

This document is licensed under the 3-clause BSD license.

===Motivation===

Since Bitcoin v0.3.1 (addressing CVE-2010-5137), Bitcoin's scripting
capabilities have been significantly restricted to mitigate known
vulnerabilities related to excessive computational time and memory usage.
These early safeguards were necessary to prevent denial-of-service attacks and
ensure the stability and reliability of the Bitcoin network.

Unfortunately, these restrictions removed much of the ability for users to
control the exact spending conditions of their outputs, which has frustrated
the long-held ideal of programmable money without third-party trust.

==Execution of Tapscript 0xC2==

If a taproot leaf has a version of 0xc2, execution of opcodes is as defined
below.  All opcodes not explicitly defined here are treated exactly as defined
by [[bip-0342.mediawiki|BIP342]].

Validation of a script fails if:
* It exceeds the remaining varops budget for the transaction.
* Any stack element exceeds 4,000,000 bytes.
* The total size of all stack (and altstack) elements exceeds 8,000,000 bytes.
* The number of stack elements (including altstack elements) exceeds 32,768.

===Rationale===

There needs to be some limit on memory usage, to avoid a memory-based denial
of service.

Putting the entire transaction on the stack is a foreseeable use case, hence
using the block size (4MB) as a limit makes sense.  However, allowing 4MB
stack elements is a significant increase in memory requirements, so a total
limit of twice that many bytes (8MB) is introduced.  Many stack operations
require making at least one copy, so this allows such use.

Putting all outputs or inputs from the transaction on the stack as separate
elements requires as much stack capacity as there are inputs or outputs.  The
smallest possible input is 41 bytes (allowing almost 24,390 inputs), and the
smallest possible output is 9 bytes (allowing almost 111,111 outputs).
However, empty outputs are rare and not economically interesting.  Thus we
consider smallest non-OP_RETURN standard output script, which is P2WPKH at 22
bytes, giving a minimum output size of 31 bytes, allowing 32,258 outputs in a
maximally-sized transaction.

This makes 32,768 a reasonable upper limit for stack elements.

===SUCCESS Opcodes===

The following opcodes are renamed OP_SUCCESSx, and cause validation to
immediately succeed:

* OP_1NEGATE = OP_SUCCESS79
* OP_NEGATE = OP_SUCCESS143
* OP_ABS = OP_SUCCESS144<ref>Anthony Towns suggested this could become an
  opcode which normalized the value on the top of the stack by truncating any
  trailing zeroes.</ref>

====Rationale====

Negative numbers are not natively supported in 0xC2 Tapscript.  Arbitrary
precision makes them difficult to manipulate and negative values are not used
meaningfully in bitcoin transactions.

===Arbitrary-length Values, Endianness, and Normalization of Results===

The restoration of bit operations means that the little-endianness of stack
values is once more exposed to the Script author, if they mix them with
arithmetic operations.  The restoration of arbitrary-length values
additionally exposes the endianness to the implementation authors (who cannot
simply load stack entries into registers), and requires explicit consideration
when considering varops costs of operations.<ref>For example, removing
trailing bytes from a stack element is almost free, whereas removing bytes
from the front involves copying all remaining bytes.</ref>

Note that only arithmetic operations (those which treat operands as numbers)
normalize their results: bit and byte operations do not.<ref>Such
non-arithmetic operations can be used to operate on values such as preimages
or (with introspection) parts of transactions, where truncation of zeros would
be unexpected.  One could argue that even arithmetic operators should not
normalize, but that would be a gratuitous and surprising change.  Note that "0
OP_ADD" can always be used to cheaply normalize the top stack element.</ref>
Thus operations such as "0 OP_ADD" and "2 OP_MUL" will never result in a top
stack entry with a trailing zero byte, but "0 OP_OR" and "1 OP_UPSHIFT"
may.<ref>The original Bitcoin implementation had a similar operational split,
but OP_LSHIFT and OP_RSHIFT did normalize, which was almost a requirement
given that they also preserved the sign of the shifted operand</ref>

To be explicit, the following operations are defined as arithmetic and will
normalize their results:

* OP_1ADD
* OP_1SUB
* OP_2MUL
* OP_2DIV
* OP_ADD
* OP_SUB
* OP_MUL
* OP_DIV
* OP_MOD
* OP_MIN
* OP_MAX

===Non-Arithmetic Opcodes Dealing With Stack Numbers===

The following opcodes are redefined in 0xC2 Tapscript to read numbers from the
stack as arbitrary-length little-endian values (instead of CScriptNum):

# OP_CHECKLOCKTIMEVERIFY
# OP_CHECKSEQUENCEVERIFY
# OP_VERIFY
# OP_PICK
# OP_ROLL
# OP_IFDUP
# OP_CHECKSIGADD

For OP_CHECKLOCKTIMEVERIFY and OP_CHECKSEQUENCEVERIFY, the operand is decoded
and costed as an arbitrary-length unsigned integer.  However, the decoded
value MUST be less than 2<sup>32</sup>, because it is compared against the
32-bit nLockTime or nSequence transaction field.  OP_CHECKSEQUENCEVERIFY
performs this range check before disable-flag handling or BIP68 masking, so
any non-zero bits above bit 31 cause failure.

These opcodes are redefined in 0xC2 Tapscript to write numbers to the stack as
minimal-length little-endian values (instead of CScriptNum):

# OP_CHECKSIGADD
# OP_DEPTH
# OP_SIZE

In addition, the [[bip-0342.mediawiki#specification|BIP-342 success
requirement]] is modified to require a non-zero variable-length unsigned
integer value (not <code>CastToBool()</code>):

Previously:

``4. (ii) If the execution results in anything but exactly one element on the
stack which evaluates to true with <code>CastToBool()</code>, fail.``

Now:

``4. (ii) If the execution results in anything but exactly one element on the
stack which contains one or more non-zero bytes, fail.``

This final success check consumes varops budget as <code>wordspan(A) * 2</code>
(COMPARINGZERO), where <code>A</code> is the remaining stack element. If this
cost exceeds the remaining transaction varops budget, fail before performing
the non-zero-byte check.

===Enabled Opcodes===

Fifteen opcodes that were removed in v0.3.1 are re-enabled in 0xC2 Tapscript.

If there are fewer than the required number of stack elements, these opcodes
fail validation.  These are popped off the stack in right-to-left order,
i.e. <nowiki>[A B]</nowiki> means pop B off the stack, then pop A off the
stack.

See [[bip-0440.mediawiki|BIP440]] for the meaning of the
annotations in the varops cost field.

====Byte Lengths and Word Spans====

This BIP uses the <code>length(...)</code> and <code>wordspan(...)</code>
convention from [[bip-0440.mediawiki|BIP440]]: <code>length(X)</code> is the
script-visible byte length of stack element <code>X</code>, while
<code>wordspan(X)</code> is that length rounded up to the 64-bit word span used
for numeric and bit-vector work.  Some formulas intentionally mix the two when
an opcode performs both word-rounded interpretation and exact byte movement.

====Splice Opcodes====

{|
! Mnemonic
! Opcode
! Input Stack
! Description
! Definition
! Varops Cost
! Varops Reason
|-
|OP_CAT
|126
|<nowiki>[A B]</nowiki>
|Append B to A
|
# Pop operands off the stack.
# Append B to A.
# Push A onto the stack.
|(length(A) + length(B)) * 3
|COPYING
|-
|OP_SUBSTR
|127
|<nowiki>[A BEGIN LEN]</nowiki>
|Extract bytes BEGIN through BEGIN+LEN of A
|
# Pop operands off the stack.
# Remove BEGIN bytes from the front of A (all bytes if BEGIN is greater than length of A).
# If length(A) is greater than value(LEN), truncate A to length value(LEN).
# Push A onto the stack.
|(wordspan(LEN) + wordspan(BEGIN)) * 2 + MIN(Value of LEN, MAX(length(A) - Value of BEGIN, 0)) * 3
|LENGTHCONV + COPYING
|-
|OP_LEFT
|128
|<nowiki>[A OFFSET]</nowiki>
|Extract the left OFFSET bytes of A
|
# Pop operands off the stack.
# If length(A) is greater than value(OFFSET), truncate A to length value(OFFSET).
# Push A onto the stack.
|wordspan(OFFSET) * 2
|LENGTHCONV
|-
|OP_RIGHT
|129
|<nowiki>[A OFFSET]</nowiki>
|Extract the rightmost OFFSET bytes of A
|
# Pop operands off the stack.
# Convert OFFSET to a bounded length value.
# If value(OFFSET) is less than length(A), remove length(A) - value(OFFSET) bytes from the front of A.
# Otherwise leave A unchanged.
# Push A onto the stack.
|wordspan(OFFSET) * 2 + MIN(Value of OFFSET, length(A)) * 3
|LENGTHCONV + COPYING
|}

=====Rationale=====

OP_CAT may require a reallocation of A (hence, COPYING A) before appending B.

OP_SUBSTR may have to copy LEN bytes, but also needs to read its two numeric
operands.  LEN is limited to the length of the operand minus BEGIN.

OP_LEFT only needs to read its OFFSET operand (truncation is free), whereas
OP_RIGHT must copy the rightmost bytes, which depends on the bounded OFFSET
value.

====Bit Operation Opcodes====

{|
! Mnemonic
! Opcode
! Input Stack
! Description
! Definition
! Varops Cost
! Varops Reason
|-
|OP_INVERT
|131
|<nowiki>[A]</nowiki>
|Bitwise invert A
|
# Pop operands off the stack.
# For each byte in A, replace it with that byte bitwise XOR 0xFF (i.e. invert the bits)
# Push A onto the stack.
|wordspan(A) * 4
|OTHER
|-
|OP_AND
|132
|<nowiki>[A B]</nowiki>
|Binary AND of A and B
|
# Pop operands off the stack.
# If B is longer than A, swap B and A.
# For each byte in A (the longer operand): bitwise AND it with the equivalent byte in B (or 0 if past end of B)
# Push A onto the stack.
|(wordspan(A) + wordspan(B)) * 2
|OTHER + ZEROING
|-
|OP_OR
|133
|<nowiki>[A B]</nowiki>
|Binary OR of A and B
|
# Pop operands off the stack.
# If B is longer than A, swap B and A.
# For each byte in B (the shorter operand): bitwise OR it into the equivalent byte in A (altering A).
# Push A onto the stack.
|MIN(wordspan(A), wordspan(B)) * 4
|OTHER
|-
|OP_XOR
|134
|<nowiki>[A B]</nowiki>
|Binary exclusive-OR of A and B
|
# Pop operands off the stack.
# If B is longer than A, swap B and A.
# For each byte in B (the shorter operand): exclusive OR it into the equivalent byte in A (altering A).
# Push A onto the stack.
|MIN(wordspan(A), wordspan(B)) * 4
|OTHER
|}

=====Rationale=====

OP_AND, OP_OR and OP_XOR are assumed to fold the results into the longer of
the two operands.  This is an OTHER operation (i.e. cost is 4 per byte), but
OP_AND needs to do this until one operand is exhausted, and then zero the rest
(ZEROING, cost 2 per byte).  OP_OR and OP_XOR can stop processing the operands
as soon as the shorter operand is exhausted.

====Bitshift Opcodes====

Note that these are raw bitshifts, unlike the sign-preserving arithmetic
shifts in Bitcoin v0.3.0, and as such they also do not truncate trailing
zeroes from results: they are renamed OP_UPSHIFT (née OP_LSHIFT) and
OP_DOWNSHIFT (née OP_RSHIFT).

{|
! Mnemonic
! Opcode
! Input Stack
! Description
! Definition
! Varops Cost
! Varops Reason
|-
|OP_UPSHIFT
|152
|<nowiki>[A BITS]</nowiki>
|Move bits of A right by BITS (numerically increase)
|
# Pop operands off the stack.
# If A shifted by value(BITS) would exceed the individual stack limit, fail.
# If value(BITS) % 8 == 0: simply prepend value(BITS) / 8 zeroes to A.
# Otherwise: prepend (value(BITS) / 8) + 1 zeroes to A, then shift A *down* (8 - (value(BITS) % 8)) bits.
# Push A onto the stack.
|wordspan(BITS) * 2 + (Value of BITS) / 8 * 2 + length(A) * 3.  If BITS % 8 != 0, add wordspan(length(A) + (Value of BITS) / 8) * 4
|LENGTHCONV + ZEROING + COPYING. If BITS % 8 != 0, + OTHER over the shifted result length.
|-
|OP_DOWNSHIFT
|153
|<nowiki>[A BITS]</nowiki>
|Move bits of A left by BITS (numerically decrease)
|
# Pop operands off the stack.
# For BITOFF from 0 to (length(A)-1) * 8 - value(BITS):
## Copy each bit in A from BITOFF + value(BITS) to BITOFF.
# Truncate A to remove value(BITS) / 8 bytes from the end (or all bytes, if value(BITS) / 8 > length(A)).
# Push A onto the stack.
|wordspan(BITS) * 2 + MAX((length(A) - (Value of BITS) / 8), 0) * 3
|LENGTHCONV + COPYING
|}

=====Rationale=====

DOWNSHIFT needs to read the value of the second operand BITS.  It then needs
to move the remainder of A (the part after offset BITS/8 bytes).  In practice
this should be implemented in word-size chunks, not bit-by-bit!

UPSHIFT also needs to read BITS.  In general, it may need to reallocate
(copying A and zeroing out remaining words).  If not moving an exact number of
bytes (BITS % 8 != 0), another pass is needed to perform the bitshift.

OP_UPSHIFT can produce huge results, and so must be checked for limits prior
to evaluation.  It is also carefully defined to avoid reallocating twice
(reallocating to prepend bytes, then again to append a single byte) which has
the practical advantage of being able to share the same downward bitshift
routine as OP_DOWNSHIFT.

====Multiply and Divide Opcodes====

{|
! Mnemonic
! Opcode
! Input Stack
! Description
! Definition
! Varops Cost
! Varops Reason
|-
|OP_2MUL
|141
|<nowiki>[A]</nowiki>
|Multiply A by 2
|
# Pop operands off the stack.
# Shift each byte in A 1 bit to the left (increasing values, equivalent to C's << operator), overflowing into the next byte.
# If the final byte overflows, append a single 1 byte.
# Otherwise, truncate A at the last non-zero byte.
# Push A onto the stack.
|wordspan(A) * 7
|OTHER + COPYING
|-
|OP_2DIV
|142
|<nowiki>[A]</nowiki>
|Divide A by 2
|
# Pop operands off the stack.
# Shift each byte in A 1 bit to the right (decreasing values, equivalent to C's >> operator), taking the next byte’s bottom bit as the value of the top bit, and tracking the last non-zero value.
# Truncate A at the last non-zero byte.
# Push A onto the stack.
|wordspan(A) * 4
|OTHER
|-
|OP_MUL
|149
|<nowiki>[A B]</nowiki>
|Multiply A by B
|
# Pop operands off the stack.
# Calculate the varops cost of the operation: if it exceeds the remaining budget, fail.
# Allocate an all-zero vector R of length equal to length(A) + length(B).
# For each word in A, multiply it by B and add it into the vector R, offset by the word offset in A.
# Truncate R at the last non-zero byte.
# Push R onto the stack.
|(length(A) + length(B)) * 3 + wordspan(A) / 8 * wordspan(B) * 27  (BEWARE OVERFLOW)
|See Appendix
|-
|OP_DIV
|150
|<nowiki>[A B]</nowiki>
|Divide A by (non-zero) B
|
# Pop operands off the stack.
# Calculate the varops cost of the operation: if it exceeds the remaining budget, fail.
# If B is empty or all zeroes, fail.
# Perform division as per Knuth's The Art of Computer Programming v2 page 272, Algorithm D "Division of non-negative integers".
# Trim trailing zeroes off the quotient.
# Push the quotient onto the stack.
|wordspan(A) * 18 + wordspan(B) * 4 + wordspan(A)^2 * 2 / 3  (BEWARE OVERFLOW)
|See Appendix
|-
|OP_MOD
|151
|<nowiki>[A B]</nowiki>
|Replace A with remainder when A divided by (non-zero) B
|
# Pop operands off the stack.
# Calculate the varops cost of the operation: if it exceeds the remaining budget, fail.
# If B is empty or all zeroes, fail.
# Perform division as per Knuth's The Art of Computer Programming v2 page 272, Algorithm D "Division of non-negative integers".
# Trim trailing zeroes off the remainder.
# Push the remainder onto the stack.
|wordspan(A) * 18 + wordspan(B) * 4 + wordspan(A)^2 * 2 / 3  (BEWARE OVERFLOW)
|See Appendix
|}

=====Rationale=====

These opcodes can be computationally intensive, which is why the varops budget must be checked before operations.  OP_2MUL and OP_2DIV are far simpler, equivalent to OP_UPSHIFT and OP_DOWNSHIFT by 1 bit, except truncating the most-significant zero bytes.

The detailed rationale for these costs can be found in Appendix A.

===Limited Hashing Opcodes===

OP_RIPEMD160 and OP_SHA1 are now defined to FAIL validation if their operands exceed 520 bytes.<ref>There seems little reason to allow large hashing with SHA1 and RIPEMD, and they are not as optimized as SHA256, so we restrict their usage to the older byte limit.</ref>

===Extended Opcodes===

The opcodes OP_ADD, OP_SUB, OP_1ADD and OP_1SUB are redefined in 0xC2 Tapscript to operate on variable-length unsigned integers.  These always produce minimal values (no trailing zero bytes).

{|
! Mnemonic
! Opcode
! Input Stack
! Description
! Definition
! Varops Cost
! Varops Reason
|-
|OP_ADD
|147
|<nowiki>[A B]</nowiki>
|Add A and B
|
# Pop operands off the stack.
# Option 1: trim trailing zeroes off A and B.
# If B is longer than A, swap A and B.
# For each byte in B, add it and previous overflow into the equivalent byte in A, remembering next overflow.
# If there was final overflow, append a 1 byte to A.
# Option 2: If there was no final overflow, remember last non-zero byte written into A, and truncate A after that point.
# Either Option 1 or Option 2 MUST be implemented.
|MAX(wordspan(A), wordspan(B)) * 9
|ARITH + COPYING
|-
|OP_1ADD
|139
|<nowiki>[A]</nowiki>
|Add one to A
|
# Pop operands off the stack.
# Let B = 1, and continue as OP_ADD.
|MAX(wordspan(1), wordspan(A)) * 9
|ARITH + COPYING
|-
|OP_SUB
|148
|<nowiki>[A B]</nowiki>
|Subtract B from A where B is <= A
|
# Pop operands off the stack.
# For each byte in B, subtract it and previous underflow from the equivalent byte in A, remembering next underflow.
# If there was final underflow, fail validation.
# Remember last non-zero byte written into A, and truncate A after that point.
|MAX(wordspan(A), wordspan(B)) * 6
|ARITH
|-
|OP_1SUB
|140
|<nowiki>[A]</nowiki>
|Subtract 1 from (non-zero) A
|
# Pop operands off the stack.
# Let B = 1, and continue as OP_SUB.
|MAX(wordspan(1), wordspan(A)) * 6
|ARITH
|}

====Rationale====

Note that the basic cost for ADD is six times the maximum operand length
(ARITH), but then considers the case where a reallocation and copy needs to
occur to append the final carry byte (COPYING, which costs 3 units per byte).

Subtraction is cheaper because underflow does not occur: that is a validation
failure, as mathematicians agree the result would not be natural.

===Misc Operators===

The following opcodes have costs below:

{|
! Opcode
! Varops Budget Cost
! Varops Reason
|-
| OP_CHECKLOCKTIMEVERIFY
| wordspan(operand) * 2
| LENGTHCONV
|-
| OP_CHECKSEQUENCEVERIFY
| wordspan(operand) * 2
| LENGTHCONV
|-
| OP_CHECKSIGADD
| MAX(wordspan(1), wordspan(number operand)) * 9 + 500,000
| ARITH + COPYING + SIGCHECK
|-
| OP_CHECKSIG
| 500,000
| SIGCHECK
|-
| OP_CHECKSIGVERIFY
| 500,000
| SIGCHECK
|}

====Rationale====

OP_CHECKSIGADD does an OP_1ADD on success, so we use the same cost as that.
For simplicity, this is charged whether the OP_CHECKSIGADD succeeds or not.

===Other Operators===

The varops costs of the following opcodes are defined in
[[bip-0440.mediawiki|BIP440]]:

* OP_VERIFY
* OP_NOT
* OP_0NOTEQUAL
* OP_EQUAL
* OP_EQUALVERIFY
* OP_2DUP
* OP_3DUP
* OP_2OVER
* OP_IFDUP
* OP_DUP
* OP_OVER
* OP_PICK
* OP_TUCK
* OP_ROLL
* OP_BOOLOR
* OP_NUMEQUAL
* OP_NUMEQUALVERIFY
* OP_NUMNOTEQUAL
* OP_LESSTHAN
* OP_GREATERTHAN
* OP_LESSTHANOREQUAL
* OP_GREATERTHANOREQUAL
* OP_MIN
* OP_MAX
* OP_WITHIN
* OP_SHA256
* OP_HASH160
* OP_HASH256

Any opcodes not mentioned in this document or the preceding list have a cost
of 0 (they do not operate on variable-length stack objects).

==Backwards compatibility==

This BIP defines a previously unused (and thus, always-successful) tapscript
version, for backwards compatibility.

==Reference Implementation==

Work in progress:

	https://github.com/jmoik/bitcoin/tree/gsr

==Changelog==

* 0.2.2: 2026-06-15: clarify wordspan costs, OP_RIGHT semantics, OP_UPSHIFT unaligned cost, CLTV/CSV bounds, and final success-check cost.
* 0.2.1: 2023-03-27: fix OP_MUL cost to round length(B) up
* 0.2.0: 2025-02-21: change costs to match those in varops budget
* 0.1.0: 2025-09-27: first public posting

==Thanks==

This BIP would not exist without the thoughtful contributions of coders who
considered all the facets carefully and thoroughly, and also my inspirational
wife Alex and my kids who have been tirelessly supportive of my
esoteric-seeming endeavors such as this!

In alphabetical order:
* Anthony Towns
* Brandon Black (aka Reardencode)
* John Light
* Jonas Nick
* Mark "Murch" Erhardt 
* Rijndael (aka rot13maxi)
* Steven Roose
* FIXME: your name here!

==Appendix A: Cost Model Calculations for Multiply and Divide==

Multiplication and division require multiple passes over the operands, meaning
a cost proportional to the square of the lengths involved, and the word size
used for that iteration makes a difference.  We assume 8 bytes (64 bits) at a
time are evaluated, and the ability to multiply two 64-bit numbers and receive
a 128-bit result, and divide a 128-bit number by a 64-bit number to receive a
128-bit quotient and remainder.  This is true on modern 64-bit CPUs (sometimes
using multiple instructions).

===Multiplication Cost===

For multiplication, the steps break down like so:
# Allocate and zero the result: cost = (length(A) + length(B)) * 2 (ZEROING)
# For each word in A:
#* Multiply by each word in B, into a scratch vector: cost = 6 * wordspan(B) (ARITH)
#* Sum scratch vector at the word offset into the result: cost = 6 * wordspan(B) (ARITH)

We increase the operand lengths to the next word boundary for the word-span
loops, using <code>wordspan(n)</code> from [[bip-0440.mediawiki|BIP440]], as
the multiplication below makes the difference from the simple byte length
significant.

Note: we do not assume Karatsuba, Toom-Cook or other optimizations.

The theoretical cost is: (length(A) + length(B)) * 2 + wordspan(A) / 8 * wordspan(B) * 12.

However, benchmarking reveals that the inner loop overhead (branch
misprediction, cache effects on small elements) is undercosted by the
theoretical model.  A 2.25× multiplier on the quadratic term accounts for
this, giving a cost of: (length(A) + length(B)) * 3 + wordspan(A) / 8 *
wordspan(B) * 27.

This is slightly asymmetric: in practice an implementation usually finds that
CPU pipelining means choosing B as the larger operand is optimal.

===Division Cost===

For division, the steps break down like so:

# Bit shift both operands to set top bit of B (OP_UPSHIFT, without overflow for B): cost = wordspan(A) * 6 + wordspan(B) * 4

# Trim trailing bytes.  This costs according to the number of byte removed, but since that is subtractive on future costs, we ignore it.

# If B is longer, the answer is 0 already.  So assume A is longer from now on (or equal length).

# Compare: cost = wordspan(A) * 2 (COMPARING)

# Subtract: cost = wordspan(A) * 6 (ARITH)

# for (wordspan(A) - NormalizedLength(B)) in words:
## Multiply word by B -> scratch: cost = NormalizedLength(B) * 6 (ARITH)
## Subtract scratch from A: cost = wordspan(A) * 6 (ARITH)
## Add B into A (no overflow): cost = wordspan(A) * 6 (ARITH)
## Shrink A by 1 word.

# OP_MOD: shift A down, trim trailing zeroes: cost = wordspan(A) * 4

# OP_DIV: trim trailing zeros: cost = wordspan(A) * 4

Note that the loop at step 6 shrinks A every time, so the *average* cost of
each iteration is (NormalizedLength(B) * 6 + wordspan(A) * 12) / 2.  The cost of
step 6 is:

	(wordspan(A) - NormalizedLength(B)) / 8 * (NormalizedLength(B) * 6 + wordspan(A) * 12) / 2

The worst case is when NormalizedLength(B) is 0: wordspan(A) * wordspan(A) * 2 / 3.

The cost for all the steps is: wordspan(A) * 18 + wordspan(B) * 4 + wordspan(A) * wordspan(A) * 2 / 3.
