Average Error: 30.0 → 0.4
Time: 17.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -92003.15010654574:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{elif}\;x \le 85033.98202656642:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -92003.15010654574:\\
\;\;\;\;\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 85033.98202656642:\\
\;\;\;\;\frac{\log \left(e^{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\\

\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}
double f(double x) {
        double r3090673 = x;
        double r3090674 = 1.0;
        double r3090675 = r3090673 + r3090674;
        double r3090676 = cbrt(r3090675);
        double r3090677 = cbrt(r3090673);
        double r3090678 = r3090676 - r3090677;
        return r3090678;
}
↓
double f(double x) {
        double r3090679 = x;
        double r3090680 = -92003.15010654574;
        bool r3090681 = r3090679 <= r3090680;
        double r3090682 = cbrt(r3090679);
        double r3090683 = r3090682 / r3090679;
        double r3090684 = 0.3333333333333333;
        double r3090685 = -0.1111111111111111;
        double r3090686 = r3090685 / r3090679;
        double r3090687 = r3090684 + r3090686;
        double r3090688 = r3090683 * r3090687;
        double r3090689 = -1.0;
        double r3090690 = cbrt(r3090689);
        double r3090691 = -r3090679;
        double r3090692 = cbrt(r3090691);
        double r3090693 = r3090690 * r3090692;
        double r3090694 = r3090682 - r3090693;
        double r3090695 = r3090688 + r3090694;
        double r3090696 = 85033.98202656642;
        bool r3090697 = r3090679 <= r3090696;
        double r3090698 = 1.0;
        double r3090699 = r3090679 + r3090698;
        double r3090700 = cbrt(r3090699);
        double r3090701 = r3090700 * r3090700;
        double r3090702 = r3090682 * r3090682;
        double r3090703 = r3090682 * r3090702;
        double r3090704 = cbrt(r3090703);
        double r3090705 = r3090704 * r3090704;
        double r3090706 = r3090701 - r3090705;
        double r3090707 = exp(r3090706);
        double r3090708 = log(r3090707);
        double r3090709 = r3090700 + r3090704;
        double r3090710 = r3090708 / r3090709;
        double r3090711 = r3090697 ? r3090710 : r3090695;
        double r3090712 = r3090681 ? r3090695 : r3090711;
        return r3090712;
}

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 < -92003.15010654574 or 85033.98202656642 < x

    1. Initial program 60.5

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

    3. Applied add-cbrt-cube60.6

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

    4. Taylor expanded around -inf 62.4

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

    5. Simplified0.7

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

    if -92003.15010654574 < x < 85033.98202656642

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube0.2

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

    5. Applied flip--0.2

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

    7. Applied add-log-exp0.2

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -92003.15010654574:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \mathbf{elif}\;x \le 85033.98202656642:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)} \cdot \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\right)}{\sqrt[3]{x + 1} + \sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{x}}{x} \cdot \left(\frac{1}{3} + \frac{\frac{-1}{9}}{x}\right) + \left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019104 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))