Average Error: 30.6 → 12.3
Time: 8.6s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.53462789897405172 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.016487902542018727:\\ \;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.53462789897405172 \cdot 10^{61}:\\
\;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\

\mathbf{elif}\;x \le 0.016487902542018727:\\
\;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\

\mathbf{else}:\\
\;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\

\end{array}
double f(double x) {
        double r292 = x;
        double r293 = 1.0;
        double r294 = r292 + r293;
        double r295 = cbrt(r294);
        double r296 = cbrt(r292);
        double r297 = r295 - r296;
        return r297;
}
↓
double f(double x) {
        double r298 = x;
        double r299 = -4.534627898974052e+61;
        bool r300 = r298 <= r299;
        double r301 = 0.3333333333333333;
        double r302 = 1.0;
        double r303 = 2.0;
        double r304 = pow(r298, r303);
        double r305 = r302 / r304;
        double r306 = 0.3333333333333333;
        double r307 = pow(r305, r306);
        double r308 = r301 * r307;
        double r309 = 0.06172839506172839;
        double r310 = 8.0;
        double r311 = pow(r298, r310);
        double r312 = r302 / r311;
        double r313 = pow(r312, r306);
        double r314 = r309 * r313;
        double r315 = r308 + r314;
        double r316 = 0.1111111111111111;
        double r317 = 5.0;
        double r318 = pow(r298, r317);
        double r319 = r302 / r318;
        double r320 = pow(r319, r306);
        double r321 = r316 * r320;
        double r322 = r315 - r321;
        double r323 = 0.016487902542018727;
        bool r324 = r298 <= r323;
        double r325 = 1.0;
        double r326 = r298 + r325;
        double r327 = cbrt(r326);
        double r328 = r327 * r327;
        double r329 = cbrt(r298);
        double r330 = r329 * r329;
        double r331 = cbrt(r330);
        double r332 = cbrt(r331);
        double r333 = cbrt(r329);
        double r334 = cbrt(r333);
        double r335 = r332 * r334;
        double r336 = r331 * r335;
        double r337 = r336 * r329;
        double r338 = r328 - r337;
        double r339 = r327 + r329;
        double r340 = r338 / r339;
        double r341 = 0.0;
        double r342 = r341 + r325;
        double r343 = r327 * r339;
        double r344 = 0.6666666666666666;
        double r345 = pow(r298, r344);
        double r346 = r343 + r345;
        double r347 = r342 / r346;
        double r348 = r324 ? r340 : r347;
        double r349 = r300 ? r322 : r348;
        return r349;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -4.534627898974052e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 40.2

      \[\leadsto \color{blue}{\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    if -4.534627898974052e+61 < x < 0.016487902542018727

    1. Initial program 5.4

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

    3. Applied flip--5.5

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

    5. Applied add-cube-cbrt5.4

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

    6. Applied cbrt-prod5.4

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

    8. Applied add-cube-cbrt5.4

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

    9. Applied cbrt-prod5.4

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

    10. Applied cbrt-prod5.3

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

    if 0.016487902542018727 < x

    1. Initial program 59.1

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

    3. Applied flip3--59.0

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

    4. Simplified1.0

      \[\leadsto \frac{\color{blue}{0 + 1}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

    5. Simplified4.4

      \[\leadsto \frac{0 + 1}{\color{blue}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification12.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.53462789897405172 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 0.016487902542018727:\\ \;\;\;\;\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \sqrt[3]{x}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x}}}\right)\right) \cdot \sqrt[3]{x}}{\sqrt[3]{x + 1} + \sqrt[3]{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))