In computer software and hardware, find first set (ffs) or find first one is a bit operation that returns the index or position of the least significant 'one' bit in its argument, counting from the least significant bit. A related operation is count trailing zeros (ctz) or number of trailing zeros (ntz), which counts the number of zero bits following the least significant one bit. The operation that finds the index of the most significant set bit is log base 2, so called because it computes the binary logarithm ⌊log2(x)⌋.[1] This is closely related to count leading zeros (clz) or number of leading zeros (nlz), which counts the number of zero bits preceding the most significant one bit.[nb 1] There are two common variants of find first set, the POSIX definition which starts indexing of bits at 1,[2] herein labelled ffs, and the variant which starts indexing of bits at zero, which is equivalent to ctz.
Most modern CPU instruction set architectures provide one or more of these as hardware operators; software emulation is usually provided for the others, either as compiler intrinsics or in system libraries.
Examples
Given the following 32-bit word:
- 0000 0000 0000 0000 1000 0000 0000 1000
The count trailing zeros operation would return 3, while the count leading zeros operation returns 16. The count leading zeros operation depends on the word size: if this 32-bit word were truncated to a 16-bit word, count leading zeros would return zero. The find first set operation would return 4, indicating the 4th position from the right. The truncated log base 2 is 15.
Similarly, given the following 32-bit word, the bitwise negation of the above word:
- 1111 1111 1111 1111 0111 1111 1111 0111
The count trailing ones operation would return 3, the count leading ones operation would return 16, and the find first zero operation ffz would return 4.
If the word is zero (no bits set), count leading zeros and count trailing zeros both return the number of bits in the word, while ffs returns zero. Both log base 2 and zero-based implementations of find first set generally return an undefined result for the zero word.
Hardware support
Many instruction set architectures include find first set or related operations. The most common operation is count leading zeros (clz); other operations can be implemented efficiently in terms of it (see Properties and relations).
| Platform | Mnemonic | Name | Operand widths | Description | On application to 0 |
|---|---|---|---|---|---|
| ARM (ARMv5T architecture and later) except Cortex-M0/M0+/M1/M23 | clz[3] | Count Leading Zeros | 32 | clz | 32 |
| ARM (ARMv8-A architecture) | clz | Count Leading Zeros | 32, 64 | clz | Operand width |
| AVR32 | clz[4] | Count Leading Zeros | 32 | clz | 32 |
| DEC Alpha | ctlz[5] | Count Leading Zeros | 64 | clz | 64 |
| cttz[5] | Count Trailing Zeros | 64 | ctz | 64 | |
| Intel 80386 and later | bsf[6] | Bit Scan Forward | 16, 32, 64 | ctz | Undefined; sets zero flag |
| bsr[6] | Bit Scan Reverse | 16, 32, 64 | Log base 2 | Undefined; sets zero flag | |
| x86 supporting BMI1 or ABM | lzcnt[7] | Count Leading Zeros | 16, 32, 64 | clz | Operand width; sets carry flag |
| x86 supporting BMI1 | tzcnt[8] | Count Trailing Zeros | 16, 32, 64 | ctz | Operand width; sets carry flag |
| Itanium | clz[9] | Count Leading Zeros | 64 | clz | 64 |
| MIPS32/MIPS64 | clz[10][11] | Count Leading Zeros in Word | 32, 64 | clz | Operand width |
| clo[10][11] | Count Leading Ones in Word | 32, 64 | clo | Operand width | |
| Motorola 68020 and later | bfffo[12] | Find First One in Bit Field | Arbitrary | Log base 2 | Field offset + field width |
| PDP-10 | jffo | Jump if Find First One | 36 | clz | 0; no operation (i.e., jumps on nonzero) |
| POWER/PowerPC/Power ISA | cntlz/cntlzw/cntlzd[13] | Count Leading Zeros | 32, 64 | clz | Operand width |
| Power ISA 3.0 and later | cnttzw/cnttzd[14] | Count Trailing Zeros | 32, 64 | ctz | Operand width |
| RISC-V ("B" Extension) | clz[15] | Count Leading Zeros | 32, 64 | clz | Operand width |
| ctz[15] | Count Trailing Zeros | 32, 64 | ctz | Operand width | |
| SPARC Oracle Architecture 2011 and later | lzcnt (synonym: lzd)[16] | Leading Zero Count | 64 | clz | 64 |
| VAX | ffs[17] | Find First Set | 0–32 | ctz | Operand width; sets zero flag |
| IBM z/Architecture | flogr[18] | Find Leftmost One | 64 | clz | 64 |
| vclz[18] | Vector Count Leading Zeroes | 8, 16, 32, 64 | clz | Operand width | |
| vctz[18] | Vector Count Trailing Zeroes | 8, 16, 32, 64 | ctz | Operand width |
On some Alpha platforms CTLZ and CTTZ are emulated in software.
Tool and library support
A number of compiler and library vendors supply compiler intrinsics or library functions to perform find first set and/or related operations, which are frequently implemented in terms of the hardware instructions above:
| Tool/library | Name | Type | Input type(s) | Notes | On application to 0 |
|---|---|---|---|---|---|
| C23 standard library[19][20] | stdc_first_trailing_one[_uc,_us,_ui,_ul,_ull]stdc_trailing_zeros[_uc,_us,_ui,_ul,_ull]stdc_first_leading_one[_uc,_us,_ui,_ul,_ull]stdc_leading_zeros[_uc,_us,_ui,_ul,_ull] | Library function | unsigned char, unsigned short, unsigned int, unsigned long, unsigned long long | Functions for zero and ones are also provided. | 0 (bit index) Operand width (count) |
| C++20 standard library[21][22] | bit_ceil bit_floorbit_widthcountl_zero countl_onecountr_zero countr_one | Library function | unsigned char, unsigned short, unsigned int, unsigned long, unsigned long long | ||
| POSIX.1 compliant libc 4.3BSD libc OS X 10.3 libc[2][23] | ffs | Library function | int | Includes glibc. POSIX does not supply the complementary log base 2 / clz. | 0 |
| FreeBSD 5.3 libc OS X 10.4 libc[23] | ffslflsflsl | Library function | int, long | fls("find last set") computes (log base 2) + 1. | 0 |
| FreeBSD 7.1 libc[24] | ffsllflsll | Library function | long long | 0 | |
| GCC 3.4.0[25][26] Clang 5.x[27][28] | __builtin_ffs[l,ll,imax]__builtin_clz[l,ll,imax]__builtin_ctz[l,ll,imax] | Built-in functions | unsigned int, unsigned long, unsigned long long, uintmax_t | GCC documentation considers result undefined clz and ctz on 0. | 0 (ffs) |
| Visual Studio 2005 | _BitScanForward[29]_BitScanReverse[30] | Compiler intrinsics | unsigned long, unsigned __int64 | Separate return value to indicate zero input | Undefined |
| Visual Studio 2008 | __lzcnt[31] | Compiler intrinsic | unsigned short, unsigned int, unsigned __int64 | Relies on hardware support for the lzcnt instruction introduced in BMI1 or ABM. | Operand width |
| Visual Studio 2012 | _arm_clz[32] | Compiler intrinsic | unsigned int | Relies on hardware support for the clz instruction introduced in the ARMv5T architecture and later. | ? |
| Intel C++ Compiler | _bit_scan_forward_bit_scan_reverse[33][34] | Compiler intrinsics | int | Undefined | |
| Nvidia CUDA[35] | __clz | Functions | 32-bit, 64-bit | Compiles to fewer instructions on the GeForce 400 series | 32 |
__ffs | 0 | ||||
| LLVM | llvm.ctlz.*llvm.cttz.*[36] | Intrinsic | 8, 16, 32, 64, 256 | LLVM assembly language | Operand width, if 2nd argument is 0; undefined otherwise |
GHC 7.10 (base 4.8), in Data.Bits[citation needed] | countLeadingZeroscountTrailingZeros | Library function | FiniteBits b => b | Haskell programming language | Operand width |
| Rust standard library[37][38][39] | highest_oneleading_zeroslowest_onetrailing_zeros | Library method | All primitive integer types and other integer types such as NonZero<T> | highest_one and lowest_one are not yet stable. | |
| Zig[40][41] | @clz@ctz | builtin function | integer or vector |
Properties and relations
If bits are labeled starting at 1 (which is the convention used in this article), then count trailing zeros and find first set operations are related by ctz(x) = ffs(x) − 1 (except when the input is zero). If bits are labeled starting at 0, then count trailing zeros and find first set are exactly equivalent operations. Given w bits per word, the log2 is easily computed from the clz and vice versa by log2(x) = w − 1 − clz(x).
As demonstrated in the example above, the find first zero, count leading ones, and count trailing ones operations can be implemented by negating the input and using find first set, count leading zeros, and count trailing zeros. The reverse is also true.
On platforms with an efficient log2 operation such as M68000, ctz can be computed by:
- ctz(x) = log2(x & −x)
where & denotes bitwise AND and −x denotes the two's complement of x. The expression x & −x clears all but the least-significant 1 bit, so that the most- and least-significant 1 bit are the same.
On platforms with an efficient count leading zeros operation such as ARM and PowerPC, ffs can be computed by:
- ffs(x) = w − clz(x & −x).
Conversely, on machines without log2 or clz operators, clz can be computed using ctz, albeit inefficiently:
- clz = w − ctz(2⌈log2(x)⌉) (which depends on ctz returning w for the zero input)
On platforms with an efficient Hamming weight (population count) operation such as SPARC's POPC[42][43] or Blackfin's ONES,[44] there is:
- ctz(x) = popcount((x & −x) − 1),[45][46] or ctz(x) = popcount(~(x | −x)),
- ffs(x) = popcount(x ^ ~−x)[42]
- clz = 32 − popcount(2⌈log2(x)⌉ − 1)
where ^ denotes bitwise exclusive-OR, | denotes bitwise OR and ~ denotes bitwise negation.
The inverse problem (given i, produce an x such that ctz(x) = i) can be computed with a left-shift (1 << i).
Find first set and related operations can be extended to arbitrarily large bit arrays in a straightforward manner by starting at one end and proceeding until a word that is not all-zero (for ffs, ctz, clz) or not all-one (for ffz, clo, cto) is encountered. A tree data structure that recursively uses bitmaps to track which words are nonzero can accelerate this.
Software emulation
Most CPUs dating from the late 1980s onward have bit operators for ffs or equivalent, but a few modern ones like some of the ARM Cortex-M processors do not. In lieu of hardware operators for ffs, clz and ctz, software can emulate them with shifts, integer arithmetic and bitwise operators. There are several approaches depending on architecture of the CPU and to a lesser extent, the programming language semantics and compiler code generation quality. The approaches may be loosely described as linear search, binary search, search+table lookup, de Bruijn multiplication, floating point conversion/exponent extract, and bit operator (branchless) methods. There are tradeoffs between execution time and storage space as well as portability and efficiency.
Software emulations are usually deterministic. They return a defined result for all input values; in particular, the result for an input of all zero bits is usually 0 for ffs, and the bit length of the operand for the other operations.
If one has a hardware clz or equivalent, ctz can be efficiently computed with bit operations, but the converse is not true: clz is not efficient to compute in the absence of a hardware operator.
2n
The function 2⌈log2(x)⌉ (round up to the nearest power of two) using shifts and bitwise ORs[47] is not efficient to compute as in this 32-bit example and even more inefficient if we have a 64-bit or 128-bit operand:
function pow2(x):
if x = 0 return invalid // invalid is implementation defined (not in [0,63])
x ← x - 1
for each y in {1, 2, 4, 8, 16}: x ← x | (x >> y)
return x + 1
FFS
Since ffs = ctz + 1 (POSIX) or ffs = ctz (other implementations), the applicable algorithms for ctz may be used, with a possible final step of adding 1 to the result, and returning 0 instead of the operand length for input of all zero bits.
CTZ
The canonical algorithm is a loop counting zeros starting at the LSB until a 1-bit is encountered:
function ctz1 (x)
if x = 0 return w
t ← 1
r ← 0
while (x & t) = 0
t ← t << 1
r ← r + 1
return r
This algorithm executes O(w) time and operations, and is impractical in practice due to a large number of conditional branches.
An exception is if the inputs are uniformly distributed. In that case, we can rely on the fact that half the return values will be 0, one quarter will be 1, and so on. The average number of loop iterations per function call is 1, and the algorithm executes in O(1) average-case time.
A lookup table can eliminate most branches:
table[1..2n-1] = ctz(i) for i in 1..2n-1
function ctz2 (x)
if x = 0 return w
r ← 0
while (x & (2n−1)) ≠ 0
x ← x >> n
r ← r + n
return r + table[x & (2n−1)]
The parameter n is fixed (typically 8) and represents a time–space tradeoff. The loop may also be fully unrolled. But as a linear lookup, this approach is still O(n) in the number of bits in the operand.
If n = 4 is chosen, the table of 16 2-bit entries can be encoded in a single 32-bit constant using SIMD within a register techniques:
// binary 000100100001001100010010000100xx
table ← 0x12131210
function ctz2a (x)
if x = 0 return w
r ← 0
while (x & 15) = 0
x ← x >> 4
r ← r + 4
return r + ((table >> 2*(x & 15)) & 3);
This technique is quite practical in applications such as the binary GCD algorithm, where the x values are uniformly distributed, so the iterative loop is not needed 15⁄16 of the time, and suffers minimal branch misprediction penalty.
A binary search implementation takes a logarithmic number of operations and branches, as in this 32-bit version:[48][49]
function ctz3 (x)
if x = 0 return 32
n ← 0
if (x & 0x0000FFFF) = 0: n ← n + 16, x ← x >> 16
if (x & 0x000000FF) = 0: n ← n + 8, x ← x >> 8
if (x & 0x0000000F) = 0: n ← n + 4, x ← x >> 4
if (x & 0x00000003) = 0: n ← n + 2, x ← x >> 2
if (x & 0x00000001) = 0: n ← n + 1
// Equivalently, n ← n + 1 - (x & 1)
return n
This algorithm can be assisted by a table as well, replacing the last 2 or 3 if statements with a 16- or 256-entry lookup table using the least significant bits of x as an index.
As mentioned in § Properties and relations, if the hardware has a clz operator, the most efficient approach to computing ctz is thus:
function ctz4 (x)
if x = 0 return w
// Isolates the LSB
x ← x & −x
return w − 1 − clz(x)
A similar technique can take advantage of a population count instruction:
function ctz4a (x)
if x = 0 return w
// Makes a mask of the least-significant bits
x ← x ^ (x − 1)
return popcount(x) − 1
An algorithm for 32-bit ctz uses a de Bruijn sequence to construct a minimal perfect hash function that eliminates all branches.[50][51] This algorithm assumes that the result of the multiplication is truncated to 32 bit.
for i from 0 to 31: table[ 0x077CB531 << i >> 27 & 31 ] ← i // table [0..31] initialized
function ctz5 (x)
if x = 0 return 32
return table[((x & −x) * 0x077CB531) >> 27 & 31]
The expression (x & −x) again isolates the least-significant 1 bit. There are then only 32 possible words, which the unsigned multiplication and shift hash to the correct position in the table. This algorithm is branch-free if it does not need to handle the zero input.
The technique can be extended to 64-bit words.[52]
A minor variant uses the expression (x & (x−1)) to compute a mask of n trailing ones, as in the popcount method, and a different minimal perfect hash multiplier.[52] This allows the lookup table to be shared with a count leading zeros implementation.
This also permits a folded implementation using a half-width multiply.[53] After computing the mask, the bitwise xor of the top and bottom halves is sufficient to uniquely identify the original mask. A suitable multiplier produces a minimal perfect hash which may be decoded by a lookup table:
for i from 0 to 31: table[ ((0xffff0000 >> i) * 0x70a7) >> 11 & 31 ] ← 31 - i // table [0..31] initialized
function ctz5 (x)
if x = 0 return 32
x ← x ^ (x - 1)
x ← x ^ (x >> 16)
return table[(x * 0x70a7) >> 11 & 31] // 0x70d3 also works
This is advantageous if the narrower multiply saves more time than the folding step adds. The technique can also be extended to 64-bit words, using the 32-bit multipliers 0x783a9b23, 0x78291d9b, 0x782c8d4f, or 0x78291acf.[53]
CLZ
The canonical algorithm examines one bit at a time starting from the MSB until a non-zero bit is found, as shown in this example. It executes in O(n) time where n is the bit-length of the operand, and is not a practical algorithm for general use.
function clz1 (x)
if x = 0 return w
t ← 1 << (w - 1)
r ← 0
while (x & t) = 0
t ← t >> 1
r ← r + 1
return r
An improvement on the previous looping approach examines eight bits at a time then uses a 256 (28) entry lookup table for the first non-zero byte. This approach, however, is still O(n) in execution time.
function clz2 (x)
if x = 0 return w
t ← 0xff << (w - 8)
r ← 0
while (x & t) = 0
t ← t >> 8
r ← r + 8
return r + table[x >> (w - 8 - r)]
Binary search can reduce execution time to O(log2n):
function clz3 (x)
if x = 0 return 32
n ← 0
if (x & 0xFFFF0000) = 0: n ← n + 16, x ← x << 16
if (x & 0xFF000000) = 0: n ← n + 8, x ← x << 8
if (x & 0xF0000000) = 0: n ← n + 4, x ← x << 4
if (x & 0xC0000000) = 0: n ← n + 2, x ← x << 2
if (x & 0x80000000) = 0: n ← n + 1
return n
The fastest portable approaches to simulate clz are a combination of binary search and table lookup: an 8-bit table lookup (28=256 1-byte entries) can replace the bottom 3 branches in binary search. 64-bit operands require an additional branch. A larger width lookup can be used but the maximum practical table size is limited by the size of L1 data cache on modern processors. Saving a branch is more than offset by the latency of an L1 cache miss.
An algorithm similar to de Bruijn multiplication for CTZ works for CLZ, but rather than isolating the most-significant bit, it rounds up to the nearest integer of the form 2n−1 using shifts and bitwise ORs:[54]
table[0..31] = {0, 9, 1, 10, 13, 21, 2, 29, 11, 14, 16, 18, 22, 25, 3, 30,
8, 12, 20, 28, 15, 17, 24, 7, 19, 27, 23, 6, 26, 5, 4, 31}
function clz4 (x)
for each y in {1, 2, 4, 8, 16}: x ← x | (x >> y)
return table[((x * 0x07C4ACDD) >> 27) % 32]
For processors with deep pipelines, like Prescott and later Intel processors, it may be faster to replace branches by bitwise AND and OR operators (even though many more instructions are required) to avoid pipeline flushes for mispredicted branches (and these types of branches are inherently unpredictable):
function clz5 (x)
r = (x > 0xFFFF) << 4; x >>= r;
q = (x > 0xFF ) << 3; x >>= q; r |= q;
q = (x > 0xF ) << 2; x >>= q; r |= q;
q = (x > 0x3 ) << 1; x >>= q; r |= q;
r |= (x >> 1);
return r;
On platforms that provide hardware conversion of integers to floating point, the exponent field can be extracted and subtracted from a constant to compute the count of leading zeros. Corrections are needed to account for rounding errors.[48][55] Floating point conversion can have substantial latency. This method is highly non-portable and not usually recommended.
int x;
int r;
union { unsigned int u[2]; double d; } t;
t.u[LE] = 0x43300000; // LE is 1 for little-endian
t.u[!LE] = x;
t.d -= 4503599627370496.0;
r = (t.u[LE] >> 20) - 0x3FF; // log2
r++; // CLZ
Applications
The count leading zeros (clz) operation can be used to efficiently implement normalization, which encodes an integer as m × 2e, where m has its most significant bit in a known position (such as the highest position). This can in turn be used to implement Newton–Raphson division, perform integer to floating point conversion in software, and other applications.[48][56]
Count leading zeros (clz) can be used to compute the 32-bit predicate "x = y" (zero if true, one if false) via the identity clz(x − y) >> 5, where ">>" is unsigned right shift.[57] It can be used to perform more sophisticated bit operations like finding the first string of n 1 bits.[58] The expression 1 << (16 − clz(x − 1)/2) is an effective initial guess for computing the square root of a 32-bit integer using Newton's method.[59] CLZ can efficiently implement null suppression, a fast data compression technique that encodes an integer as the number of leading zero bytes together with the nonzero bytes.[60] It can also efficiently generate exponentially distributed integers by taking the clz of uniformly random integers.[48]
The log base 2 can be used to anticipate whether a multiplication will overflow, since ⌈log2(xy)⌉ ≤ ⌈log2(x)⌉ + ⌈log2(y)⌉.[61]
Count leading zeros and count trailing zeros can be used together to implement Gosper's loop-detection algorithm,[62] which can find the period of a function of finite range using limited resources.[49]
The binary GCD algorithm spends many cycles removing trailing zeros; this can be replaced by a count trailing zeros (ctz) followed by a shift. A similar loop appears in computations of the hailstone sequence.
A bit array can be used to implement a priority queue. In this context, find first set (ffs) is useful in implementing the "pop" or "pull highest priority element" operation efficiently. The Linux kernel real-time scheduler internally uses sched_find_first_bit() for this purpose.[63]
The count trailing zeros operation gives a simple optimal solution to the Tower of Hanoi problem: the disks are numbered from zero, and at move k, disk number ctz(k) is moved the minimum possible distance to the right (circling back around to the left as needed). It can also generate a Gray code by taking an arbitrary word and flipping bit ctz(k) at step k.[49]
See also
- Bit Manipulation Instruction Sets – Type of computer instructions
- Trailing zero
- Leading zero
- Trailing digit
- Leading digit
- Bit-length
Notes
- ↑ These four operations also have (much less common) negated versions:
- find first zero (ffz), which identifies the index of the least significant zero bit;
- count trailing ones, which counts the number of one bits following the least significant zero bit.
- count leading ones, which counts the number of one bits preceding the most significant zero bit;
- find the index of the most significant zero bit, which is an inverted version of the binary logarithm.
References
- ↑ Anderson. Find the log base 2 of an integer with the MSB N set in O(N) operations (the obvious way).
- 1 2 "FFS(3)". Linux Programmer's Manual. The Linux Kernel Archives. Retrieved 2012-01-02.
- ↑ "ARM Instruction Reference > ARM general data processing instructions > CLZ". ARM Developer Suite Assembler Guide. ARM. Retrieved 2012-01-03.
- ↑ "AVR32 Architecture Document" (PDF) (CORP072610 ed.). Atmel Corporation. 2011. 32000D–04/201. Archived from the original (PDF) on 2017-10-25. Retrieved 2016-10-22.
- 1 2 Alpha Architecture Reference Manual (PDF). Compaq. 2002. pp. 4-32, 4-34.
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- ↑ "AMD64 Architecture Programmer's Manual, Volume 3: General-Purpose and System Instructions" (PDF). AMD64 Technology (Version 3.28 ed.). Advanced Micro Devices (AMD). September 2019 [2013]. Publication No. 24594. Archived (PDF) from the original on 2019-09-30. Retrieved 2014-01-02.
- ↑ Intel Itanium Architecture Software Developer's Manual. Volume 3: Intel Itanium Instruction Set. Vol. 3. Intel. 2010. pp. 3:38. Archived from the original on 2019-06-26.
- 1 2 MIPS Architecture For Programmers. Volume II-A: The MIPS32 Instruction Set (Revision 3.02 ed.). MIPS Technologies. 2011. pp. 101–102. Archived from the original on 2017-11-07. Retrieved 2012-01-04.
- 1 2 MIPS Architecture For Programmers. Volume II-A: The MIPS64 Instruction Set (Revision 3.02 ed.). MIPS Technologies. 2011. pp. 105, 107, 122, 123. Archived from the original on 2017-11-07. Retrieved 2012-01-04.
- ↑ M68000 Family Programmer's Reference Manual (Includes CPU32 Instructions) (PDF) (revision 1 ed.). Motorola. 1992. pp. 4-43–4-45. M68000PRM/AD. Archived from the original (PDF) on 2019-12-08.
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- ↑ "Chapter 3.3.13 Fixed-Point Logical Instructions - Chapter 3.3.13.1 64-bit Fixed-Point Logical Instructions". Power ISA Version 3.0B. IBM. pp. 95, 98.
- 1 2 Wolf, Clifford (2019-03-22). "RISC-V "B" Bit Manipulation Extension for RISC-V" (PDF). Github (Draft) (v0.37 ed.). Retrieved 2020-01-09.
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- ↑ Meneide, JeanHeyd; Wiedijk, Freek (2024-02-22). Information technology — Programming languages — C, N3220 working draft (PDF). ISO/IEC. pp. 305–308. Retrieved 2025-08-07.
- ↑ "Standard library header <stdbit.h> (C23)". cppreference.com. Retrieved 2025-08-07.
- ↑ Smith, Richard (2020-04-01). N4861 Working Draft, Standard for Programming Language C++ (PDF). ISO/IEC. pp. 1150–1153. Retrieved 2020-05-25.
- ↑ "Standard library header <bit>". cppreference.com. Retrieved 2020-05-25.
- 1 2 "FFS(3)". Mac OS X Developer Library. Apple, Inc. 1994-04-19. Retrieved 2012-01-04.
- ↑ "FFS(3)". FreeBSD Library Functions Manual. The FreeBSD Project. Retrieved 2012-01-04.
- ↑ "Other built-in functions provided by GCC". Using the GNU Compiler Collection (GCC). Free Software Foundation, Inc. Retrieved 2015-11-14.
- ↑ "GCC 3.4.0 ChangeLog". GCC 3.4.0. Free Software Foundation, Inc. Retrieved 2015-11-14.
- ↑ "Clang Language Extensions - chapter Builtin Functions". The Clang Team. Retrieved 2017-04-09.
Clang supports a number of builtin library functions with the same syntax as GCC
- ↑ "Source code of Clang". LLVM Team, University of Illinois at Urbana-Champaign. Retrieved 2017-04-09.
- ↑ "_BitScanForward, _BitScanForward64". Visual Studio 2008: Visual C++: Compiler Intrinsics. Microsoft. 2012-11-16. Retrieved 2018-05-21.
- ↑ "_BitScanReverse, _BitScanReverse64". Visual Studio 2008: Visual C++: Compiler Intrinsics. Microsoft. 2012-11-16. Retrieved 2018-05-21.
- ↑ "__lzcnt16, __lzcnt, __lzcnt64". Visual Studio 2008: Visual C++: Compiler Intrinsics. Microsoft. Retrieved 2012-01-03.
- ↑ "ARM intrinsics". Visual Studio 2012: Visual C++: Compiler Intrinsics. Microsoft. 2012-08-20. Retrieved 2022-05-09.
- ↑ "Intel Intrinsics Guide". Intel. Retrieved 2020-04-03.
- ↑ Intel C++ Compiler for Linux Intrinsics Reference. Intel. 2006. p. 21.
- ↑ NVIDIA CUDA Programming Guide (PDF) (Version 3.0 ed.). NVIDIA. 2010. p. 92.
- ↑ "'llvm.ctlz.*' Intrinsic, 'llvm.cttz.*' Intrinsic". LLVM Language Reference Manual. The LLVM Compiler Infrastructure. Retrieved 2016-02-23.
- ↑ "i32 - Rust". doc.rust-lang.org. Retrieved 2025-11-18.
- ↑ "u32 - Rust". doc.rust-lang.org. Retrieved 2025-11-18.
- ↑ "NonZero in std::num - Rust". doc.rust-lang.org. Retrieved 2025-11-18.
- ↑ "Documentation - The Zig Programming Language". ziglang.org. Retrieved 2025-11-18.
- ↑ "Documentation - The Zig Programming Language". ziglang.org. Retrieved 2025-11-18.
- 1 2 SPARC International, Inc. (1992). "A.41: Population Count. Programming Note". The SPARC architecture manual: version 9 (PDF) (Version 9 ed.). Englewood Cliffs, New Jersey, USA: Prentice Hall. pp. 205. ISBN 978-0-13-825001-0.
- ↑ Warren, Jr., Henry S. (2013) [2002]. Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. ISBN 978-0-321-84268-8. 0-321-84268-5.
- ↑ Blackfin Instruction Set Reference (Preliminary ed.). Analog Devices. 2001. pp. 8–24. Part Number 82-000410-14.
- ↑ Dietz, Henry Gordon. "The Aggregate Magic Algorithms". University of Kentucky. Archived from the original on 2019-10-31.
- ↑ Isenberg, Gerd (2019-11-03) [2012]. "BitScan: Index of LS1B by Popcount". Chess Programming Wiki (CPW). Archived from the original on 2020-01-09. Retrieved 2020-01-09.
- ↑ Anderson. Round up to the next highest power of 2.
- 1 2 3 4 Warren. Chapter 5-3: Counting Leading 0's.
- 1 2 3 Warren. Chapter 5-4: Counting Trailing 0's.
- ↑ Leiserson, Charles E.; Prokop, Harald; Randall, Keith H. (1998-07-07). "Using de Bruijn Sequences to Index a 1 in a Computer Word" (PDF). MIT Laboratory for Computer Science, Cambridge, MA, USA. Archived (PDF) from the original on 2020-01-09. Retrieved 2020-01-09.
- ↑ Busch, Philip (2009-03-01) [2009-02-21]. "Computing Trailing Zeros HOWTO" (PDF). Archived (PDF) from the original on 2016-08-01. Retrieved 2020-01-09.
- 1 2 Isenberg, Gerd (2019-11-03) [2012]. "BitScan: De Bruijn Multiplication". Chess Programming Wiki (CPW). Archived from the original on 2020-01-09. Retrieved 2020-01-09.
- 1 2 Isenberg, Gerd (2019-11-03) [2012]. "BitScan: Matt Taylor's Folding trick". Chess Programming Wiki (CPW). Archived from the original on 2020-01-09. Retrieved 2026-04-17.
- ↑ Anderson. Find the log base 2 of an N-bit integer in O(lg(N)) operations with multiply and lookup.
- ↑ Anderson. Find the integer log base 2 of an integer with a 64-bit IEEE float.
- ↑ Sloss, Andrew N.; Symes, Dominic; Wright, Chris (2004). ARM system developer's guide designing and optimizing system software (1 ed.). San Francisco, CA, USA: Morgan Kaufmann. pp. 212–213. ISBN 978-1-55860-874-0.
- ↑ Warren. Chapter 2-11: Comparison Predicates.
- ↑ Warren. Chapter 6-2: Find First String of 1-Bits of a Given Length.
- ↑ Warren. Chapter 11-1: Integer Square Root.
- ↑ Schlegel, Benjamin; Gemulla, Rainer; Lehner, Wolfgang [in German] (June 2010). "Fast integer compression using SIMD instructions". Proceedings of the Sixth International Workshop on Data Management on New Hardware. pp. 34–40. doi:10.1145/1869389.1869394. ISBN 978-1-45030189-3. S2CID 7545142.
- ↑ Warren. Chapter 2-12: Overflow Detection.
- ↑ Gosper, Bill (April 1995) [1972-02-29]. Baker, Henry Givens Jr. (ed.). "Loop detector". HAKMEM (retyped & converted ed.). Cambridge, Massachusetts, USA: Artificial Intelligence Laboratory, Massachusetts Institute of Technology (MIT). AI Memo 239 Item 132. Archived from the original on 2019-10-08. Retrieved 2020-01-09.
- ↑ Aas, Josh (2005-02-17). Understanding the Linux 2.6.8.1 CPU Scheduler (PDF). Silicon Graphics, Inc. (SGI). p. 19. Archived (PDF) from the original on 2017-05-19. Retrieved 2020-01-09.
Further reading
- Warren, Jr., Henry S. (2013) [2002]. Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. ISBN 978-0-321-84268-8. 0-321-84268-5.
- Anderson, Sean Eron (2005) [1997]. "Bit Twiddling Hacks". Stanford University. Archived from the original on 2020-01-08. Retrieved 2012-01-03. (NB. Lists several efficient public domain C implementations for count trailing zeros and log base 2.)
External links
- Intel Intrinsics Guide
- Chess Programming Wiki: BitScan: A detailed explanation of a number of implementation methods for ffs (the index of the least significant 1 bit "LS1B") and log base 2 (the index of the most significant 1 bit "MS1B") of 64-bit values.