Average Error: 29.9 → 8.9
Time: 9.8s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3435.29454990893873:\\ \;\;\;\;\mathsf{fma}\left(0.333333333333333315, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \le 0.10532502913683747:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3435.29454990893873:\\
\;\;\;\;\mathsf{fma}\left(0.333333333333333315, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\

\mathbf{elif}\;x \le 0.10532502913683747:\\
\;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\

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

\end{array}
double f(double x) {
        double r66025 = x;
        double r66026 = 1.0;
        double r66027 = r66025 + r66026;
        double r66028 = cbrt(r66027);
        double r66029 = cbrt(r66025);
        double r66030 = r66028 - r66029;
        return r66030;
}
↓
double f(double x) {
        double r66031 = x;
        double r66032 = -3435.2945499089387;
        bool r66033 = r66031 <= r66032;
        double r66034 = 0.3333333333333333;
        double r66035 = 1.0;
        double r66036 = 2.0;
        double r66037 = pow(r66031, r66036);
        double r66038 = r66035 / r66037;
        double r66039 = cbrt(r66038);
        double r66040 = 8.0;
        double r66041 = pow(r66031, r66040);
        double r66042 = r66035 / r66041;
        double r66043 = cbrt(r66042);
        double r66044 = 0.06172839506172839;
        double r66045 = r66043 * r66044;
        double r66046 = fma(r66034, r66039, r66045);
        double r66047 = 0.1111111111111111;
        double r66048 = 5.0;
        double r66049 = pow(r66031, r66048);
        double r66050 = r66035 / r66049;
        double r66051 = cbrt(r66050);
        double r66052 = r66047 * r66051;
        double r66053 = r66046 - r66052;
        double r66054 = 0.10532502913683747;
        bool r66055 = r66031 <= r66054;
        double r66056 = r66031 * r66031;
        double r66057 = 1.0;
        double r66058 = r66057 * r66057;
        double r66059 = r66056 - r66058;
        double r66060 = cbrt(r66059);
        double r66061 = r66031 - r66057;
        double r66062 = cbrt(r66061);
        double r66063 = r66060 / r66062;
        double r66064 = cbrt(r66031);
        double r66065 = r66063 - r66064;
        double r66066 = 0.6666666666666666;
        double r66067 = pow(r66031, r66066);
        double r66068 = r66057 + r66031;
        double r66069 = cbrt(r66068);
        double r66070 = r66064 + r66069;
        double r66071 = r66069 * r66070;
        double r66072 = r66067 + r66071;
        double r66073 = r66057 / r66072;
        double r66074 = r66055 ? r66065 : r66073;
        double r66075 = r66033 ? r66053 : r66074;
        return r66075;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -3435.2945499089387

    1. Initial program 60.2

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

    3. Applied add-cbrt-cube60.2

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

    4. Simplified60.2

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

    5. Taylor expanded around inf 44.9

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

    6. Simplified30.7

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

    if -3435.2945499089387 < x < 0.10532502913683747

    1. Initial program 0.1

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

    3. Applied flip-+0.1

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

    4. Applied cbrt-div0.1

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

    if 0.10532502913683747 < x

    1. Initial program 59.5

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

    3. Applied add-cbrt-cube59.5

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

    4. Simplified59.5

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

    6. Applied flip3--59.5

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

    7. Applied cube-div59.5

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

    8. Applied cbrt-div59.5

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

    9. Simplified30.7

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

    10. Simplified4.4

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3435.29454990893873:\\ \;\;\;\;\mathsf{fma}\left(0.333333333333333315, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - 0.1111111111111111 \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\\ \mathbf{elif}\;x \le 0.10532502913683747:\\ \;\;\;\;\frac{\sqrt[3]{x \cdot x - 1 \cdot 1}}{\sqrt[3]{x - 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \sqrt[3]{1 + x} \cdot \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))