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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3659.711310353127:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}}\\ \mathbf{elif}\;x \le 2853.6107128789863:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}}\\ \end{array}\]

double f(double x) {
        double r2323015 = x;
        double r2323016 = 1.0;
        double r2323017 = r2323015 + r2323016;
        double r2323018 = cbrt(r2323017);
        double r2323019 = cbrt(r2323015);
        double r2323020 = r2323018 - r2323019;
        return r2323020;
}

↓

double f(double x) {
        double r2323021 = x;
        double r2323022 = -3659.711310353127;
        bool r2323023 = r2323021 <= r2323022;
        double r2323024 = 1.0;
        double r2323025 = r2323024 / r2323021;
        double r2323026 = r2323025 / r2323021;
        double r2323027 = r2323021 * r2323021;
        double r2323028 = r2323026 / r2323027;
        double r2323029 = cbrt(r2323028);
        double r2323030 = -0.1111111111111111;
        double r2323031 = 0.6666666666666666;
        double r2323032 = cbrt(r2323025);
        double r2323033 = 7.0;
        double r2323034 = pow(r2323021, r2323033);
        double r2323035 = r2323024 / r2323034;
        double r2323036 = cbrt(r2323035);
        double r2323037 = 0.04938271604938271;
        double r2323038 = r2323036 * r2323037;
        double r2323039 = fma(r2323031, r2323032, r2323038);
        double r2323040 = fma(r2323029, r2323030, r2323039);
        double r2323041 = r2323024 + r2323021;
        double r2323042 = cbrt(r2323041);
        double r2323043 = cbrt(r2323021);
        double r2323044 = r2323043 * r2323043;
        double r2323045 = cbrt(r2323043);
        double r2323046 = r2323045 * r2323045;
        double r2323047 = r2323045 * r2323046;
        double r2323048 = r2323044 * r2323047;
        double r2323049 = cbrt(r2323048);
        double r2323050 = r2323042 + r2323049;
        double r2323051 = r2323040 / r2323050;
        double r2323052 = 2853.6107128789863;
        bool r2323053 = r2323021 <= r2323052;
        double r2323054 = r2323042 * r2323042;
        double r2323055 = cbrt(r2323054);
        double r2323056 = cbrt(r2323042);
        double r2323057 = -r2323049;
        double r2323058 = fma(r2323055, r2323056, r2323057);
        double r2323059 = r2323053 ? r2323058 : r2323051;
        double r2323060 = r2323023 ? r2323051 : r2323059;
        return r2323060;
}

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -3659.711310353127:\\
\;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}}\\

\mathbf{elif}\;x \le 2853.6107128789863:\\
\;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}\right))_*\\

\mathbf{else}:\\
\;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}}\\

\end{array}

Derivation

Split input into 2 regimes
if x < -3659.711310353127 or 2853.6107128789863 < x
1. Initial program 60.3
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Using strategy rm
3. Applied add-cbrt-cube60.4
  \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\]
4. Using strategy rm
5. Applied add-cube-cbrt60.6
  \[\leadsto \sqrt[3]{x + 1} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}\]
6. Using strategy rm
7. Applied flip--60.6
  \[\leadsto \color{blue}{\frac{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)} \cdot \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}{\sqrt[3]{x + 1} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}}\]
8. Taylor expanded around inf 33.6
  \[\leadsto \frac{\color{blue}{\left(\frac{4}{81} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + \frac{2}{3} \cdot {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}}}{\sqrt[3]{x + 1} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}\]
9. Simplified1.1
  \[\leadsto \frac{\color{blue}{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}}{\sqrt[3]{x + 1} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}\]
if -3659.711310353127 < x < 2853.6107128789863
1. Initial program 0.1
  \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
2. Using strategy rm
3. Applied add-cbrt-cube0.1
  \[\leadsto \sqrt[3]{x + 1} - \color{blue}{\sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}\]
4. Using strategy rm
5. Applied add-cube-cbrt0.1
  \[\leadsto \sqrt[3]{x + 1} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}}\]
6. Using strategy rm
7. Applied add-cube-cbrt0.1
  \[\leadsto \sqrt[3]{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\]
8. Applied cbrt-prod0.1
  \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\]
9. Applied fma-neg0.1
  \[\leadsto \color{blue}{(\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}\right)}\right))_*}\]
Recombined 2 regimes into one program.
Final simplification0.6
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3659.711310353127:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}}\\ \mathbf{elif}\;x \le 2853.6107128789863:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{(\left(\sqrt[3]{\frac{\frac{\frac{1}{x}}{x}}{x \cdot x}}\right) \cdot \frac{-1}{9} + \left((\frac{2}{3} \cdot \left(\sqrt[3]{\frac{1}{x}}\right) + \left(\sqrt[3]{\frac{1}{{x}^{7}}} \cdot \frac{4}{81}\right))_*\right))_*}{\sqrt[3]{1 + x} + \sqrt[3]{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right)}}\\ \end{array}\]

Error

Derivation

`if x < -3659.711310353127 or 2853.6107128789863 < x`

`if -3659.711310353127 < x < 2853.6107128789863`

Reproduce