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<title>Tak (function)</title>

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Tak (function)

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In <a href="Computer_science" title="Computer science">computer science</a>, the Tak function is a <a href="Recursion_(computer_science)" title="Recursion (computer science)">recursive function</a>, named after Ikuo Takeuchi. It is defined as follows:

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<mtext>otherwise</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \tau (x,y,z)={\begin{cases}\tau (\tau (x-1,y,z),\tau (y-1,z,x),\tau (z-1,x,y))&{\text{if }}y<x\\z&{\text{otherwise}}\end{cases}}}</annotation>

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<div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def tak(x: int, y: int, z: int) -> int:

if y < x:

return tak(

tak(x - 1, y, z),

tak(y - 1, z, x),

tak(z - 1, x, y)

)

else:

return z

</pre></div>

This function is often used as a <a href="Benchmark_(computing)" title="Benchmark (computing)">benchmark</a> for languages with optimization for <a href="Recursion" title="Recursion">recursion</a>.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a><a href="#cite_note-acornuser198606-3">[3]</a><a href="#cite_note-acornuser198611-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="tak()_vs._tarai()">tak() vs. tarai()</h2></div>

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The original definition by Takeuchi was as follows:

<div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def tarai(x: int, y: int, z: int) -> int:

if y < x:

return tarai(

tarai(x - 1, y, z),

tarai(y - 1, z, x),

tarai(z - 1, x, y)

)

else:

return y # not z!

</pre></div>

tarai is short for たらい回し (tarai mawashi, "to pass around") in Japanese.

<a href="John_McCarthy_(computer_scientist)" title="John McCarthy (computer scientist)">John McCarthy</a> named this function tak() after Takeuchi.<a href="#cite_note-5">[5]</a>

However, in certain later references, the y somehow got turned into the z. This is a small, but significant difference because the original version benefits significantly from <a href="Lazy_evaluation" title="Lazy evaluation">lazy evaluation</a>.

Though written in exactly the same manner as others, the <a href="Haskell_(programming_language)" class="mw-redirect" title="Haskell (programming language)">Haskell</a> code below runs much faster.

<div class="mw-highlight mw-highlight-lang-haskell mw-content-ltr" dir="ltr"><pre>tarai :: Int -> Int -> Int -> Int

tarai x y z

| x <= y = y

| otherwise = tarai (tarai (x-1) y z)

(tarai (y-1) z x)

(tarai (z-1) x y)

</pre></div>

One can easily accelerate this function via <a href="Memoization" title="Memoization">memoization</a> yet lazy evaluation still wins.

The best known way to optimize tarai is to use a mutually recursive helper function as follows.

<div class="mw-highlight mw-highlight-lang-python mw-content-ltr" dir="ltr"><pre>def laziest_tarai(x: int, y: int, zx: int, zy: int, zz: int) -> int:

if not y < x:

return y

else:

return laziest_tarai(

tarai(x-1, y, z),

tarai(y-1, z, x),

tarai(zx, zy, zz)-1, x, y)

def tarai(x: int, y: int, z: int) -> int:

if not y < x:

return y

else:

return laziest_tarai(

tarai(x-1, y, z),

tarai(y-1, z, x),

z-1, x, y)

</pre></div>

Here is an efficient implementation of tarai() in C:

<div class="mw-highlight mw-highlight-lang-c mw-content-ltr" dir="ltr"><pre>int tarai(int x, int y, int z)

{

while (x > y) {

int oldx = x, oldy = y;

x = tarai(x - 1, y, z);

y = tarai(y - 1, z, oldx);

if (x <= y) break;

z = tarai(z - 1, oldx, oldy);

}

return y;

}

</pre></div>

Note the additional check for (<code>x <= y</code>) before z (the third argument) is evaluated, avoiding unnecessary recursive evaluation.

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

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</style><cite id="CITEREFPeter_Coffee1996" class="citation journal cs1">Peter Coffee (1996). "Tak test stands the test of time". PC Week. 13 (39).</cite>

</li>

<a rel="nofollow" class="external text" href="http://buildingblocksjava.com/recursive-methods/">"Recursive Methods"</a>

by Elliotte Rusty Harold

</li>

<li id="cite_note-acornuser198606-3"><a href="#cite_ref-acornuser198606_3-0">^</a> <cite id="CITEREFJohnson-Davies1986" class="citation news cs1">Johnson-Davies, David (June 1986). <a rel="nofollow" class="external text" href="https://archive.org/details/AcornUser047-Jun86/page/n180/mode/1up">"Six of the Best Against the Clock"</a>. Acorn User. pp. 179, 181–182. Retrieved 28 October 2020.</cite>

</li>

<li id="cite_note-acornuser198611-4"><a href="#cite_ref-acornuser198611_4-0">^</a> <cite id="CITEREFJohnson-Davies1986" class="citation news cs1">Johnson-Davies, David (November 1986). <a rel="nofollow" class="external text" href="https://archive.org/details/AcornUser052-Nov86/page/n198/mode/1up">"Testing the Tak"</a>. Acorn User. pp. 197, 199. Retrieved 28 October 2020.</cite>

</li>

<cite id="CITEREFJohn_McCarthy1979" class="citation journal cs1">John McCarthy (December 1979). "An Interesting LISP Function". ACM Lisp Bulletin (3): 6–8. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F1411829.1411833">10.1145/1411829.1411833</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:31639459">31639459</a>.</cite>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/TAKFunction.html">"TAK Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.nue.org/nue/#tak-function">TAK Function</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070912060931/http://www.nue.org/nue/#tak-function">Archived</a> 2007-09-12 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li></ul>

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<ul><li><a href="Free_software" title="Free software">Free software</a></li>

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<li><a href="Performance_per_watt" title="Performance per watt">Performance per watt</a></li>

<li><a href="Data_center_infrastructure_efficiency" title="Data center infrastructure efficiency">Data center infrastructure efficiency</a></li>

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<li><a href="Transaction_Processing_Performance_Council" title="Transaction Processing Performance Council">Transaction Processing Performance Council</a> (TPC)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Processor</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Floating-point_unit" title="Floating-point unit">Floating-point unit</a> (<a href="FLOPS" class="mw-redirect" title="FLOPS">FLOPS</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

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<li><a href="LINPACK_benchmarks" title="LINPACK benchmarks">LINPACK</a></li>

<li><a href="LAPACK" title="LAPACK">LAPACK</a></li>

<li><a href="Prime95" title="Prime95">Prime95</a></li>

<li><a href="Super_PI" title="Super PI">Super PI</a></li>

<li><a href="SuperPrime" title="SuperPrime">SuperPrime</a></li>

<li><a href="Whetstone_(benchmark)" title="Whetstone (benchmark)">Whetstone</a></li>

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<ul><li><a href="DEISA_Benchmark_Suite" class="mw-redirect" title="DEISA Benchmark Suite">DEISA Benchmark Suite</a></li>

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<li><a href="HPC_Challenge_Benchmark" title="HPC Challenge Benchmark">HPC Challenge Benchmark</a></li>

<li><a href="Princeton_Application_Repository_for_Shared-Memory_Computers" title="Princeton Application Repository for Shared-Memory Computers">Princeton Application Repository for Shared-Memory Computers</a> (PARSEC)</li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Peripherals" class="mw-redirect" title="Peripherals">Peripherals</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computer_network" title="Computer network">Network</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>BreakingPoint Systems</li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Filesystems" class="mw-redirect" title="Filesystems">Filesystems</a> and storage</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bonnie%2B%2B" title="Bonnie++">Bonnie++</a></li>

<li><a href="IOzone" title="IOzone">IOzone</a></li>

<li><a href="Diskspd" class="mw-redirect" title="Diskspd">Diskspd</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Computer_memory" title="Computer memory">Computer memory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>BSS Random Access benchmark</li>

<li><a href="HPC_Challenge_Benchmark" title="HPC Challenge Benchmark">HPC Challenge Random Memory Access</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Input/output" title="Input/output">Input/output</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Iometer" title="Iometer">Iometer</a></li>

<li>Ioblazer</li>

<li><a href="IBM_iSeries_benchmarks" class="mw-redirect" title="IBM iSeries benchmarks">IBM iSeries benchmarks</a> (Commercial Processing Workload)</li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Computer system (entire)</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hierarchical_INTegration" title="Hierarchical INTegration">Hierarchical INTegration</a> (HINT)</li>

<li><a href="NBench" title="NBench">NBench</a> (CPU, memory)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Energy consumption</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Average_CPU_power" class="mw-redirect" title="Average CPU power">Average CPU power</a> (<a href="X86" title="X86">x86</a>)</li>

<li><a href="EEMBC" title="EEMBC">EEMBC</a> (<a href="Embedded_systems" class="mw-redirect" title="Embedded systems">embedded systems</a>)</li>

<li><a href="Data_center_infrastructure_efficiency" title="Data center infrastructure efficiency">Data center infrastructure efficiency</a></li>

<li><a href="SPECpower" title="SPECpower">SPECpower</a> (<a href="Java_(software_platform)" title="Java (software platform)">Java</a> software)</li>

<li><a href="Server_Efficiency_Rating_Tool" title="Server Efficiency Rating Tool">Server Efficiency Rating Tool</a> (SERT)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Software</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="JavaScript" title="JavaScript">JavaScript</a> engine</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Browser_speed_test" class="mw-redirect" title="Browser speed test">Browser speed test</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Cryptography" title="Cryptography">Cryptography</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cycles_per_byte" class="mw-redirect" title="Cycles per byte">Cycles per byte</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Multiuser" class="mw-redirect" title="Multiuser">Multiuser</a> system</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="AIM_Multiuser_Benchmark" title="AIM Multiuser Benchmark">AIM Multiuser Benchmark</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Virtual_machine" title="Virtual machine">Virtual machine</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="VMmark" title="VMmark">VMmark</a></li>

<li><a href="SPECvirt" title="SPECvirt">SPECvirt</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Recursion" title="Recursion">Recursion</a> performance</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Database_transaction" title="Database transaction">Database transactions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="TATP_Benchmark" title="TATP Benchmark">TATP Benchmark</a></li>

<li><a href="Transaction_Processing_over_XML" title="Transaction Processing over XML">Transaction Processing over XML</a> (TPoX)</li>

<li><a href="YCSB" title="YCSB">YCSB</a> (<a href="NoSQL" title="NoSQL">NoSQL</a>)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Web_server_benchmarking" title="Web server benchmarking">Web server benchmarking</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Apache_JMeter" title="Apache JMeter">Apache JMeter</a></li>

<li><a href="Curl-loader" title="Curl-loader">Curl-loader</a></li>

<li><a href="Httperf" title="Httperf">httperf</a></li>

<li><a href="OpenSTA" title="OpenSTA">OpenSTA</a></li>

<li><a href="Tsung" title="Tsung">Tsung</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Platform specific</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adjusted_Peak_Performance" title="Adjusted Peak Performance">Adjusted Peak Performance</a> (<a href="Nuclear_weapon" title="Nuclear weapon">Nuclear weapon</a> simulation)</li>

<li><a href="AnTuTu" title="AnTuTu">AnTuTu</a> (<a href="ARM_architecture_family" title="ARM architecture family">ARM</a>)</li>

<li><a href="BogoMips" title="BogoMips">BogoMips</a> (<a href="Linux" title="Linux">Linux</a>)</li>

<li><a href="Coremark" title="Coremark">Coremark</a> (<a href="Embedded_systems" class="mw-redirect" title="Embedded systems">embedded systems</a>)</li>

<li><a href="ICOMP_(index)" title="ICOMP (index)">iCOMP (index)</a> (Intel)</li>

<li><a href="Novabench" title="Novabench">Novabench</a> (Windows and <a href="MacOS" title="MacOS">macOS</a>)</li>

<li><a href="Phoronix_Test_Suite" title="Phoronix Test Suite">Phoronix Test Suite</a> (Linux)</li>

<li><a href="Performance_Rating" title="Performance Rating">Performance Rating</a> (AMD)</li>

<li><a href="Sysinfo" title="Sysinfo">Sysinfo</a> & <a href="SysSpeed" class="mw-redirect" title="SysSpeed">SysSpeed</a> (<a href="Motorola_68k" class="mw-redirect" title="Motorola 68k">Motorola 68k</a>)</li>

<li><a href="WorldBench" title="WorldBench">WorldBench</a> (Windows)</li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-22" href="https://en.wikipedia.org/wiki/?title=Tak_(function)&oldid=1301932557">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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