Average Error: 29.4 → 8.4
Time: 5.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4426.84223240725714:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 4.9505821770774615 \cdot 10^{-17}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4426.84223240725714:\\
\;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\

\mathbf{elif}\;x \le 4.9505821770774615 \cdot 10^{-17}:\\
\;\;\;\;\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\

\end{array}
double f(double x) {
        double r66091 = x;
        double r66092 = 1.0;
        double r66093 = r66091 + r66092;
        double r66094 = cbrt(r66093);
        double r66095 = cbrt(r66091);
        double r66096 = r66094 - r66095;
        return r66096;
}
↓
double f(double x) {
        double r66097 = x;
        double r66098 = -4426.842232407257;
        bool r66099 = r66097 <= r66098;
        double r66100 = 0.03292181069958847;
        double r66101 = 1.0;
        double r66102 = 4.0;
        double r66103 = pow(r66097, r66102);
        double r66104 = r66101 / r66103;
        double r66105 = r66100 * r66104;
        double r66106 = 0.037037037037037035;
        double r66107 = 3.0;
        double r66108 = pow(r66097, r66107);
        double r66109 = r66101 / r66108;
        double r66110 = r66106 * r66109;
        double r66111 = r66105 - r66110;
        double r66112 = r66106 / r66097;
        double r66113 = r66112 / r66097;
        double r66114 = r66111 + r66113;
        double r66115 = cbrt(r66114);
        double r66116 = 4.9505821770774615e-17;
        bool r66117 = r66097 <= r66116;
        double r66118 = 1.0;
        double r66119 = r66097 + r66118;
        double r66120 = cbrt(r66119);
        double r66121 = cbrt(r66120);
        double r66122 = r66121 * r66121;
        double r66123 = r66122 * r66121;
        double r66124 = cbrt(r66097);
        double r66125 = r66123 - r66124;
        double r66126 = 0.0;
        double r66127 = r66126 + r66118;
        double r66128 = r66120 + r66124;
        double r66129 = r66120 * r66128;
        double r66130 = 0.6666666666666666;
        double r66131 = pow(r66097, r66130);
        double r66132 = r66129 + r66131;
        double r66133 = r66127 / r66132;
        double r66134 = r66117 ? r66125 : r66133;
        double r66135 = r66099 ? r66115 : r66134;
        return r66135;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x < -4426.842232407257

    1. Initial program 60.1

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

    3. Applied add-cbrt-cube60.1

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

    4. Simplified60.1

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

    5. Taylor expanded around inf 30.5

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

    6. Simplified29.9

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

    if -4426.842232407257 < x < 4.9505821770774615e-17

    1. Initial program 0.0

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

    3. Applied add-cube-cbrt0.1

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

    if 4.9505821770774615e-17 < x

    1. Initial program 56.6

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

    3. Applied flip3--56.4

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

    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. Simplified4.3

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4426.84223240725714:\\ \;\;\;\;\sqrt[3]{\left(0.0329218106995884732 \cdot \frac{1}{{x}^{4}} - 0.037037037037037035 \cdot \frac{1}{{x}^{3}}\right) + \frac{\frac{0.037037037037037035}{x}}{x}}\\ \mathbf{elif}\;x \le 4.9505821770774615 \cdot 10^{-17}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020059 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))