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65-552: NTZ may mean "Number of trailing zeros", see find first set NTZ Stadium, homeground of SKA-Sverdlovsk Natuzzi (NYSE ticker symbol) the ISO code of the Saudi Arabian–Iraqi neutral zone Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title NTZ . If an internal link led you here, you may wish to change

130-460: A 256 (2 ) entry lookup table for the first non-zero byte. This approach, however, is still O(n) in execution time. Binary search can reduce execution time to O(log 2 n): The fastest portable approaches to simulate clz are a combination of binary search and table lookup: an 8-bit table lookup (2 =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

195-876: A Report on the EDVAC proposal for an electronic stored-program digital computer. The 1949 EDSAC , which was inspired by the First Draft , used two's complement representation of negative binary integers. Many early computers, including the CDC 6600 , the LINC , the PDP-1 , and the UNIVAC 1107, use ones' complement notation; the descendants of the UNIVAC 1107, the UNIVAC 1100/2200 series , continued to do so. The IBM 700/7000 series scientific machines use sign/magnitude notation, except for

260-417: A binary representation), a two's complement for the number 3 ( 011 2 ) is 5 ( 101 2 ), because summed to the original it gives 2 = 1000 2 = 011 2 + 101 2 . Where this correspondence is employed for representing negative numbers, it effectively means, using an analogy with decimal digits and a number-space only allowing eight non-negative numbers 0 through 7, dividing the number-space in two sets:

325-454: 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. The count trailing zeros operation gives

390-406: 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. The function 2 (round up to

455-505: A given negative number in binary digits: For example, to calculate the decimal number −6 in binary from the number 6 : To verify that 1010 indeed has a value of −6 , add the place values together, but subtract the sign value from the final calculation. Because the most significant value is the sign value, it must be subtracted to produce the correct result: 1010 = − ( 1 ×2 ) + ( 0 ×2 ) + ( 1 ×2 ) + ( 0 ×2 ) = 1 ×−8 + 0 + 1 ×2 + 0 = −6. Note that steps 2 and 3 together are

520-467: 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. 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. It can be used to perform more sophisticated bit operations like finding

585-435: 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. Most CPUs dating from

650-407: A nonzero number equal to its own negation is forced by the fact that zero is its own negation, and that the total number of numbers is even. Proof: there are 2^n - 1 nonzero numbers (an odd number). Negation would partition the nonzero numbers into sets of size 2, but this would result in the set of nonzero numbers having even cardinality. So at least one of the sets has size 1, i.e., a nonzero number

715-407: A number to its two's complement results in the N lowest bits set to 0 and the carry bit 1, where the latter has the weight (reading it as an unsigned binary number) of 2 . Hence, in the unsigned binary arithmetic the value of two's-complement negative number x * of a positive x satisfies the equality x * = 2 − x . For example, to find the four-bit representation of −5 (subscripts denote

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780-410: A signed integer. Both shifting and doubling the precision are important for some multiplication algorithms. Note that unlike addition and subtraction, width extension and right shifting are done differently for signed and unsigned numbers. With only one exception, starting with any number in two's-complement representation, if all the bits are flipped and 1 added, the two's-complement representation of

845-610: 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 . Two%27s complement Two's complement is the most common method of representing signed (positive, negative, and zero) integers on computers, and more generally, fixed point binary values. Two's complement uses

910-405: A two's-complement number with a certain number of bits into one with more bits (e.g., when copying from a one-byte variable to a two-byte variable), the most-significant bit must be repeated in all the extra bits. Some processors do this in a single instruction; on other processors, a conditional must be used followed by code to set the relevant bits or bytes. Similarly, when a number is shifted to

975-512: A valid method to compute the additive inverse − n {\displaystyle -n} of any (positive or negative) integer n {\displaystyle n} where both input and output are in two's complement format. An alternative to compute − n {\displaystyle -n} is to use subtraction 0 − n {\displaystyle 0-n} . See below for subtraction of integers in two's complement format. Two's complement

1040-404: Is 1, so the value represented is negative. The two's complement of a negative number is the corresponding positive value, except in the special case of the most negative number . For example, inverting the bits of −5 (above) gives: And adding one gives the final value: Likewise, the two's complement of zero is zero: inverting gives all ones, and adding one changes the ones back to zeros (since

1105-464: Is a loop counting zeros starting at the LSB until a 1-bit is encountered: This algorithm executes O ( n ) time and operations, and is impractical in practice due to a large number of conditional branches. A lookup table can eliminate most branches: 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

1170-484: Is also used in computer science as another method of signed number representation and is called a Ones' complement (named that because summing such a number with the original gives the 'all 1s'). Compared to other systems for representing signed numbers ( e.g., ones' complement ), the two's complement has the advantage that the fundamental arithmetic operations of addition , subtraction , and multiplication are identical to those for unsigned binary numbers (as long as

1235-420: Is an example of a radix complement . The 'two' in the name refers to the term which, expanded fully in an N -bit system, is actually "two to the power of N" - 2 (the only case where exactly 'two' would be produced in this term is N = 1 , so for a 1-bit system, but these do not have capacity for both a sign and a zero), and it is only this full term in respect to which the complement is calculated. As such,

1300-404: Is an exception, it is a valid number in regular two's complement systems. All arithmetic operations work with it both as an operand and (unless there was an overflow) a result. Given a set of all possible N -bit values, we can assign the lower (by the binary value) half to be the integers from 0 to (2 − 1) inclusive and the upper half to be −2 to −1 inclusive. The upper half (again, by

1365-399: Is arbitrary, but by convention all negative numbers have a left-most bit ( most significant bit ) of one. Therefore, the most positive four-bit number is 0111 (7.) and the most negative is 1000 (−8.). Because of the use of the left-most bit as the sign bit, the absolute value of the most negative number (|−8.| = 8.) is too large to represent. Negating a two's complement number

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1430-494: Is easily computed from the clz and vice versa by log 2 ( 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 2 operation such as M68000, ctz can be computed by: where & denotes bitwise AND and −x denotes

1495-414: Is its own negation. The presence of the most negative number can lead to unexpected programming bugs where the result has an unexpected sign, or leads to an unexpected overflow exception, or leads to completely strange behaviors. For example, In the C and C++ programming languages, the above behaviours are undefined and not only may they return strange results, but the compiler is free to assume that

1560-405: Is simple: Invert all the bits and add one to the result. For example, negating 1111, we get 0000 + 1 = 1 . Therefore, 1111 in binary must represent −1 in decimal. The system is useful in simplifying the implementation of arithmetic on computer hardware. Adding 0011 (3.) to 1111 (−1.) at first seems to give the incorrect answer of 10010. However, the hardware can simply ignore

1625-434: Is still O(n) in the number of bits in the operand. A binary search implementation takes a logarithmic number of operations and branches, as in this 32-bit version: This algorithm can be assisted by a table as well, replacing the bottom three "if" statements with a 256 entry lookup table using the first non-zero byte encountered as an index. If the hardware has a clz operator, the most efficient approach to computing ctz

1690-444: Is the negative of the corresponding power of two. The value  w of an N -bit integer a N − 1 a N − 2 … a 0 {\displaystyle a_{N-1}a_{N-2}\dots a_{0}} is given by the following formula: The most significant bit determines the sign of the number and is sometimes called the sign bit . Unlike in sign-and-magnitude representation,

1755-419: Is thus: An algorithm for 32-bit ctz uses de Bruijn sequences to construct a minimal perfect hash function that eliminates all branches. This algorithm assumes that the result of the multiplication is truncated to 32 bit. 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

1820-456: Is usually provided for any that aren't available, either as compiler intrinsics or in system libraries . Given the following 32-bit word: 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

1885-420: The base of the representation ): Hence, with N = 4 : The calculation can be done entirely in base 10, converting to base 2 at the end: A shortcut to manually convert a binary number into its two's complement is to start at the least significant bit (LSB), and copy all the zeros, working from LSB toward the most significant bit (MSB) until the first 1 is reached; then copy that 1, and flip all

1950-399: The binary digit with the greatest value as the sign to indicate whether the binary number is positive or negative; when the most significant bit is 1 the number is signed as negative and when the most significant bit is 0 the number is signed as positive. As a result, non-negative numbers are represented as themselves: 6 is 0110, zero is 0000, and -6 is 1010 (~6 + 1). Note that while

2015-754: 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: Conversely, on machines without log 2 or clz operators, clz can be computed using ctz , albeit inefficiently: On platforms with an efficient Hamming weight (population count) operation such as SPARC 's POPC or Blackfin 's ONES , there is: 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

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2080-400: 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: 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

2145-474: The binary two's complement of a positive number essentially means subtracting the number from the 2 . But as can be seen for the three-bit example and the four-bit 1000 2 ( 2 ), the number 2 will not itself be representable in a system limited to N bits, as it is just outside the N bits space (the number is nevertheless the reference point of the "Two's complement" in an N -bit system). Because of this, systems with maximally N -bits must break

2210-424: The binary value) can be used to represent negative integers from −2 to −1 because, under addition modulo 2 they behave the same way as those negative integers. That is to say that, because i + j mod 2 = i + ( j + 2 ) mod 2 , any value in the set {  j + k  2 | k is an integer }  can be used in place of  j . For example, with eight bits, the unsigned bytes are 0 to 255. Subtracting 256 from

2275-478: The clz of uniformly random integers. The log base 2 can be used to anticipate whether a multiplication will overflow, since ⌈log 2 (xy)⌉ ≤ ⌈log 2 (x)⌉ + ⌈log 2 (y)⌉ . Count leading zeros and count trailing zeros can be used together to implement Gosper's loop-detection algorithm , which can find the period of a function of finite range using limited resources. The binary GCD algorithm spends many cycles removing trailing zeros; this can be replaced by

2340-480: The computer industry. The first minicomputer, the PDP-8 introduced in 1965, uses two's complement arithmetic, as do the 1969 Data General Nova , the 1970 PDP-11 , and almost all subsequent minicomputers and microcomputers. A two's-complement number system encodes positive and negative numbers in a binary number representation. The weight of each bit is a power of two, except for the most significant bit , whose weight

2405-444: The desired property that the sign of integers can be reversed by taking the complement of its binary representation, but two's complement has an exception - the lowest negative, as can be seen in the tables. The method of complements had long been used to perform subtraction in decimal adding machines and mechanical calculators . John von Neumann suggested use of two's complement binary representation in his 1945 First Draft of

2470-468: 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. Floating point conversion can have substantial latency. This method is highly non-portable and not usually recommended. The count leading zeros (clz) operation can be used to efficiently implement normalization , which encodes an integer as m  × 2 , where m has its most significant bit in

2535-485: The first four of the numbers 0 1 2 3 remain the same, while the remaining four encode negative numbers, maintaining their growing order, so making 4 encode -4, 5 encode -3, 6 encode -2 and 7 encode -1. A binary representation has an additional utility however, because the most significant bit also indicates the group (and the sign): it is 0 for the first group of non-negatives, and 1 for the second group of negatives. The tables at right illustrate this property. Calculation of

2600-478: The first string of n 1 bits. The expression clz(x − y)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 . 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. It can also efficiently generate exponentially distributed integers by taking

2665-403: The hardware instructions above: 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 2

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2730-573: The index registers which are two's complement. Early commercial computers storing negative values in two's complement form include the English Electric DEUCE (1955) and the Digital Equipment Corporation PDP-5 (1963) and PDP-6 (1964). The System/360 , introduced in 1964 by IBM , then the dominant player in the computer industry, made two's complement the most widely used binary representation in

2795-428: The inputs are represented in the same number of bits as the output, and any overflow beyond those bits is discarded from the result). This property makes the system simpler to implement, especially for higher-precision arithmetic. Additionally, unlike ones' complement systems, two's complement has no representation for negative zero , and thus does not suffer from its associated difficulties. Otherwise, both schemes have

2860-890: The late 1980s onward have bit operators for ffs or equivalent, but a few modern ones like some of the ARM-Mx series 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

2925-485: The left-most bit to give the correct answer of 0010 (2.). Overflow checks still must exist to catch operations such as summing 0100 and 0100. The system therefore allows addition of negative operands without a subtraction circuit or a circuit that detects the sign of a number. Moreover, that addition circuit can also perform subtraction by taking the two's complement of a number (see below), which only requires an additional cycle or its own adder circuit. To perform this,

2990-477: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=NTZ&oldid=933022821 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Find first set Most modern CPU instruction set architectures provide one or more of these as hardware operators; software emulation

3055-807: The maximum practical table size is limited by the size of L1 data cache on modern processors, which is 32 KB for many. 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 2 −1 using shifts and bitwise ORs: 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): On platforms that provide hardware conversion of integers to floating point,

3120-445: The nearest power of two) using shifts and bitwise ORs 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: 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. The canonical algorithm

3185-403: The negative of that number is obtained. Positive 12 becomes negative 12, positive 5 becomes negative 5, zero becomes zero(+overflow), etc. Taking the two's complement (negation) of the minimum number in the range will not have the desired effect of negating the number. For example, the two's complement of −128 in an eight-bit system is −128 , as shown in the table to

3250-408: The number of binary bits is fixed throughout a computation it is otherwise arbitrary. Unlike the ones' complement scheme, the two's complement scheme has only one representation for zero. Furthermore, arithmetic implementations can be used on signed as well as unsigned integers and differ only in the integer overflow situations. The following is the procedure for obtaining the two's complement of

3315-722: 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. 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 ). On some Alpha platforms CTLZ and CTTZ are emulated in software. 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

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3380-458: The overflow is ignored). The two's complement of the most negative number representable (e.g. a one as the most-significant bit and all other bits zero) is itself. Hence, there is an 'extra' negative number for which two's complement does not give the negation, see § Most negative number below. The sum of a number and its ones' complement is an N -bit word with all 1 bits, which is (reading as an unsigned binary number) 2 − 1 . Then adding

3445-436: The pattern represents a non-negative value. To convert to −5 in two's-complement notation, first, all bits are inverted, that is: 0 becomes 1 and 1 becomes 0: At this point, the representation is the ones' complement of the decimal value −5. To obtain the two's complement, 1 is added to the result, giving: The result is a signed binary number representing the decimal value −5 in two's-complement form. The most significant bit

3510-456: The precise definition of the Two's complement of an N -bit number is the complement of that number with respect to 2 . The defining property of being a complement to a number with respect to 2 is simply that the summation of this number with the original produce 2 . For example, using binary with numbers up to three-bits (so N = 3 and 2 = 2 = 8 = 1000 2 , where ' 2 ' indicates

3575-400: The programmer has ensured that undefined numerical operations never happen, and make inferences from that assumption. This enables a number of optimizations, but also leads to a number of strange bugs in programs with these undefined calculations. This most negative number in two's complement is sometimes called "the weird number" , because it is the only exception. Although the number

3640-452: The remaining bits (Leave the MSB as a 1 if the initial number was in sign-and-magnitude representation). This shortcut allows a person to convert a number to its two's complement without first forming its ones' complement. For example: in two's complement representation, the negation of "0011 1100" is "1100 0 100 ", where the underlined digits were unchanged by the copying operation (while

3705-416: The rest of the digits were flipped). In computer circuitry, this method is no faster than the "complement and add one" method; both methods require working sequentially from right to left, propagating logic changes. The method of complementing and adding one can be sped up by a standard carry look-ahead adder circuit; the LSB towards MSB method can be sped up by a similar logic transformation. When turning

3770-399: The right . Although the expected result from negating −128 is +128 , there is no representation of +128 with an eight bit two's complement system and thus it is in fact impossible to represent the negation. Note that the two's complement being the same number is detected as an overflow condition since there was a carry into but not out of the most-significant bit. Having

3835-419: The right, the most-significant bit, which contains the sign information, must be maintained. However, when shifted to the left, a bit is shifted out. These rules preserve the common semantics that left shifts multiply the number by two and right shifts divide the number by two. However, if the most-significant bit changes from 0 to 1 (and vice versa), overflow is said to occur in the case that the value represents

3900-427: The sign bit also has the weight −(2 ) shown above. Using N bits, all integers from −(2 ) to 2 − 1 can be represented. In two's complement notation, a non-negative number is represented by its ordinary binary representation ; in this case, the most significant bit is 0. Though, the range of numbers represented is not the same as with unsigned binary numbers. For example, an 8-bit unsigned number can represent

3965-400: The subtraction into two operations: first subtract from the maximum number in the N -bit system, that is 2 -1 (this term in binary is actually a simple number consisting of 'all 1s', and a subtraction from it can be done simply by inverting all bits in the number also known as the bitwise NOT operation ) and then adding the one. Coincidentally, that intermediate number before adding the one

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4030-449: The table. (This algorithm does not handle the zero input.) 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. An improvement on the previous looping approach examines eight bits at a time then uses

4095-528: The top half (128 to 255) yields the signed bytes −128 to −1. The relationship to two's complement is realised by noting that 256 = 255 + 1 , and (255 − x ) is the ones' complement of  x . For example, an 8 bit number can only represent every integer from −128. to 127., inclusive, since (2 = 128.) . −95. modulo 256. is equivalent to 161. since Fundamentally, the system represents negative integers by counting backward and wrapping around . The boundary between positive and negative numbers

4160-399: The two's complement of a negative binary number, all bits are inverted, or "flipped", by using the bitwise NOT operation; the value of 1 is then added to the resulting value, ignoring the overflow which occurs when taking the two's complement of 0. For example, using 1 byte (=8 bits), the decimal number 5 is represented by The most significant bit (the leftmost bit in this case) is 0, so

4225-425: The values 0 to 255 (11111111). However a two's complement 8-bit number can only represent non-negative integers from 0 to 127 (01111111), because the rest of the bit combinations with the most significant bit as '1' represent the negative integers −1 to −128. The two's complement operation is the additive inverse operation, so negative numbers are represented by the two's complement of the absolute value . To get

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