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copied!<blockquote> <p>After that, however, it subtracted (dividend/2^31) from the result.</p> </blockquote> <p>Actually, it subtracts <code>dividend >> 31</code>, which is <code>-1</code> for negative <code>dividend</code>, and 0 for non-negative dividend, when right-shifting negative integers is arithmetic right-shift (and <code>int</code> is 32 bits wide).</p> <pre><code>0x6666667 = (2^34 + 6)/10 </code></pre> <p>So for <code>x < 0</code>, we have, writing <code>x = 10*k + r</code> with <code>-10 < r <= 0</code>,</p> <pre><code>0x66666667 * (10*k+r) = (2^34+6)*k + (2^34 + 6)*r/10 = 2^34*k + 6*k + (2^34+6)*r/10 </code></pre> <p>Now, arithmetic right shift of negative integers yields the floor of <code>v / 2^n</code>, so</p> <pre><code>(0x66666667 * x) >> 34 </code></pre> <p>results in</p> <pre><code>k + floor((6*k + (2^34+6)*r/10) / 2^34) </code></pre> <p>So we need to see that</p> <pre><code>-2^34 < 6*k + (2^34+6)*r/10 < 0 </code></pre> <p>The right inequality is easy, both <code>k</code> and <code>r</code> are non-positive, and not both are 0.</p> <p>For the left inequality, a bit more analysis is needed.</p> <pre><code>r >= -9 </code></pre> <p>so the absolute value of <code>(2^34+6)*r/10</code> is at most <code>2^34+6 - (2^34+6)/10</code>.</p> <pre><code>|k| <= 2^31/10, </code></pre> <p>so <code>|6*k| <= 3*2^31/5</code>.</p> <p>And it remains to verify that</p> <pre><code>6 + 3*2^31/5 < (2^34+6)/10 1288490194 < 1717986919 </code></pre> <p>Yup, true.</p>

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