Average Error: 29.7 → 9.1
Time: 17.0s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3568.684213158036072854883968830108642578:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\ \mathbf{elif}\;x \le 0.04970871362191315451228845745390572119504:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, {x}^{\frac{2}{3}}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3568.684213158036072854883968830108642578:\\
\;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\

\mathbf{elif}\;x \le 0.04970871362191315451228845745390572119504:\\
\;\;\;\;\mathsf{fma}\left(\sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{x}\right)\\

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

\end{array}
double f(double x) {
        double r47640 = x;
        double r47641 = 1.0;
        double r47642 = r47640 + r47641;
        double r47643 = cbrt(r47642);
        double r47644 = cbrt(r47640);
        double r47645 = r47643 - r47644;
        return r47645;
}
↓
double f(double x) {
        double r47646 = x;
        double r47647 = -3568.684213158036;
        bool r47648 = r47646 <= r47647;
        double r47649 = 0.3333333333333333;
        double r47650 = 1.0;
        double r47651 = 2.0;
        double r47652 = pow(r47646, r47651);
        double r47653 = r47650 / r47652;
        double r47654 = cbrt(r47653);
        double r47655 = 8.0;
        double r47656 = pow(r47646, r47655);
        double r47657 = r47650 / r47656;
        double r47658 = cbrt(r47657);
        double r47659 = 0.06172839506172839;
        double r47660 = r47658 * r47659;
        double r47661 = fma(r47649, r47654, r47660);
        double r47662 = 5.0;
        double r47663 = pow(r47646, r47662);
        double r47664 = r47650 / r47663;
        double r47665 = cbrt(r47664);
        double r47666 = 0.1111111111111111;
        double r47667 = r47665 * r47666;
        double r47668 = r47661 - r47667;
        double r47669 = 0.049708713621913155;
        bool r47670 = r47646 <= r47669;
        double r47671 = 1.0;
        double r47672 = r47646 + r47671;
        double r47673 = cbrt(r47672);
        double r47674 = r47673 * r47673;
        double r47675 = cbrt(r47674);
        double r47676 = sqrt(r47675);
        double r47677 = r47676 * r47676;
        double r47678 = cbrt(r47673);
        double r47679 = cbrt(r47646);
        double r47680 = -r47679;
        double r47681 = fma(r47677, r47678, r47680);
        double r47682 = r47671 + r47646;
        double r47683 = cbrt(r47682);
        double r47684 = r47683 + r47679;
        double r47685 = 0.6666666666666666;
        double r47686 = pow(r47646, r47685);
        double r47687 = fma(r47684, r47683, r47686);
        double r47688 = r47671 / r47687;
        double r47689 = r47670 ? r47681 : r47688;
        double r47690 = r47648 ? r47668 : r47689;
        return r47690;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -3568.684213158036

    1. Initial program 60.1

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

    2. Taylor expanded around inf 45.7

      \[\leadsto \color{blue}{\left(0.3333333333333333148296162562473909929395 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.06172839506172839163511412152729462832212 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111049432054187491303309798 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    3. Simplified31.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798}\]

    if -3568.684213158036 < x < 0.049708713621913155

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied cbrt-prod0.1

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

    5. Applied fma-neg0.1

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

    7. Applied add-sqr-sqrt0.1

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

    if 0.049708713621913155 < x

    1. Initial program 59.3

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

    3. Applied flip3--59.2

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

    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. Simplified4.5

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

  4. Final simplification9.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3568.684213158036072854883968830108642578:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\ \mathbf{elif}\;x \le 0.04970871362191315451228845745390572119504:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}} \cdot \sqrt{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, {x}^{\frac{2}{3}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))