2cbrt (problem 3.3.4)

Average Error: 29.9 → 0.5

Time: 57.7s

Precision: 64

Math

\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -82890.2927719847:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{elif}\;x \le 65240.119470548154:\\ \;\;\;\;\frac{\frac{\frac{{\left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right)\right)}^{3} - {\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}^{3}}{\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right)\right)\right) + \left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right)\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right)\right)}}{\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \end{array}\]

C

double f(double x) {
        double r2889249 = x;
        double r2889250 = 1.0;
        double r2889251 = r2889249 + r2889250;
        double r2889252 = cbrt(r2889251);
        double r2889253 = cbrt(r2889249);
        double r2889254 = r2889252 - r2889253;
        return r2889254;
}

↓

double f(double x) {
        double r2889255 = x;
        double r2889256 = -82890.2927719847;
        bool r2889257 = r2889255 <= r2889256;
        double r2889258 = cbrt(r2889255);
        double r2889259 = r2889258 / r2889255;
        double r2889260 = 0.3333333333333333;
        double r2889261 = -0.1111111111111111;
        double r2889262 = r2889261 / r2889255;
        double r2889263 = r2889260 + r2889262;
        double r2889264 = r2889259 * r2889263;
        double r2889265 = -1.0;
        double r2889266 = cbrt(r2889265);
        double r2889267 = -r2889255;
        double r2889268 = cbrt(r2889267);
        double r2889269 = r2889266 * r2889268;
        double r2889270 = r2889258 - r2889269;
        double r2889271 = r2889264 + r2889270;
        double r2889272 = 65240.119470548154;
        bool r2889273 = r2889255 <= r2889272;
        double r2889274 = 1.0;
        double r2889275 = r2889255 + r2889274;
        double r2889276 = cbrt(r2889275);
        double r2889277 = cbrt(r2889276);
        double r2889278 = r2889276 * r2889276;
        double r2889279 = cbrt(r2889278);
        double r2889280 = r2889277 * r2889279;
        double r2889281 = r2889280 * r2889280;
        double r2889282 = r2889281 * r2889281;
        double r2889283 = 3.0;
        double r2889284 = pow(r2889282, r2889283);
        double r2889285 = r2889258 * r2889258;
        double r2889286 = r2889285 * r2889285;
        double r2889287 = pow(r2889286, r2889283);
        double r2889288 = r2889284 - r2889287;
        double r2889289 = r2889286 * r2889286;
        double r2889290 = r2889286 * r2889282;
        double r2889291 = r2889289 + r2889290;
        double r2889292 = r2889282 * r2889282;
        double r2889293 = r2889291 + r2889292;
        double r2889294 = r2889288 / r2889293;
        double r2889295 = r2889281 + r2889285;
        double r2889296 = r2889294 / r2889295;
        double r2889297 = r2889258 + r2889280;
        double r2889298 = r2889296 / r2889297;
        double r2889299 = r2889273 ? r2889298 : r2889271;
        double r2889300 = r2889257 ? r2889271 : r2889299;
        return r2889300;
}

Error

Bits error versus x

Derivation

1. Split input into 2 regimes

2. if x < -82890.2927719847 or 65240.119470548154 < x

   1. Initial program 60.5

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 60.6

      \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - \sqrt[3]{x}\]

   4. Applied cbrt-prod 60.7

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]

   5. Taylor expanded around -inf 62.4

      \[\leadsto \color{blue}{\left(e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \left({\left(x \cdot -1\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1} + \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}\right)}\]

   6. Simplified 0.7

      \[\leadsto \color{blue}{\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)}\]

   if -82890.2927719847 < x < 65240.119470548154

   1. Initial program 0.2

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 0.2

      \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - \sqrt[3]{x}\]

   4. Applied cbrt-prod 0.2

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
   5. Using strategy rm

   6. Applied flip-- 0.2

      \[\leadsto \color{blue}{\frac{\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) - \sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} + \sqrt[3]{x}}}\]
   7. Using strategy rm

   8. Applied flip-- 0.3

      \[\leadsto \frac{\color{blue}{\frac{\left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right) - \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}{\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} + \sqrt[3]{x}}\]
   9. Using strategy rm

   10. Applied flip3-- 0.3

       \[\leadsto \frac{\frac{\color{blue}{\frac{{\left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right)\right)}^{3} - {\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}^{3}}{\left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right)\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right)\right) + \left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)\right)\right) \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)\right)}}}{\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} + \sqrt[3]{x}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -82890.2927719847:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{elif}\;x \le 65240.119470548154:\\ \;\;\;\;\frac{\frac{\frac{{\left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right)\right)}^{3} - {\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right)}^{3}}{\left(\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) + \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right)\right)\right) + \left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right)\right) \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)\right)\right)}}{\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) + \sqrt[3]{x} \cdot \sqrt[3]{x}}}{\sqrt[3]{x} + \sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))