1693:
682:
1688:{\displaystyle {\begin{aligned}\operatorname {NOT} x&=\sum _{n=0}^{\lfloor \log _{2}(x)\rfloor }2^{n}\left=2^{\left\lfloor \log _{2}(x)\right\rfloor +1}-1-x\\x\operatorname {AND} y&=\sum _{n=0}^{\lfloor \log _{2}(x)\rfloor }2^{n}\left(\left\lfloor {\frac {x}{2^{n}}}\right\rfloor {\bmod {2}}\right)\left(\left\lfloor {\frac {y}{2^{n}}}\right\rfloor {\bmod {2}}\right)\\x\operatorname {OR} y&=\sum _{n=0}^{\lfloor \log _{2}(x)\rfloor }2^{n}\left(\left(\left\lfloor {\frac {x}{2^{n}}}\right\rfloor {\bmod {2}}\right)+\left(\left\lfloor {\frac {y}{2^{n}}}\right\rfloor {\bmod {2}}\right)-\left(\left\lfloor {\frac {x}{2^{n}}}\right\rfloor {\bmod {2}}\right)\left(\left\lfloor {\frac {y}{2^{n}}}\right\rfloor {\bmod {2}}\right)\right)\\x\operatorname {XOR} y&=\sum _{n=0}^{\lfloor \log _{2}(x)\rfloor }2^{n}\left(\left{\bmod {2}}\right)=\sum _{n=0}^{\lfloor \log _{2}(x)\rfloor }2^{n}\left\end{aligned}}}
465:
2257:
2478:
2468:
2364:
2354:
2249:
25:
2436:
2594:, so care must be taken when using them. The result of shifting by a bit count greater than or equal to the word's size is undefined behavior in C and C++. Right-shifting a negative value is implementation-defined and not recommended by good coding practice; the result of left-shifting a signed value is undefined if the result cannot be represented in the result type.
2426:
260:
392:
2451:, the bits are "rotated" as if the left and right ends of the register were joined. The value that is shifted into the right during a left-shift is whatever value was shifted out on the left, and vice versa for a right-shift operation. This is useful if it is necessary to retain all the existing bits, and is frequently used in digital
2527:, because if a large number is stored in two registers, the bit that is shifted off one end of the first register must come in at the other end of the second. With rotate-through-carry, that bit is "saved" in the carry flag during the first shift, ready to shift in during the second shift without any extra preparation.
503:
The bitwise XOR may be used to invert selected bits in a register (also called toggle or flip). Any bit may be toggled by XORing it with 1. For example, given the bit pattern 0010 (decimal 2) the second and fourth bits may be toggled by a bitwise XOR with a bit pattern containing 1 in the second and
2822:
where the shift amount is tested to ensure that it does not introduce undefined behavior. However, the branch adds an additional code path and presents an opportunity for timing analysis and attack, which is often not acceptable in high-integrity software. In addition, the code compiles to multiple
2180:
If the width of the register (frequently 32 or even 64) is larger than the number of bits (usually 8) of the smallest addressable unit, frequently called byte, the shift operations induce an addressing scheme from the bytes to the bits. Thereby the orientations "left" and "right" are taken from the
163:
On simple low-cost processors, typically, bitwise operations are substantially faster than division, several times faster than multiplication, and sometimes significantly faster than addition. While modern processors usually perform addition and multiplication just as fast as bitwise operations due
3025:
If the promoted type of the left-hand operand is int, only the five lowest-order bits of the right-hand operand are used as the shift distance. It is as if the right-hand operand were subjected to a bitwise logical AND operator & with the mask value 0x1f (0b11111). The shift distance actually
354:
For example, 0110 (decimal 6) can be considered a set of four flags numbered from right to left, where the first and fourth flags are clear (0), and the second and third flags are set (1). The third flag may be cleared by using a bitwise AND with the pattern that has a zero only in the third bit:
3029:
If the promoted type of the left-hand operand is long, then only the six lowest-order bits of the right-hand operand are used as the shift distance. It is as if the right-hand operand were subjected to a bitwise logical AND operator & with the mask value 0x3f (0b111111). The shift distance
3633:
It can be hard to solve for variables in
Boolean algebra, because unlike regular algebra, several operations do not have inverses. Operations without inverses lose some of the original data bits when they are performed, and it is not possible to recover this missing information.
2171:
in a computer processor have a fixed width, so some bits will be "shifted out" of the register at one end, while the same number of bits are "shifted in" from the other end; the differences between bit shift operators lie in how they determine the values of the shifted-in bits.
484:
operation on each pair of corresponding bits. The result in each position is 1 if only one of the bits is 1, but will be 0 if both are 0 or both are 1. In this we perform the comparison of two bits, being 1 if the two bits are different, and 0 if they are the same. For example:
251:, which can be visualized on a graph as a downward line that effectively "flips" an increasing range from 0 to 255, to a decreasing range from 255 to 0. A simple but illustrative example use is to invert a grayscale image where each pixel is stored as an unsigned integer.
283:
operation on each pair of the corresponding bits. Thus, if both bits in the compared position are 1, the bit in the resulting binary representation is 1 (1 × 1 = 1); otherwise, the result is 0 (1 × 0 = 0 and 0 × 0 = 0). For example:
180:
In the explanations below, any indication of a bit's position is counted from the right (least significant) side, advancing left. For example, the binary value 0001 (decimal 1) has zeroes at every position but the first (i.e., the rightmost) one.
2189:
orientation. Disregarding the boundary effects at both ends of the register, arithmetic and logical shift operations behave the same, and a shift by 8 bit positions transports the bit pattern by 1 byte position in the following way:
3209:
Although machines often have efficient built-in instructions for performing arithmetic and logical operations, all these operations can be performed by combining the bitwise operators and zero-testing in various ways. For example, here is a
429:
The bitwise OR may be used to set to 1 the selected bits of the register described above. For example, the fourth bit of 0010 (decimal 2) may be set by performing a bitwise OR with the pattern with only the fourth bit set:
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is a variant of the rotate operation, where the bit that is shifted in (on either end) is the old value of the carry flag, and the bit that is shifted out (on the other end) becomes the new value of the carry flag.
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However, as the logical right-shift inserts value 0 bits into the most significant bit, instead of copying the sign bit, it is ideal for unsigned binary numbers, while the arithmetic right-shift is ideal for signed
547:
to zero. Performing XOR on a value against itself always yields zero, and on many architectures this operation requires fewer clock cycles and less memory than loading a zero value and saving it to the register.
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In the first case, the leftmost digit was shifted past the end of the register, and a new 0 was shifted into the rightmost position. In the second case, the rightmost 1 was shifted out (perhaps into the
687:
2297:), and a new 1 was copied into the leftmost position, preserving the sign of the number. Multiple shifts are sometimes shortened to a single shift by some number of digits. For example:
247:, the bitwise complement of a number is the "mirror reflection" of the number across the half-way point of the unsigned integer's range. For example, for 8-bit unsigned integers,
600:
310:(0). For example, given a bit pattern 0011 (decimal 3), to determine whether the second bit is set we use a bitwise AND with a bit pattern containing 1 only in the second bit:
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are sometimes considered bitwise operations, because they treat a value as a series of bits rather than as a numerical quantity. In these operations, the digits are moved, or
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Bitwise operations are necessary particularly in lower-level programming such as device drivers, low-level graphics, communications protocol packet assembly, and decoding.
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In C#, the right-shift is an arithmetic shift when the first operand is an int or long. If the first operand is of type uint or ulong, the right-shift is a logical shift.
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Sometimes it is useful to simplify complex expressions made up of bitwise operations, for example when writing compilers. The goal of a compiler is to translate a
2936:. Clang provides some rotate intrinsics for Microsoft compatibility that suffers the problems above. GCC does not offer rotate intrinsics. Intel also provides x86
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behaves like a logical shift, and does not copy the sign bit. The number of places to shift is given as the second argument. For example, the following assigns
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Throughout this article, 0xFFFF means that all the bits in your data type need to be set to 1. The exact number of bits depends on the width of the data type.
2268:, the bits that are shifted out of either end are discarded. In a left arithmetic shift, zeros are shifted in on the right; in a right arithmetic shift, the
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can simulate a logical or arithmetic shift of one position by setting up the carry flag beforehand. For example, if the carry flag contains 0, then
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411:
operation on each pair of corresponding bits. The result in each position is 0 if both bits are 0, while otherwise the result is 1. For example:
340:, portions that should not be altered or portions that are not of interest. In this case, the 0 values mask the bits that are not of interest.)
2964:" to perform logical right shifts, but since the logical and arithmetic left-shift operations are identical for signed integer, there is no "
2185:, such that a left shift increases and a right shift decreases the value of the number ― if the left digits are read first, this makes up a
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42:
2834:, the following is recommended. The pattern is recognized by many compilers, and the compiler will emit a single rotate instruction:
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89:
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3215:
2379:, zeros are shifted in to replace the discarded bits. Therefore, the logical and arithmetic left-shifts are exactly the same.
61:
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46:
3978:"Constant not propagated into inline assembly, results in "constraint 'I' expects an integer constant expression""
2613:), but one can be synthesized from the shift operators. Care must be taken to ensure the statement is well formed to avoid
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160:. Most bitwise operations are presented as two-operand instructions where the result replaces one of the input operands.
68:
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556:
3060:. If the byte value is negative, the highest bit is one, then ones are used to fill up the extra bytes in the int. So
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in software with security requirements. For example, a naive implementation that left-rotates a 32-bit unsigned value
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Because the result 0010 is non-zero, we know the second bit in the original pattern was set. This is often called
35:
156:. It is a fast and simple action, basic to the higher-level arithmetic operations and directly supported by the
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3679:
2554:" for right shift. The number of places to shift is given as the second argument to the operator. For example,
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Rotate through carry is especially useful when performing shifts on numbers larger than the processor's native
57:
351:. This technique is an efficient way to store a number of Boolean values using as little memory as possible.
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Another example is a pseudocode implementation of addition, showing how to calculate a sum of two integers
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design choices, bitwise operations do commonly use less power because of the reduced use of resources.
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3684:
Operations at the top of this list are executed first. See the main article for a more complete list.
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of the given binary value. Bits that are 0 become 1, and those that are 1 become 0. For example:
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The type of the shift expression is the promoted type of the left-hand operand. For example,
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machine instructions, which is often less efficient than the processor's native instruction.
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of a binary number by checking the value of the lowest valued bit. Using the example above:
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This technique may be used to manipulate bit patterns representing sets of
Boolean states.
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is a logical right-shift, and if the carry flag contains a copy of the sign bit, then
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is an arithmetic right-shift. For this reason, some microcontrollers such as low end
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676:, for the non-negative integers, the bitwise operations can be written as follows:
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Additionally, XOR can be composed using the 3 basic operations (AND, OR, NOT)
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possible. Boolean algebra is used to simplify complex bitwise expressions.
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The C-family of languages lack a rotate operator (although C++20 provides
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to the left by two bits, which is equivalent to a multiplication by four.
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This example uses an 8-bit register, interpreted as two's complement:
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of the value minus one. If two's complement arithmetic is used, then
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that takes two equal-length binary representations and performs the
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2331:, then the same right-shift operation results in division by 2 and
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Because 6 AND 1 is zero, 6 is divisible by two and therefore even.
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The operation may be used to determine whether a particular bit is
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actually used is therefore always in the range 0 to 63, inclusive.
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2728:
is outside the range 0–31 inclusive. A second try might result in
2520:, and don't bother with arithmetic or logical shift instructions.
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3912:"Near constant time rotate that does not violate the standards?"
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bits results in undefined behavior in the right-hand expression
2960:" operators perform arithmetic shifts. Java adds the operator "
2311:
is equivalent to multiplying by 2 (provided the value does not
1718:
Here is the bitwise equivalent operations of two bits P and Q:
4048:"Synthesizing arithmetic operations using bit-shifting tricks"
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a right shift by 8 positions increases the byte address by 1.
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a right shift by 8 positions decreases the byte address by 1.
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a left shift by 8 positions decreases the byte address by 1,
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a left shift by 8 positions increases the byte address by 1,
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that takes two bit patterns of equal length and performs the
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that takes two bit patterns of equal length and performs the
3869:. Software Engineering Institute, Carnegie Mellon University
152:(considered as a bit string) at the level of its individual
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sometimes use XOR as a short-cut to setting the value of a
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0110 (decimal 6) AND 0001 (decimal 1) = 0000 (decimal 0)
347:) of a register in which each bit represents an individual
3863:"INT13-C. Use bitwise operators only on unsigned operands"
121:"Binary shift" redirects here. For the excess-3 code, see
3026:
used is therefore always in the range 0 to 31, inclusive.
2546:
In C and C++ languages, the logical shift operators are "
4130:" by Enrique Zeleny, The Wolfram Demonstrations Project.
3118:
uses bitwise operations to evaluate each of two or more
2590:
Shifts can result in implementation-defined behavior or
3954:"Circular rotate that does not violate C/C++ standard?"
638:, then vector addition corresponds to the bitwise XOR.
373:
Because of this property, it becomes easy to check the
343:
The bitwise AND may be used to clear selected bits (or
229:
NOT 10101011 (decimal 171) = 01010100 (decimal 84)
2327:
of division by 2. If the binary number is treated as
685:
654:
612:
572:
3138:), the logical left and right shift operators are "
3130:In Pascal, as well as in all its dialects (such as
3103:
3061:
3034:
2999:
2826:To avoid the undefined behavior and branches under
49:. Unsourced material may be challenged and removed.
2537:Circular shift § Implementing circular shifts
2300:00010111 (decimal +23) LEFT-SHIFT-BY-TWO = 010111
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666:
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594:
3680:Operators in C and C++ § Operator precedence
3906:
3904:
3218:showing how to multiply two arbitrary integers
4008:
4006:
8:
3933:"Poor optimization of portable rotate idiom"
3826:JTC1/SC22/WG14 N843 "C programming language"
2278:00010111 (decimal +23) LEFT-SHIFT = 0010111
1699:Truth table for all binary logical operators
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1101:
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4128:Plots Of Compositions Of Bitwise Operations
3613:x & (y ^ z) = (x & y) ^ (x & z)
3608:x & (y | z) = (x & y) | (x & z)
3146:", respectively. Even for signed integers,
3618:x + y = (x ^ y) + ((x & y) << 1)
3345:using bitwise operators and zero-testing:
551:If the set of bit strings of fixed length
3997:The Java Language Specification, section
3839:"Arithmetic operators - cppreference.com"
3458:x & (y & z) = (x & y) & z
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109:Learn how and when to remove this message
2952:, all integer types are signed, so the "
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3603:x | (y & z) = (x | y) & (x | z)
2285:10010111 (decimal −105) RIGHT-SHIFT =
4107:, a tool for bitwise-XOR files/streams
3972:
3970:
3048:In bit and shift operations, the type
2987:(unsigned right shift) are called the
2971:More details of Java shift operators:
3234:) using only bitshifts and addition:
2315:), while a right arithmetic shift by
7:
3821:
3819:
2447:In this operation, sometimes called
47:adding citations to reliable sources
4085:- is subtraction here, not negation
4076:- is negation here, not subtraction
3571:a ^ b = (a & ~b) | (~a & b)
2506:x RIGHT-ROTATE-THROUGH-CARRY-BY-ONE
2502:x RIGHT-ROTATE-THROUGH-CARRY-BY-ONE
3045:bit positions with zero-extension.
2323:value is equivalent to taking the
14:
3803:"CMicrotek Low-power Design Blog"
2916:There are also compiler-specific
2181:standard writing of numbers in a
595:{\displaystyle {\bf {F}}_{2}^{n}}
4101:supports Bitwise AND, OR and XOR
616:
576:
23:
3566:a ^ b = (a | b) & (~a | ~b)
3437:high-level programming language
3216:ancient Egyptian multiplication
125:. For the general concept, see
34:needs additional citations for
4121:Wolfram Demonstrations Project
3629:Inverses and solving equations
2439:Right circular shift or rotate
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1123:
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1:
3867:CERT: Secure Coding Standards
2429:Left circular shift or rotate
2401:Another form of shift is the
630:{\displaystyle {\bf {F}}_{2}}
468:Bitwise XOR of 4-bit integers
4050:. Bisqwit.iki.fi. 2014-02-15
395:Bitwise OR of 4-bit integers
3054:is implicitly converted to
2307:A left arithmetic shift by
539:programmers and optimizing
232:The result is equal to the
4170:
3677:
3428:
2983:(signed right shift), and
2534:
2481:Right rotate through carry
2394:
2342:
2241:
332:. (By analogy, the use of
188:
120:
4099:Online Bitwise Calculator
4014:"Chapter 15. Expressions"
3888:"Operator (C# Reference)"
3536:x ^ (y ^ z) = (x ^ y) ^ z
3492:x | (y | z) = (x | y) | z
3158:to the left by two bits:
2471:Left rotate through carry
2047:
1779:
1733:
211:on each bit, forming the
4117:Bitwise Operations Mod N
4111:Division using bitshifts
3956:. Intel Developer Forums
3914:. Stack Exchange Network
3439:into the most efficient
3347:
3236:
3160:
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2631:
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2542:In C family of languages
2167:, to the left or right.
642:Mathematical equivalents
4149:Operators (programming)
3752:Bitwise operations in C
3154:the result of shifting
2583:the result of shifting
2531:In high-level languages
667:{\displaystyle x\geq y}
3999:15.19. Shift Operators
3598:~(x & y) = ~x | ~y
3593:~(x | y) = ~x & ~y
3192:, used in cryptography
2550:" for left shift and "
2482:
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2289:1001011 (decimal −53)
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2260:Right arithmetic shift
2253:
1703:There are 16 possible
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559:) is thought of as an
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268:
16:Computer science topic
4119:" by Enrique Zeleny,
3737:Arithmetic logic unit
3453:x & y = y & x
2714:(x >> (32 - n))
2535:Further information:
2480:
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2438:
2428:
2395:Further information:
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2252:Left arithmetic shift
2251:
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366:11 (decimal 11) = 0
262:
222:111 (decimal 7) =
166:instruction pipelines
123:Shifted binary (code)
4031:"JavaScript Bitwise"
2996:aByte >>> 2
2968:" operator in Java.
2708:However, a shift by
2629:positions is simply
2518:rotate through carry
2498:rotate through carry
2490:Rotate through carry
2459:Rotate through carry
2333:rounding toward zero
2183:place-value notation
683:
652:
610:
570:
482:logical exclusive OR
409:logical inclusive OR
362:10 (decimal 6) AND 1
321:0 (decimal 2) = 00
317:1 (decimal 3) AND 00
134:computer programming
43:improve this article
3843:en.cppreference.com
3782:Primitive data type
3674:Order of operations
3588:x & (x | y) = x
3583:x | (x & y) = x
3502:x | 0xFFFF = 0xFFFF
3196:count leading zeros
2367:Right logical shift
591:
522:0 (decimal 10) =
496:1 (decimal 3) = 0
492:1 (decimal 5) XOR 0
295:(decimal 3) = 000
291:(decimal 5) AND 001
58:"Bitwise operation"
3463:x & 0xFFFF = x
3214:implementation of
2615:undefined behavior
2592:undefined behavior
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2357:Left logical shift
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514:0 (decimal 2) XOR
504:fourth positions:
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201:bitwise complement
4144:Binary arithmetic
3935:. GNU GCC Project
3709:<< >>
3623:x - y = ~(~x + y)
2998:is equivalent to
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537:Assembly language
448:0 (decimal 8) =
440:0 (decimal 2) OR
226:000 (decimal 8)
176:Bitwise operators
138:bitwise operation
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3922:
3920:
3919:
3908:
3899:
3898:
3896:
3895:
3884:
3878:
3877:
3875:
3874:
3859:
3853:
3852:
3850:
3849:
3835:
3829:
3823:
3814:
3813:
3811:
3810:
3799:
3777:Logical operator
3742:Bit manipulation
3725:
3720:
3715:
3710:
3705:
3700:
3695:
3690:
3624:
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3572:
3567:
3557:
3552:
3547:
3542:
3537:
3532:
3520:
3508:
3503:
3498:
3493:
3488:
3476:
3469:
3464:
3459:
3454:
3420:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3344:
3340:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3270:
3267:
3264:
3261:
3258:
3255:
3252:
3249:
3246:
3243:
3240:
3233:
3229:
3225:
3221:
3179:
3176:
3173:
3170:
3167:
3164:
3149:
3145:
3141:
3105:
3101:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3059:
3053:
3036:
3035:n >>> s
3021:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2997:
2986:
2982:
2978:
2967:
2963:
2959:
2955:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2773:
2770:
2767:
2764:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2727:
2723:
2719:
2715:
2711:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2628:
2624:
2612:
2608:
2586:
2582:
2575:
2572:
2569:
2566:
2563:
2560:
2553:
2549:
2507:
2503:
2463:
2421:
2407:bitwise rotation
2387:binary numbers.
2385:two's complement
2349:
2329:ones' complement
2321:two's complement
2266:arithmetic shift
2244:Arithmetic shift
2238:Arithmetic shift
2195:
2194:
2146:
2141:
2134:
2129:
2122:
2117:
2110:
2105:
2098:
2093:
2086:
2079:
2072:
2065:
2058:
2053:
1721:
1709:binary variables
1694:
1692:
1691:
1686:
1684:
1680:
1676:
1675:
1674:
1665:
1661:
1660:
1656:
1654:
1653:
1641:
1632:
1628:
1626:
1625:
1613:
1597:
1596:
1586:
1570:
1569:
1556:
1538:
1534:
1533:
1532:
1523:
1519:
1518:
1514:
1513:
1512:
1503:
1499:
1497:
1496:
1484:
1470:
1466:
1465:
1464:
1455:
1451:
1449:
1448:
1436:
1415:
1414:
1404:
1388:
1387:
1374:
1336:
1332:
1331:
1327:
1326:
1325:
1316:
1312:
1310:
1309:
1297:
1286:
1282:
1281:
1280:
1271:
1267:
1265:
1264:
1252:
1238:
1234:
1233:
1232:
1223:
1219:
1217:
1216:
1204:
1190:
1186:
1185:
1184:
1175:
1171:
1169:
1168:
1156:
1140:
1139:
1129:
1113:
1112:
1099:
1061:
1057:
1056:
1055:
1046:
1042:
1040:
1039:
1027:
1016:
1012:
1011:
1010:
1001:
997:
995:
994:
982:
971:
970:
960:
944:
943:
930:
880:
879:
872:
868:
855:
854:
832:
828:
827:
826:
817:
813:
806:
805:
796:
792:
790:
789:
777:
761:
760:
750:
734:
733:
720:
675:
673:
671:
670:
665:
636:
634:
633:
628:
626:
625:
620:
619:
601:
599:
598:
593:
590:
585:
580:
579:
478:binary operation
422:(decimal 3) = 0
418:(decimal 5) OR 0
405:binary operation
277:binary operation
250:
239:
234:two's complement
213:ones' complement
209:logical negation
191:Ones' complement
164:to their longer
114:
107:
103:
100:
94:
92:
51:
27:
19:
4169:
4168:
4164:
4163:
4162:
4160:
4159:
4158:
4154:Boolean algebra
4134:
4133:
4095:
4090:
4089:
4084:
4080:
4075:
4071:
4066:
4062:
4053:
4051:
4046:
4045:
4041:
4029:
4025:
4012:
4011:
4004:
3996:
3992:
3983:
3981:
3976:
3975:
3968:
3959:
3957:
3952:
3951:
3947:
3938:
3936:
3931:
3930:
3926:
3917:
3915:
3910:
3909:
3902:
3893:
3891:
3886:
3885:
3881:
3872:
3870:
3861:
3860:
3856:
3847:
3845:
3837:
3836:
3832:
3828:, section 6.5.7
3824:
3817:
3808:
3806:
3801:
3800:
3796:
3791:
3786:
3732:
3723:
3718:
3713:
3708:
3703:
3698:
3693:
3688:
3682:
3676:
3631:
3622:
3617:
3612:
3607:
3602:
3597:
3592:
3587:
3582:
3579:
3570:
3565:
3556:x ^ 0xFFFF = ~x
3555:
3550:
3545:
3540:
3535:
3530:
3527:
3518:
3515:
3506:
3501:
3496:
3491:
3486:
3483:
3474:
3467:
3462:
3457:
3452:
3449:
3433:
3431:Boolean algebra
3427:
3425:Boolean algebra
3422:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3342:
3338:
3335:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3231:
3227:
3223:
3219:
3204:
3186:
3181:
3180:
3177:
3174:
3171:
3168:
3165:
3162:
3147:
3143:
3139:
3136:Standard Pascal
3128:
3113:
3102:will result in
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3055:
3049:
3018:
3015:
3012:
3009:
3006:
3003:
3000:
2995:
2989:shift operators
2984:
2980:
2976:
2965:
2961:
2957:
2953:
2946:
2930:_rotr8, _rotr16
2926:_rotl8, _rotl16
2922:circular shifts
2914:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2820:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2725:
2721:
2717:
2713:
2709:
2706:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2633:
2626:
2622:
2610:
2606:
2603:
2601:Circular shifts
2584:
2580:
2577:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2551:
2547:
2544:
2539:
2533:
2505:
2501:
2461:
2449:rotate no carry
2419:
2399:
2393:
2347:
2341:
2305:
2290:
2283:
2246:
2240:
2178:
2157:
2136:
2124:
2112:
2100:
2088:
2081:
2074:
2067:
2060:
2049:
1705:truth functions
1701:
1682:
1681:
1645:
1636:
1617:
1608:
1607:
1603:
1602:
1598:
1588:
1561:
1488:
1479:
1478:
1474:
1440:
1431:
1430:
1426:
1425:
1421:
1420:
1416:
1406:
1379:
1353:
1338:
1337:
1301:
1292:
1291:
1287:
1256:
1247:
1246:
1242:
1208:
1199:
1198:
1194:
1160:
1151:
1150:
1146:
1145:
1141:
1131:
1104:
1078:
1063:
1062:
1031:
1022:
1021:
1017:
986:
977:
976:
972:
962:
935:
909:
894:
893:
846:
845:
841:
836:
781:
772:
771:
767:
766:
762:
752:
725:
699:
681:
680:
650:
649:
647:
644:
613:
608:
607:
568:
567:
531:
501:
462:
457:
456:0 (decimal 10)
427:
389:
381:
371:
370:10 (decimal 2)
326:
300:
263:Bitwise AND of
257:
249:NOT x = 255 - x
248:
237:
230:
227:
205:unary operation
193:
187:
178:
130:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
4167:
4165:
4157:
4156:
4151:
4146:
4136:
4135:
4132:
4131:
4124:
4113:
4108:
4102:
4094:
4093:External links
4091:
4088:
4087:
4078:
4069:
4060:
4039:
4023:
4002:
3990:
3980:. LLVM Project
3966:
3945:
3924:
3900:
3879:
3854:
3830:
3815:
3793:
3792:
3790:
3787:
3785:
3784:
3779:
3774:
3769:
3764:
3762:Find first set
3759:
3754:
3749:
3744:
3739:
3733:
3731:
3728:
3727:
3726:
3721:
3716:
3711:
3706:
3701:
3696:
3691:
3678:Main article:
3675:
3672:
3671:
3670:
3669:
3668:
3665:
3662:
3659:
3653:
3652:
3651:
3648:
3645:
3642:
3630:
3627:
3626:
3625:
3620:
3615:
3610:
3605:
3600:
3595:
3590:
3585:
3578:
3575:
3574:
3573:
3568:
3559:
3558:
3553:
3548:
3543:
3538:
3533:
3526:
3523:
3522:
3521:
3514:
3511:
3510:
3509:
3504:
3499:
3494:
3489:
3482:
3479:
3478:
3477:
3471:
3470:
3465:
3460:
3455:
3448:
3445:
3429:Main article:
3426:
3423:
3348:
3237:
3203:
3200:
3199:
3198:
3193:
3185:
3182:
3161:
3127:
3124:
3112:
3109:
3108:
3107:
3046:
3041:right-shifted
3031:
3027:
3023:
2992:
2979:(left shift),
2975:The operators
2945:
2942:
2837:
2731:
2632:
2619:timing attacks
2602:
2599:
2557:
2543:
2540:
2532:
2529:
2485:
2484:
2474:
2460:
2457:
2443:
2442:
2432:
2418:
2415:
2403:circular shift
2397:Circular shift
2392:
2391:Circular shift
2389:
2371:
2370:
2360:
2343:Main article:
2340:
2337:
2304:(decimal +92)
2299:
2284:
2282:(decimal +46)
2277:
2242:Main article:
2239:
2236:
2235:
2234:
2231:
2230:
2227:
2224:
2223:
2220:
2213:
2212:
2209:
2206:
2205:
2202:
2177:
2176:Bit addressing
2174:
2156:
2153:
2150:
2149:
2144:
2139:
2132:
2127:
2120:
2115:
2108:
2103:
2096:
2091:
2084:
2077:
2070:
2063:
2056:
2051:
2045:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1989:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1933:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1877:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1837:
1834:
1831:
1828:
1825:
1821:
1820:
1815:
1810:
1805:
1800:
1795:
1790:
1785:
1780:
1778:
1773:
1768:
1761:
1756:
1749:
1744:
1739:
1734:
1732:
1727:
1700:
1697:
1696:
1695:
1679:
1673:
1669:
1664:
1659:
1652:
1648:
1644:
1639:
1635:
1631:
1624:
1620:
1616:
1611:
1606:
1601:
1595:
1591:
1585:
1582:
1579:
1576:
1573:
1568:
1564:
1560:
1555:
1552:
1549:
1545:
1541:
1537:
1531:
1527:
1522:
1517:
1511:
1507:
1502:
1495:
1491:
1487:
1482:
1477:
1473:
1469:
1463:
1459:
1454:
1447:
1443:
1439:
1434:
1429:
1424:
1419:
1413:
1409:
1403:
1400:
1397:
1394:
1391:
1386:
1382:
1378:
1373:
1370:
1367:
1363:
1359:
1356:
1354:
1352:
1349:
1346:
1343:
1340:
1339:
1335:
1330:
1324:
1320:
1315:
1308:
1304:
1300:
1295:
1290:
1285:
1279:
1275:
1270:
1263:
1259:
1255:
1250:
1245:
1241:
1237:
1231:
1227:
1222:
1215:
1211:
1207:
1202:
1197:
1193:
1189:
1183:
1179:
1174:
1167:
1163:
1159:
1154:
1149:
1144:
1138:
1134:
1128:
1125:
1122:
1119:
1116:
1111:
1107:
1103:
1098:
1095:
1092:
1088:
1084:
1081:
1079:
1077:
1074:
1071:
1068:
1065:
1064:
1060:
1054:
1050:
1045:
1038:
1034:
1030:
1025:
1020:
1015:
1009:
1005:
1000:
993:
989:
985:
980:
975:
969:
965:
959:
956:
953:
950:
947:
942:
938:
934:
929:
926:
923:
919:
915:
912:
910:
908:
905:
902:
899:
896:
895:
892:
889:
886:
883:
878:
875:
871:
867:
864:
861:
858:
853:
849:
844:
839:
835:
831:
825:
821:
816:
812:
809:
804:
800:
795:
788:
784:
780:
775:
770:
765:
759:
755:
749:
746:
743:
740:
737:
732:
728:
724:
719:
716:
713:
709:
705:
702:
700:
698:
695:
692:
689:
688:
663:
660:
657:
643:
640:
624:
618:
589:
584:
578:
530:0 (decimal 8)
506:
500:0 (decimal 6)
487:
461:
458:
432:
413:
388:
385:
379:
357:
325:0 (decimal 2)
312:
286:
256:
253:
238:NOT x = -x − 1
228:
217:
207:that performs
186:
183:
177:
174:
150:binary numeral
140:operates on a
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
4166:
4155:
4152:
4150:
4147:
4145:
4142:
4141:
4139:
4129:
4125:
4122:
4118:
4114:
4112:
4109:
4106:
4103:
4100:
4097:
4096:
4092:
4082:
4079:
4073:
4070:
4064:
4061:
4049:
4043:
4040:
4036:
4035:W3Schools.com
4032:
4027:
4024:
4019:
4015:
4009:
4007:
4003:
4000:
3994:
3991:
3979:
3973:
3971:
3967:
3955:
3949:
3946:
3934:
3928:
3925:
3913:
3907:
3905:
3901:
3889:
3883:
3880:
3868:
3864:
3858:
3855:
3844:
3840:
3834:
3831:
3827:
3822:
3820:
3816:
3804:
3798:
3795:
3788:
3783:
3780:
3778:
3775:
3773:
3770:
3768:
3765:
3763:
3760:
3758:
3757:Double dabble
3755:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3734:
3729:
3722:
3717:
3712:
3707:
3702:
3697:
3692:
3687:
3686:
3685:
3681:
3673:
3666:
3663:
3660:
3657:
3656:
3654:
3649:
3646:
3643:
3640:
3639:
3637:
3636:
3635:
3628:
3621:
3616:
3611:
3606:
3601:
3596:
3591:
3586:
3581:
3580:
3576:
3569:
3564:
3563:
3562:
3554:
3549:
3546:x ^ y ^ y = x
3544:
3539:
3534:
3531:x ^ y = y ^ x
3529:
3528:
3524:
3517:
3516:
3512:
3505:
3500:
3495:
3490:
3487:x | y = y | x
3485:
3484:
3480:
3475:x & x = x
3473:
3472:
3468:x & 0 = 0
3466:
3461:
3456:
3451:
3450:
3446:
3444:
3442:
3438:
3432:
3424:
3346:
3235:
3230:greater than
3217:
3213:
3207:
3201:
3197:
3194:
3191:
3188:
3187:
3183:
3159:
3157:
3153:
3137:
3133:
3132:Object Pascal
3125:
3123:
3121:
3117:
3110:
3058:
3052:
3047:
3044:
3040:
3033:The value of
3032:
3028:
3024:
2993:
2990:
2974:
2973:
2972:
2969:
2951:
2943:
2941:
2939:
2935:
2932:in Microsoft
2931:
2927:
2923:
2920:implementing
2919:
2835:
2833:
2829:
2824:
2729:
2630:
2620:
2616:
2600:
2598:
2595:
2593:
2588:
2555:
2541:
2538:
2530:
2528:
2526:
2521:
2519:
2515:
2511:
2499:
2494:
2491:
2479:
2475:
2469:
2465:
2464:
2458:
2456:
2454:
2450:
2437:
2433:
2427:
2423:
2422:
2416:
2414:
2412:
2408:
2404:
2398:
2390:
2388:
2386:
2380:
2378:
2377:logical shift
2365:
2361:
2355:
2351:
2350:
2346:
2345:Logical shift
2339:Logical shift
2338:
2336:
2334:
2330:
2326:
2322:
2318:
2314:
2310:
2303:
2298:
2296:
2288:
2281:
2276:
2273:
2271:
2267:
2258:
2250:
2245:
2237:
2228:
2226:
2225:
2221:
2218:
2215:
2214:
2210:
2208:
2207:
2203:
2200:
2199:Little-endian
2197:
2196:
2193:
2192:
2191:
2188:
2184:
2175:
2173:
2170:
2166:
2162:
2154:
2145:
2140:
2133:
2128:
2121:
2116:
2109:
2104:
2097:
2092:
2085:
2078:
2071:
2064:
2057:
2052:
2046:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1990:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1938:
1935:
1934:
1930:
1927:
1924:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1878:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1823:
1822:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1789:
1786:
1784:
1781:
1777:
1774:
1772:
1769:
1767:
1766:
1762:
1760:
1757:
1755:
1754:
1750:
1748:
1745:
1743:
1740:
1738:
1735:
1731:
1728:
1726:
1723:
1722:
1719:
1716:
1714:
1710:
1706:
1698:
1677:
1671:
1662:
1657:
1650:
1646:
1642:
1637:
1633:
1629:
1622:
1618:
1614:
1609:
1604:
1599:
1593:
1589:
1577:
1571:
1566:
1562:
1553:
1550:
1547:
1543:
1539:
1535:
1529:
1520:
1515:
1509:
1500:
1493:
1489:
1485:
1480:
1475:
1471:
1467:
1461:
1452:
1445:
1441:
1437:
1432:
1427:
1422:
1417:
1411:
1407:
1395:
1389:
1384:
1380:
1371:
1368:
1365:
1361:
1357:
1355:
1350:
1347:
1344:
1341:
1333:
1328:
1322:
1313:
1306:
1302:
1298:
1293:
1288:
1283:
1277:
1268:
1261:
1257:
1253:
1248:
1243:
1239:
1235:
1229:
1220:
1213:
1209:
1205:
1200:
1195:
1191:
1187:
1181:
1172:
1165:
1161:
1157:
1152:
1147:
1142:
1136:
1132:
1120:
1114:
1109:
1105:
1096:
1093:
1090:
1086:
1082:
1080:
1075:
1072:
1069:
1066:
1058:
1052:
1043:
1036:
1032:
1028:
1023:
1018:
1013:
1007:
998:
991:
987:
983:
978:
973:
967:
963:
951:
945:
940:
936:
927:
924:
921:
917:
913:
911:
906:
903:
900:
897:
890:
887:
884:
881:
876:
873:
869:
862:
856:
851:
847:
842:
837:
833:
829:
823:
814:
810:
807:
802:
793:
786:
782:
778:
773:
768:
763:
757:
753:
741:
735:
730:
726:
717:
714:
711:
707:
703:
701:
696:
693:
690:
679:
678:
677:
661:
658:
655:
641:
639:
637:
622:
605:
587:
582:
566:
563:-dimensional
562:
558:
557:machine words
554:
549:
546:
542:
538:
534:
529:
525:
521:
517:
513:
509:
505:
499:
495:
491:
486:
483:
479:
475:
466:
459:
455:
451:
447:
443:
439:
435:
431:
425:
421:
417:
412:
410:
406:
402:
393:
386:
384:
378:
376:
369:
365:
361:
356:
352:
350:
349:Boolean state
346:
341:
339:
335:
331:
324:
320:
316:
311:
309:
305:
298:
294:
290:
285:
282:
278:
274:
266:
261:
254:
252:
246:
243:For unsigned
241:
235:
225:
221:
216:
214:
210:
206:
202:
198:
192:
184:
182:
175:
173:
171:
170:architectural
167:
161:
159:
155:
151:
147:
143:
139:
135:
128:
127:Offset binary
124:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
4081:
4072:
4063:
4052:. Retrieved
4042:
4034:
4026:
4017:
3993:
3982:. Retrieved
3958:. Retrieved
3948:
3937:. Retrieved
3927:
3916:. Retrieved
3892:. Retrieved
3882:
3871:. Retrieved
3866:
3857:
3846:. Retrieved
3842:
3833:
3807:. Retrieved
3797:
3767:Karnaugh map
3683:
3650:Rotate right
3638:Has inverse
3632:
3560:
3441:machine code
3434:
3336:
3208:
3205:
3202:Applications
3155:
3151:
3129:
3114:
3056:
3050:
3042:
3038:
3016:>>>
2988:
2985:>>>
2970:
2966:<<<
2962:>>>
2947:
2915:
2825:
2821:
2707:
2604:
2596:
2589:
2578:
2545:
2522:
2517:
2513:
2497:
2495:
2489:
2488:
2453:cryptography
2448:
2446:
2411:bit rotation
2410:
2406:
2402:
2400:
2381:
2376:
2374:
2316:
2308:
2306:
2301:
2291:
2286:
2279:
2274:
2265:
2263:
2179:
2164:
2160:
2158:
2050:equivalents
1764:
1752:
1729:
1724:
1717:
1702:
645:
565:vector space
560:
552:
550:
535:
532:
527:
523:
519:
515:
511:
507:
502:
497:
493:
489:
473:
471:
453:
449:
445:
441:
437:
433:
428:
426:(decimal 7)
423:
419:
415:
400:
398:
382:
372:
367:
363:
359:
353:
342:
337:
334:masking tape
329:
327:
322:
318:
314:
307:
303:
301:
299:(decimal 1)
296:
292:
288:
272:
270:
242:
231:
223:
219:
200:
196:
194:
179:
162:
137:
131:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3890:. Microsoft
3805:. CMicrotek
3667:Shift right
3655:No inverse
3647:Rotate left
3122:to 1 or 0.
3120:units place
1713:truth table
474:bitwise XOR
336:covers, or
330:bit masking
281:logical AND
273:bitwise AND
197:bitwise NOT
99:August 2018
4138:Categories
4054:2014-03-08
4018:oracle.com
3984:2015-08-11
3960:2015-08-12
3939:2015-08-11
3918:2015-08-12
3894:2013-07-14
3873:2015-09-07
3848:2016-07-06
3809:2015-08-12
3789:References
3772:Logic gate
3699:* / %
3664:Shift left
3212:pseudocode
3116:JavaScript
3111:JavaScript
2938:intrinsics
2934:Visual C++
2918:intrinsics
2512:just have
2295:carry flag
2219:ordering:
2217:Big-endian
2201:ordering:
2187:big-endian
2161:bit shifts
2155:Bit shifts
401:bitwise OR
189:See also:
168:and other
142:bit string
69:newspapers
3551:x ^ x = 0
3541:x ^ 0 = x
3519:~(~x) = x
3507:x | x = x
3497:x | 0 = x
2611:std::rotr
2607:std::rotl
2525:word size
2496:A single
2169:Registers
2113:(p XOR q)
2101:(p AND q)
1584:⌋
1572:
1559:⌊
1544:∑
1402:⌋
1390:
1377:⌊
1362:∑
1348:
1240:−
1127:⌋
1115:
1102:⌊
1087:∑
1073:
958:⌋
946:
933:⌊
918:∑
904:
888:−
882:−
857:
748:⌋
736:
723:⌊
708:∑
694:
659:≥
646:Assuming
602:over the
541:compilers
158:processor
146:bit array
3747:Bitboard
3730:See also
3190:popcount
2981:>>
2977:<<
2958:>>
2954:<<
2893:>>
2875:<<
2860:uint32_t
2839:uint32_t
2793:>>
2775:<<
2754:uint32_t
2733:uint32_t
2716:because
2688:>>
2670:<<
2655:uint32_t
2634:uint32_t
2579:assigns
2568:<<
2552:>>
2548:<<
2313:overflow
2270:sign bit
2123:(NOT p)
2066:(NOT p)
2061:(p OR q)
2048:Bitwise
1658:⌋
1638:⌊
1630:⌋
1610:⌊
1501:⌋
1481:⌊
1453:⌋
1433:⌊
1314:⌋
1294:⌊
1269:⌋
1249:⌊
1221:⌋
1201:⌊
1173:⌋
1153:⌊
1044:⌋
1024:⌊
999:⌋
979:⌊
870:⌋
843:⌊
794:⌋
774:⌊
545:register
267:integers
245:integers
3142:" and "
3104:i == -5
2956:" and "
2924:, like
2165:shifted
2137:(NOT q)
2106:p AND q
2094:p XOR q
2082:(NOT q)
1808:Then/if
1798:If/then
1707:of two
674:
648:
308:cleared
306:(1) or
203:, is a
83:scholar
4105:XORcat
3577:Others
3416:return
3329:return
3126:Pascal
3096:0x0200
2724:, and
2718:32 - 0
2514:rotate
2417:Rotate
2264:In an
2142:p OR q
2080:p AND
555:(i.e.
375:parity
85:
78:
71:
64:
56:
3714:&
3395:shift
3350:while
3317:shift
3314:right
3302:shift
3248:while
3184:Other
3010:aByte
2905:&
2832:Clang
2375:In a
2325:floor
2319:of a
2135:p OR
2068:AND q
604:field
476:is a
403:is a
345:flags
338:masks
275:is a
265:4-bit
199:, or
148:or a
90:JSTOR
76:books
3392:left
3341:and
3299:left
3222:and
3134:and
3063:byte
3051:byte
2950:Java
2944:Java
2857:...;
2848:...,
2830:and
2751:...;
2742:...,
2652:...;
2643:...,
2617:and
2609:and
2516:and
2510:PICs
2159:The
2125:OR q
2111:NOT
2099:NOT
2087:NOT
2073:NOT
2059:NOT
1788:XNOR
1776:NAND
218:NOT
195:The
154:bits
144:, a
136:, a
62:news
3704:+ -
3694:~ -
3689:( )
3658:AND
3644:XOR
3641:NOT
3525:XOR
3513:NOT
3447:AND
3386:xor
3371:and
3269:and
3172:shl
3148:shr
3144:shr
3140:shl
3081:int
3057:int
3037:is
3004:int
2948:In
2911:));
2828:GCC
2720:is
2703:));
2625:by
2409:or
1783:AND
1771:XOR
1742:NOR
1668:mod
1563:log
1526:mod
1506:mod
1458:mod
1381:log
1345:XOR
1319:mod
1274:mod
1226:mod
1178:mod
1106:log
1049:mod
1004:mod
937:log
901:AND
848:log
820:mod
799:mod
727:log
691:NOT
460:XOR
424:111
420:011
416:101
304:set
287:010
255:AND
185:NOT
132:In
45:by
4140::
4033:.
4016:.
4005:^
3969:^
3903:^
3865:.
3841:.
3818:^
3661:OR
3481:OR
3401:by
3323:by
3308:by
3260:if
3166::=
3090:b1
3066:b1
3001:((
2940:.
2928:,
2908:31
2808:))
2799:32
2726:32
2722:32
2694:32
2455:.
2413:.
2405:,
2335:.
2302:00
2043:1
1995:0
1987:1
1939:1
1931:1
1883:0
1875:1
1827:1
1813:OR
1765:¬q
1753:¬p
1747:Xq
1715:.
1070:OR
498:11
494:01
490:10
472:A
399:A
387:OR
313:00
271:A
240:.
4126:"
4123:.
4115:"
4057:.
4037:.
4020:.
3987:.
3963:.
3942:.
3921:.
3897:.
3876:.
3851:.
3812:.
3724:|
3719:^
3419:b
3413:c
3410:←
3407:a
3404:1
3398:c
3389:a
3383:b
3380:←
3377:b
3374:a
3368:b
3365:←
3362:c
3359:0
3356:≠
3353:a
3343:b
3339:a
3332:c
3326:1
3320:b
3311:1
3305:a
3296:a
3293:+
3290:c
3287:←
3284:c
3281:0
3278:≠
3275:)
3272:1
3266:b
3263:(
3257:0
3254:≠
3251:b
3245:0
3242:←
3239:c
3232:b
3228:a
3226:(
3224:b
3220:a
3178:;
3175:2
3169:y
3163:x
3156:y
3152:x
3106:.
3099:;
3093:|
3087:=
3084:i
3078:;
3075:5
3072:-
3069:=
3043:s
3039:n
3022:.
3019:2
3013:)
3007:)
2991:.
2902:n
2899:-
2896:(
2890:x
2887:(
2884:|
2881:)
2878:n
2872:x
2869:(
2866:=
2863:y
2854:=
2851:n
2845:=
2842:x
2817:;
2814:x
2811::
2805:n
2802:-
2796:(
2790:x
2787:(
2784:|
2781:)
2778:n
2772:x
2769:(
2766:?
2763:n
2760:=
2757:y
2748:=
2745:n
2739:=
2736:x
2710:0
2700:n
2697:-
2691:(
2685:x
2682:(
2679:|
2676:)
2673:n
2667:x
2664:(
2661:=
2658:y
2649:=
2646:n
2640:=
2637:x
2627:n
2623:x
2585:y
2581:x
2574:;
2571:2
2565:y
2562:=
2559:x
2317:n
2309:n
2287:1
2280:0
2147:1
2130:p
2118:q
2089:q
2075:p
2054:0
2040:0
2037:1
2034:0
2031:1
2028:0
2025:1
2022:0
2019:1
2016:0
2013:1
2010:0
2007:1
2004:0
2001:1
1998:0
1992:0
1984:1
1981:0
1978:0
1975:1
1972:1
1969:0
1966:0
1963:1
1960:1
1957:0
1954:0
1951:1
1948:1
1945:0
1942:0
1936:0
1928:1
1925:1
1922:1
1919:0
1916:0
1913:0
1910:0
1907:1
1904:1
1901:1
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1672:2
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129:.
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
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