A note on smaller divisors

blog/division-is-hard-but-it-does-not-have-to-be/index.html

@@ -38,7 +38,7 @@
     <span class=hljs-keyword>lea</span>     <span class=hljs-built_in>rcx</span>, [<span class=hljs-built_in>rax</span> + <span class=hljs-number>59</span>]
     <span class=hljs-keyword>cmovb</span>   <span class=hljs-built_in>rax</span>, <span class=hljs-built_in>rcx</span>
     <span class=hljs-keyword>ret</span>
-</code></pre><hr><p>Oh, and it’s not like hard-coding <eq><math><mrow><msup><mn>2</mn><mn>64</mn></msup><mo>−</mo></mrow><mrow><mn>59</mn></mrow></math></eq> was necessary. Two iterations suffice for any divisor <eq><math><mrow><mo>≥</mo></mrow><mrow><msup><mn>2</mn><mn>64</mn></msup><mo>−</mo></mrow><mrow><msup><mn>2</mn><mn>32</mn></msup><mo>+</mo></mrow><mrow><mn>1</mn></mrow></math></eq>. Need more primes? Choose away, there’s a lot of them in the <eq><math><msup><mn>2</mn><mn>32</mn></msup></math></eq>-long region.<p>And this method works for division too, not just modulo:<pre><code class=language-rust><span class=hljs-keyword>fn</span> <span class="hljs-title function_">divide</span>(<span class=hljs-keyword>mut</span> n: <span class=hljs-type>u128</span>) <span class=hljs-punctuation>-></span> <span class=hljs-type>u128</span> {
+</code></pre><hr><p>Oh, and it’s not like hard-coding <eq><math><mrow><msup><mn>2</mn><mn>64</mn></msup><mo>−</mo></mrow><mrow><mn>59</mn></mrow></math></eq> was necessary. Two iterations suffice for any divisor <eq><math><mrow><mo>≥</mo></mrow><mrow><msup><mn>2</mn><mn>64</mn></msup><mo>−</mo></mrow><mrow><msup><mn>2</mn><mn>32</mn></msup><mo>+</mo></mrow><mrow><mn>1</mn></mrow></math></eq>. Need more primes? Choose away, there’s a lot of them in the <eq><math><msup><mn>2</mn><mn>32</mn></msup></math></eq>-long region.<p>Need a smaller divisor? Three iterations work for <eq><math><mrow><mi>n</mi><mo>≥</mo></mrow><mrow><msup><mn>2</mn><mn>64</mn></msup><mo>−</mo></mrow><mrow><mn>6981461082631</mn></mrow></math></eq> (42.667 bits compared to 32 for two iterations), four for <eq><math><mrow><mi>n</mi><mo>≥</mo></mrow><mrow><msup><mn>2</mn><mn>64</mn></msup><mo>−</mo></mrow><mrow><mn>281472113362716</mn></mrow></math></eq> (48 bits). Sounds like a lot? That’s still better than <code>__umodti3</code>.<p>And this method works for division too, not just modulo:<pre><code class=language-rust><span class=hljs-keyword>fn</span> <span class="hljs-title function_">divide</span>(<span class=hljs-keyword>mut</span> n: <span class=hljs-type>u128</span>) <span class=hljs-punctuation>-></span> <span class=hljs-type>u128</span> {
     <span class=hljs-keyword>let</span> <span class=hljs-keyword>mut </span><span class=hljs-variable>quotient</span> = n >> <span class=hljs-number>64</span>;
     n = n % (<span class=hljs-number>1</span> << <span class=hljs-number>64</span>) + (n >> <span class=hljs-number>64</span>) * <span class=hljs-number>59</span>;
     quotient += n >> <span class=hljs-number>64</span>;

blog/division-is-hard-but-it-does-not-have-to-be/index.md

@@ -83,6 +83,8 @@ modulo:
 
 Oh, and it's not like hard-coding $2^{64} - 59$ was necessary. Two iterations suffice for any divisor $\ge 2^{64} - 2^{32} + 1$. Need more primes? Choose away, there's a lot of them in the $2^{32}$-long region.
 
+Need a smaller divisor? Three iterations work for $n \ge 2^{64} - 6981461082631$ (42.667 bits compared to 32 for two iterations), four for $n \ge 2^{64} - 281472113362716$ (48 bits). Sounds like a lot? That's still better than `__umodti3`.
+
 And this method works for division too, not just modulo:
 
 ```rust