Average Error: 29.6 → 15.2
Time: 5.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3965.7270635849572 \lor \neg \left(x \le 5204.30305478661921\right):\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(-0.037037037037037035, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.0329218106995884732, \frac{1}{{x}^{4}}, \frac{\frac{0.037037037037037035}{x}}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3965.7270635849572 \lor \neg \left(x \le 5204.30305478661921\right):\\
\;\;\;\;\sqrt[3]{\mathsf{fma}\left(-0.037037037037037035, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.0329218106995884732, \frac{1}{{x}^{4}}, \frac{\frac{0.037037037037037035}{x}}{x}\right)\right)}\\

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

\end{array}
double f(double x) {
        double r74714 = x;
        double r74715 = 1.0;
        double r74716 = r74714 + r74715;
        double r74717 = cbrt(r74716);
        double r74718 = cbrt(r74714);
        double r74719 = r74717 - r74718;
        return r74719;
}
↓
double f(double x) {
        double r74720 = x;
        double r74721 = -3965.7270635849572;
        bool r74722 = r74720 <= r74721;
        double r74723 = 5204.303054786619;
        bool r74724 = r74720 <= r74723;
        double r74725 = !r74724;
        bool r74726 = r74722 || r74725;
        double r74727 = 0.037037037037037035;
        double r74728 = -r74727;
        double r74729 = 1.0;
        double r74730 = 3.0;
        double r74731 = pow(r74720, r74730);
        double r74732 = r74729 / r74731;
        double r74733 = 0.03292181069958847;
        double r74734 = 4.0;
        double r74735 = pow(r74720, r74734);
        double r74736 = r74729 / r74735;
        double r74737 = r74727 / r74720;
        double r74738 = r74737 / r74720;
        double r74739 = fma(r74733, r74736, r74738);
        double r74740 = fma(r74728, r74732, r74739);
        double r74741 = cbrt(r74740);
        double r74742 = 1.0;
        double r74743 = r74720 + r74742;
        double r74744 = cbrt(r74743);
        double r74745 = cbrt(r74720);
        double r74746 = cbrt(r74745);
        double r74747 = r74746 * r74746;
        double r74748 = r74747 * r74746;
        double r74749 = r74744 - r74748;
        double r74750 = r74726 ? r74741 : r74749;
        return r74750;
}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3965.7270635849572 or 5204.303054786619 < x

    1. Initial program 60.3

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

    3. Applied add-cbrt-cube60.3

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

    4. Simplified60.3

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

    5. Taylor expanded around inf 31.6

      \[\leadsto \sqrt[3]{\color{blue}{\left(0.037037037037037035 \cdot \frac{1}{{x}^{2}} + 0.0329218106995884732 \cdot \frac{1}{{x}^{4}}\right) - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}}}\]

    6. Simplified30.8

      \[\leadsto \sqrt[3]{\color{blue}{\mathsf{fma}\left(-0.037037037037037035, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.0329218106995884732, \frac{1}{{x}^{4}}, \frac{\frac{0.037037037037037035}{x}}{x}\right)\right)}}\]

    if -3965.7270635849572 < x < 5204.303054786619

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

  4. Final simplification15.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3965.7270635849572 \lor \neg \left(x \le 5204.30305478661921\right):\\ \;\;\;\;\sqrt[3]{\mathsf{fma}\left(-0.037037037037037035, \frac{1}{{x}^{3}}, \mathsf{fma}\left(0.0329218106995884732, \frac{1}{{x}^{4}}, \frac{\frac{0.037037037037037035}{x}}{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))