Average Error: 29.8 → 8.1
Time: 9.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le 1.8586209989721385 \cdot 10^{154}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x + 1}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le 1.8586209989721385 \cdot 10^{154}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\right)}^{3}}\\

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

\end{array}
double f(double x) {
        double r67225 = x;
        double r67226 = 1.0;
        double r67227 = r67225 + r67226;
        double r67228 = cbrt(r67227);
        double r67229 = cbrt(r67225);
        double r67230 = r67228 - r67229;
        return r67230;
}
↓
double f(double x) {
        double r67231 = x;
        double r67232 = 1.8586209989721385e+154;
        bool r67233 = r67231 <= r67232;
        double r67234 = 1.0;
        double r67235 = exp(r67234);
        double r67236 = log(r67235);
        double r67237 = r67231 + r67234;
        double r67238 = cbrt(r67237);
        double r67239 = r67238 * r67238;
        double r67240 = cbrt(r67231);
        double r67241 = r67238 + r67240;
        double r67242 = r67240 * r67241;
        double r67243 = r67239 + r67242;
        double r67244 = r67236 / r67243;
        double r67245 = 3.0;
        double r67246 = pow(r67244, r67245);
        double r67247 = cbrt(r67246);
        double r67248 = 0.6666666666666666;
        double r67249 = pow(r67231, r67248);
        double r67250 = r67241 * r67238;
        double r67251 = r67249 + r67250;
        double r67252 = r67234 / r67251;
        double r67253 = r67233 ? r67247 : r67252;
        return r67253;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < 1.8586209989721385e+154

    1. Initial program 25.5

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

    3. Applied add-cbrt-cube25.6

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

    4. Simplified25.6

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

    6. Applied flip3--25.5

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

    7. Simplified24.6

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

    8. Simplified24.7

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

    10. Applied add-log-exp27.4

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

    11. Applied add-log-exp27.4

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

    12. Applied add-log-exp27.4

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

    13. Applied sum-log27.4

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

    14. Applied diff-log27.4

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

    15. Simplified8.5

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

    if 1.8586209989721385e+154 < x

    1. Initial program 60.9

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

    3. Applied flip3--60.9

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

    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. Simplified5.4

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

  4. Final simplification8.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.8586209989721385 \cdot 10^{154}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\log \left(e^{1}\right)}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) \cdot \sqrt[3]{x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))