Find first set

In software, find first set (ffs) or find first one is a bit operation that, given an unsigned machine word, identifies the least significant index or position of the bit set to one in the word. A nearly equivalent 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 complementary operation that finds the index or position of the most significant set bit is log base 2, so called because it computes the binary logarithm .^[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. These four operations also have 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;
• The operation that finds the index of the most significant zero bit, which is a rounded version of the binary logarithm.

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 and so will be called by that name.

Examples

Given the following 32-bit word:

00000000000000001000000000001000

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 log base 2 is 15.

Similarly, given the following 32-bit word, the bitwise negation of the above word:

11111111111111110111111111110111

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 architectures include instructions to rapidly perform find first set and/or related operations, listed below. The most common operation is count leading zeros (clz), likely because all other operations can be implemented efficiently in terms of it (see Properties and relations).

Notes: 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:

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 log base 2 is easily computed from the clz and vice versa by lg(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 log base 2 operation such as M68000, ctz can be computed by:

ctz(x) = lg(x & (−x))

where "&" denotes bitwise AND and "−x" denotes the negative of x treating x as a signed integer in two's complement arithmetic. 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, clz can be computed using ctz by first rounding up to the nearest power of two using shifts and bitwise ORs,^[34] as in this 32-bit example (note that this example depends on ctz returning 32 for the zero input):

function clz(x):
    for each y in {1, 2, 4, 8, 16}: x ← x | (x >> y)
    return 32 − ctz(x + 1)

On platforms with an efficient Hamming weight (population count) operation such as SPARC's POPC^[35][36] or Blackfin's ONES,^[37] ctz can be computed using the identity:^[38][39]

ctz(x) = pop((x & (−x)) − 1),

ffs can be computed using:^[40]

ffs(x) = pop(x ^ (~(−x)))

where "^" denotes bitwise xor, and clz can be computed by:

function clz(x):
    for each y in {1, 2, 4, 8, 16}: x ← x | (x >> y)
    return 32 − pop(x)

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.

Algorithms

function ffs (x)
    if x = 0 return 0
    t ← 1
    r ← 1
    while (x & t) = 0
        t ← t << 1
        r ← r + 1
    return r

table[0..2n-1] = ffs(i) for i in 0..2n-1
function ffs_table (x)
    if x = 0 return 0
    r ← 0
    loop
        if (x & (2n-1)) ≠ 0
            return r + table[x & (2n-1)]
        x ← x >> n
        r ← r + n

table[0..31] initialized by: for i from 0 to 31: table[ ( 0x077CB531 * ( 1 << i ) ) >> 27 ] ← i
function ctz_debruijn (x)
    return table[((x & (-x)) * 0x077CB531) >> 27]

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 lg_debruijn (x)
    for each y in {1, 2, 4, 8, 16}: x ← x | (x >> y)
    return table[(x * 0x07C4ACDD) >> 27]

function ctz (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
    return n

/* uint8_t is a 8-bit unsigned integer, uint16_t is a 16-bit unsigned integer, uint32_t is a 32-bit unsigned integer */
#include <stdint.h> /* exists in C99 compatible C/C++ compilers */

int clz1( uint32_t x )
{
  int n;
  if (x == 0) return 32;
  for (n = 0; ((x & 0x80000000) == 0); n++, x <<= 1);
  return n;
}

static const uint8_t clz_table_4bit[16] = { 4, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 };
int clz2( uint32_t x )
{
  int n;
  if (x == 0) return 32;
  for (n = 0; ((x & 0xF0000000) == 0); n += 4, x <<= 4);
  n += (int)clz_table_4bit[x >> (32-4)];
  return n;
}

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

static const uint8_t clz_table_4bit[16] = { 4, 3, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0 };
int clz4( uint32_t x )
{
  int n;
  if ((x & 0xFFFF0000) == 0) {n  = 16; x <<= 16;} else {n = 0;}
  if ((x & 0xFF000000) == 0) {n +=  8; x <<=  8;}
  if ((x & 0xF0000000) == 0) {n +=  4; x <<=  4;}
  n += (int)clz_table_4bit[x >> (32-4)];
  return n;
}

/* The table MUST be calculated before calling this function */
static uint8_t clz_table_16bit[65536];
int clz5( uint32_t x )
{
  if ((x & 0xFFFF0000) == 0)
    return (int)clz_table_16bit[x] + 16;
  else
    return (int)clz_table_16bit[x >> 16];
}

/* Note: Tables MUST be calculated before calling each function or macro that reads from it */

uint8_t clz_table_8bit[256];
int clz8f( uint8_t x )
{
  return (int)clz_table_8bit[x];
}
#define clz8d(x) (clz_table_8bit[x])

uint8_t clz_table_16bit[65536];
int clz16f( uint16_t x )
{
  return (int)clz_table_16bit[x];
}
#define clz16d(x) (clz_table_16bit[x])

uint8_t clz_table_32bit[4294967296]; /* conceptual, but 4GB array is not practical */
int clz32f( uint32_t x )
{
  return (int)clz_table_32bit[x];
}
#define clz32d(x) (clz_table_32bit[x])

Applications

The count leading zeros (clz) operation can be used to efficiently implement normalization, which encodes an integer as m × 2^e, 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.^[45][48]

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.^[49] It can be used to perform more sophisticated bit operations like finding the first string of n 1 bits.^[50] 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.^[51] 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.^[52] It can also efficiently generate exponentially distributed integers by taking the clz of uniformly random integers.^[45]

The log base 2 can be used to anticipate whether a multiplication will overflow, since .^[53]

Count leading zeros and count trailing zeros can be used together to implement Gosper's loop-detection algorithm,^[54] which can find the period of a function of finite range using limited resources.^[46]

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.^[55]

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.^[46]

See also

• Bit Manipulation Instruction Sets for Intel and AMD x86-based processors

References

Further reading

• Warren Jr., Henry S. (2013). Hacker's Delight (2 ed.). Addison Wesley - Pearson Education, Inc. ISBN 978-0-321-84268-8.
• Anderson, Sean Eron. "Bit Twiddling Hacks". Sean Eron Anderson student homepage. Stanford University. Retrieved 3 January 2012. (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 (called LS1B) and log base 2 (called MS1B).