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<title>Arithmetic shift</title>

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Arithmetic shift

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<caption>Arithmetic shift operators in various programming languages and processors

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<th>Language or processor</th>

<th>Right

</th></tr>

<tr>

<td style="max-width:195px"><a href="ActionScript" title="ActionScript">ActionScript</a> 3, <a href="Java_(programming_language)" title="Java (programming language)">Java</a>, <a href="JavaScript" title="JavaScript">JavaScript</a>, <a href="Python_(programming_language)" title="Python (programming language)">Python</a>, <a href="PHP" title="PHP">PHP</a>, <a href="Ruby_(programming_language)" title="Ruby (programming language)">Ruby</a>, <a href="C_(programming_language)" title="C (programming language)">C</a>, <a href="C%2B%2B" title="C++">C++</a>,<a href="#cite_note-1">[1]</a><a href="D_(programming_language)" title="D (programming language)">D</a>, <a href="C_Sharp_(programming_language)" title="C Sharp (programming language)">C#</a>, <a href="Go_(programming_language)" title="Go (programming language)">Go</a>, <a href="Julia_(programming_language)" title="Julia (programming language)">Julia</a>, <a href="Rust_(programming_language)" title="Rust (programming language)">Rust</a> (signed types only)<a href="#cite_note-2">[2]</a>, <a href="Swift_(programming_language)" title="Swift (programming language)">Swift</a> (signed types only)<a href="#cite_note-3">[note 1]</a></td>

</td></tr>

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<td><code> Shift_Left </code><a href="#cite_note-4">[3]</a></td>

<td><code> Shift_Right_Arithmetic </code>

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<td><a href="Kotlin_(programming_language)" title="Kotlin (programming language)">Kotlin</a></td>

</td></tr>

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<td><a href="Fortran" title="Fortran">Fortran</a></td>

<td><code> SHIFTL </code></td>

<td><code> SHIFTA </code><a href="#cite_note-5">[note 2]</a>

</td></tr>

<tr>

<td><a href="Standard_ML" title="Standard ML">Standard ML</a></td>

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<td><a href="Verilog" title="Verilog">Verilog</a></td>

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<td><a href="OpenVMS" title="OpenVMS">OpenVMS</a> macro language

</td>

</td></tr>

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<td><a href="Scheme_(programming_language)" title="Scheme (programming language)">Scheme</a>

</td>

<td colspan="2" align="center"><code>arithmetic-shift</code><a href="#cite_note-scheme-9">[note 5]</a>

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<td><a href="Common_Lisp" title="Common Lisp">Common Lisp</a>

</td>

</td></tr>

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<td><a href="OCaml" title="OCaml">OCaml</a></td>

</td></tr>

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<td><a href="Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a>

</td>

<td colspan="2" align="center"><code>Data.Bits.shift</code><a href="#cite_note-10">[note 6]</a>

</td></tr>

<tr>

</td></tr>

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<td>Assembly: <a href="Z80" class="mw-redirect" title="Z80">Z80</a></td>

</td></tr>

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<td>Assembly: <a href="X86_assembly_language" title="X86 assembly language">x86</a></td>

</td></tr>

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<td>Assembly: <a href="Motorola_68000_series" title="Motorola 68000 series">68k</a></td>

</td></tr>

<tr>

<td>Assembly: <a href="RISC-V" title="RISC-V">RISC-V</a></td>

</td></tr></tbody></table>

In <a href="Computer_programming" title="Computer programming">computer programming</a>, an arithmetic shift is a <a href="Bitwise_operation#Bit_shifts" title="Bitwise operation">shift operator</a>, sometimes termed a signed shift (though it is not restricted to signed operands). The two basic types are the arithmetic left shift and the arithmetic right shift. For <a href="Binary_numeral_system" class="mw-redirect" title="Binary numeral system">binary numbers</a> it is a <a href="Bitwise_operation" title="Bitwise operation">bitwise operation</a> that shifts all of the bits of its operand; every bit in the operand is simply moved a given number of bit positions, and the vacant bit-positions are filled in. Instead of being filled with all 0s, as in <a href="Logical_shift" title="Logical shift">logical shift</a>, when shifting to the right, the leftmost bit (usually the <a href="Sign_bit" title="Sign bit">sign bit</a> in signed <a href="Integer" title="Integer">integer</a> representations) is replicated to fill in all the vacant positions (this is a kind of <a href="Sign_extension" title="Sign extension">sign extension</a>).

Some authors prefer the terms sticky right-shift and zero-fill right-shift for arithmetic and logical shifts respectively.<a href="#cite_note-14">[7]</a>

Arithmetic shifts can be useful as efficient ways to perform multiplication or division of signed integers by powers of two. Shifting left by n bits on a signed or unsigned binary number has the effect of multiplying it by 2n. Shifting right by n bits on a <a href="Two's_complement" title="Two's complement">two's complement</a> signed binary number has the effect of dividing it by 2n, but it always rounds down (towards negative infinity). This is different from the way rounding is usually done in signed integer division (which rounds towards 0). This discrepancy has led to bugs in a number of compilers.<a href="#cite_note-15">[8]</a>

For example, in the <a href="X86_instruction_listings" title="X86 instruction listings">x86 instruction set</a>, the SAR instruction (arithmetic right shift) divides a signed number by a <a href="Power_of_two" title="Power of two">power of two</a>, rounding towards negative infinity.<a href="#cite_note-FOOTNOTEHyde1996§_6.6.2.2_SAR-16">[9]</a> However, the IDIV instruction (signed divide) divides a signed number, rounding towards zero. So a SAR instruction cannot be substituted for an IDIV by power of two instruction nor vice versa.

<div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2></div>

The formal definition of an arithmetic shift, from <a href="Federal_Standard_1037C" title="Federal Standard 1037C">Federal Standard 1037C</a> is that it is:

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</style><blockquote class="templatequote">A shift, applied to the representation of a number in a fixed <a href="Radix" title="Radix">radix</a> numeration system and in a <a href="Fixed-point_arithmetic" title="Fixed-point arithmetic">fixed-point</a> representation system, and in which only the characters representing the fixed-point part of the number are moved. An arithmetic shift is usually equivalent to multiplying the number by a positive or a negative integral power of the radix, except for the effect of any rounding; compare the <a href="Logical_shift" title="Logical shift">logical shift</a> with the arithmetic shift, especially in the case of <a href="Floating-point" class="mw-redirect" title="Floating-point">floating-point</a> representation.</blockquote>

An important word in the FS 1073C definition is "usually".

<div class="mw-heading mw-heading3"><h3 id="Non-equivalence_of_arithmetic_right_shift_and_division">Non-equivalence of arithmetic right shift and division</h3></div>

However, arithmetic right shifts are major traps for the unwary, specifically in treating rounding of negative integers. For example, in the usual <a href="Two's_complement" title="Two's complement">two's complement</a> representation of negative integers, −1 is represented as all 1's. For an 8-bit signed integer this is 1111 1111. An arithmetic right-shift by 1 (or 2, 3, ..., 7) yields 1111 1111 again, which is still −1. This corresponds to rounding down (towards negative infinity), but is not the usual convention for division.

It is frequently stated that arithmetic right shifts are equivalent to <a href="Division_(mathematics)" title="Division (mathematics)">division</a> by a (positive, integral) power of the radix (e.g., a division by a power of 2 for binary numbers), and hence that division by a power of the radix can be optimized by implementing it as an arithmetic right shift. (A shifter is much simpler than a divider. On most processors, shift instructions will execute faster than division instructions.) Large number of 1960s and 1970s programming handbooks, manuals, and other specifications from companies and institutions such as <a href="Digital_Equipment_Corporation" title="Digital Equipment Corporation">DEC</a>, <a href="IBM" title="IBM">IBM</a>, <a href="Data_General" title="Data General">Data General</a>, and <a href="American_National_Standards_Institute" title="American National Standards Institute">ANSI</a> make such incorrect statements<a href="#cite_note-FOOTNOTESteele1977-17">[10]</a>.

Logical right shifts are equivalent to division by a power of the radix (usually 2) only for positive or unsigned numbers. Arithmetic right shifts are equivalent to logical right shifts for positive signed numbers. Arithmetic right shifts for negative numbers in N's complement (usually <a href="Two's_complement" title="Two's complement">two's complement</a>) is roughly equivalent to division by a power of the radix (usually 2), where for odd numbers rounding downwards is applied (not towards 0 as usually expected).

Arithmetic right shifts for negative numbers are equivalent to division using rounding towards 0 in <a href="Ones'_complement" title="Ones' complement">ones' complement</a> representation of signed numbers as was used by some historic computers, but this is no longer in general use.

<div class="mw-heading mw-heading4"><h4 id="Handling_the_issue_in_programming_languages">Handling the issue in programming languages</h4></div>

The (1999) ISO standard for the programming language <a href="C_(programming_language)" title="C (programming language)">C</a> defines the right shift operator in terms of divisions by powers of 2.<a href="#cite_note-FOOTNOTEISOIEC98991999§_6.5.7_Bitwise_shift_operators-18">[11]</a> Because of the above-stated non-equivalence, the standard explicitly excludes from that definition the right shifts of signed numbers that have negative values. It does not specify the behaviour of the right shift operator in such circumstances, but instead requires each individual C compiler to define the behaviour of shifting negative values right.<a href="#cite_note-20">[note 8]</a>

Like C, C++ had an implementation-defined right shift for signed integers until <a href="C%2B%2B20" title="C++20">C++20</a>. Starting in the C++20 standard, right shift of a signed integer is defined to be an arithmetic shift.<a href="#cite_note-FOOTNOTEISOCPP202020§_7.6.7_Shift_operators-21">[13]</a>

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

In applications where consistent rounding down is desired, arithmetic right shifts for signed values are useful. An example is in <a href="Downscaling" title="Downscaling">downscaling</a> raster coordinates by a power of two, which maintains even spacing. For example, right shift by 1 sends 0, 1, 2, 3, 4, 5, ... to 0, 0, 1, 1, 2, 2, ..., and −1, −2, −3, −4, ... to −1, −1, −2, −2, ..., maintaining even spacing as −2, −2, −1, −1, 0, 0, 1, 1, 2, 2, ... In contrast, integer division with rounding towards zero sends −1, 0, and 1 all to 0 (3 points instead of 2), yielding −2, −1, −1, 0, 0, 0, 1, 1, 2, 2, ... instead, which is irregular at 0.

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> The <code> >> </code> operator in C and C++ is not necessarily an arithmetic shift. Usually it is only an arithmetic shift if used with a signed integer type on its left-hand side. If it is used on an unsigned integer type instead, it will be a logical shift.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Fortran 2008.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> The Verilog arithmetic right shift operator only actually performs an arithmetic shift if the first operand is signed. If the first operand is unsigned, the operator actually performs a logical right shift.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> In the <a href="OpenVMS" title="OpenVMS">OpenVMS</a> macro language, whether an arithmetic shift is left or right is determined by whether the second operand is positive or negative. This is unusual. In most programming languages the two directions have distinct operators, with the operator specifying the direction, and the second operand is implicitly positive. (Some languages, such as Verilog, require that negative values be converted to unsigned positive values. Some languages, such as C and C++, have no defined behaviour if negative values are used.)<a href="#cite_note-FOOTNOTEHP2001-7">[4]</a>

</li>

<li id="cite_note-scheme-9"><a href="#cite_ref-scheme_9-0">^</a> In Scheme <code>arithmetic-shift</code> can be both left and right shift, depending on the second operand, very similar to the OpenVMS macro language, although R6RS Scheme adds both <code>-right</code> and <code>-left</code> variants.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> The <code>Bits</code> class from Haskell's <code>Data.Bits</code> module defines both <code>shift</code> taking a signed argument and <code>shiftL</code>/<code>shiftR</code> taking unsigned arguments. These are <a href="Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a>; for new definitions the programmer need provide only one of the two forms and the other form will be automatically defined in terms of the provided one.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> The VHDL arithmetic left shift operator is unusual. Instead of filling the LSB of the result with zero, it copies the original LSB into the new LSB. While this is an exact mirror image of the arithmetic right shift, it is not the conventional definition of the operator, and is not equivalent to multiplication by a power of 2. In the VHDL 2008 standard this strange behavior was left unchanged (for backward compatibility) for argument types that do not have forced numeric interpretation (e.g., BIT_VECTOR) but 'SLA' for unsigned and signed argument types behaves in the expected way (i.e., rightmost positions are filled with zeros). VHDL's shift left logical (SLL) function does implement the aforementioned 'standard' arithmetic shift.

</li>

<li id="cite_note-20"><a href="#cite_ref-20">^</a> The C standard was intended to not restrict the C language to either ones' complement or two's complement architectures. In cases where the behaviours of ones' complement and two's complement representations differ, such as this, the standard requires individual C compilers to document the behaviour of their target architectures. The documentation for <a href="GNU_Compiler_Collection" title="GNU Compiler Collection">GNU Compiler Collection</a> (GCC), for example, documents its behaviour as employing sign-extension.<a href="#cite_note-FOOTNOTEFSF2008§_4.5_Integers_implementation-19">[12]</a>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Cross-reference">Cross-reference</h3></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://tour.dlang.org/tour/en/gems/bit-manipulation">"Bit manipulation - Dlang Tour"</a>. tour.dlang.org. Retrieved 2019-06-23.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://doc.rust-lang.org/reference/expressions/operator-expr.html#arithmetic-and-logical-binary-operators">"Operator Expressions: Arithmetic and Logical Binary Operators"</a>. doc.rust-lang.org. Retrieved 2022-11-13.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.adaic.org/resources/add_content/standards/12aarm/html/AA-B-2.html">"Annotated Ada 2012 Reference Manual"</a>.</cite>

</li>

<li id="cite_note-FOOTNOTEHP2001-7"><a href="#cite_ref-FOOTNOTEHP2001_7-0">^</a> <a href="#CITEREFHP2001">HP 2001</a>.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.z80.info/z80syntx.htm#SLA">"Z80 Assembler Syntax"</a>.</cite>

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://github.com/riscv/riscv-isa-manual/releases/download/Ratified-IMAFDQC/riscv-spec-20191213.pdf">"The RISC-V Instruction Set Manual, Volume I: Unprivileged ISA"</a> (PDF). <a href="GitHub" title="GitHub">GitHub</a>. 2019-12-13. pp. 18–20. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/https://github.com/riscv/riscv-isa-manual/releases/download/Ratified-IMAFDQC/riscv-spec-20191213.pdf">Archived</a> (PDF) from the original on 2022-10-09. Retrieved 2021-08-07.</cite>

</li>

Thomas R. Cain and Alan T. Sherman.

<a rel="nofollow" class="external text" href="https://web.archive.org/web/20140109191811/http://www.cisa.umbc.edu/papers/ShermanCryptologia97.pdf">"How to break Gifford's cipher"</a>.

Section 8.1: "Sticky versus Non-Sticky Bit-shifting".

Cryptologia.

1997.

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFSteele_Jr" class="citation web cs1">Steele Jr, Guy. <a rel="nofollow" class="external text" href="http://dspace.mit.edu/bitstream/handle/1721.1/6090/AIM-378.pdf">"Arithmetic Shifting Considered Harmful"</a> (PDF). MIT AI Lab. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://dspace.mit.edu/bitstream/handle/1721.1/6090/AIM-378.pdf">Archived</a> (PDF) from the original on 2022-10-09. Retrieved 20 May 2013.</cite>

</li>

<li id="cite_note-FOOTNOTEHyde1996§_6.6.2.2_SAR-16"><a href="#cite_ref-FOOTNOTEHyde1996§_6.6.2.2_SAR_16-0">^</a> <a href="#CITEREFHyde1996">Hyde 1996</a>, § 6.6.2.2 SAR.

</li>

<li id="cite_note-FOOTNOTESteele1977-17"><a href="#cite_ref-FOOTNOTESteele1977_17-0">^</a> <a href="#CITEREFSteele1977">Steele 1977</a>.

</li>

<li id="cite_note-FOOTNOTEISOIEC98991999§_6.5.7_Bitwise_shift_operators-18"><a href="#cite_ref-FOOTNOTEISOIEC98991999§_6.5.7_Bitwise_shift_operators_18-0">^</a> <a href="#CITEREFISOIEC98991999">ISOIEC9899 1999</a>, § 6.5.7 Bitwise shift operators.

</li>

<li id="cite_note-FOOTNOTEFSF2008§_4.5_Integers_implementation-19"><a href="#cite_ref-FOOTNOTEFSF2008§_4.5_Integers_implementation_19-0">^</a> <a href="#CITEREFFSF2008">FSF 2008</a>, § 4.5 Integers implementation.

</li>

<li id="cite_note-FOOTNOTEISOCPP202020§_7.6.7_Shift_operators-21"><a href="#cite_ref-FOOTNOTEISOCPP202020§_7.6.7_Shift_operators_21-0">^</a> <a href="#CITEREFISOCPP202020">ISOCPP20 2020</a>, § 7.6.7 Shift operators.

</li>

</ol></div>

<div class="mw-heading mw-heading3"><h3 id="Sources_used">Sources used</h3></div>

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</style> This article incorporates <a href="Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <cite class="citation cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20220122224547/https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">Federal Standard 1037C</a>. <a href="General_Services_Administration" title="General Services Administration">General Services Administration</a>. Archived from <a rel="nofollow" class="external text" href="https://www.its.bldrdoc.gov/fs-1037/fs-1037c.htm">the original</a> on 2022-01-22.</cite>

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