Average Error: 30.4 → 11.6
Time: 5.9s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.4424562017088699 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 2.22816725821993897 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.4424562017088699 \cdot 10^{61}:\\
\;\;\;\;\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}}\\

\mathbf{elif}\;x \le 2.22816725821993897 \cdot 10^{-7}:\\
\;\;\;\;e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\right)}\\

\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 r67222 = x;
        double r67223 = 1.0;
        double r67224 = r67222 + r67223;
        double r67225 = cbrt(r67224);
        double r67226 = cbrt(r67222);
        double r67227 = r67225 - r67226;
        return r67227;
}
↓
double f(double x) {
        double r67228 = x;
        double r67229 = -4.44245620170887e+61;
        bool r67230 = r67228 <= r67229;
        double r67231 = 0.3333333333333333;
        double r67232 = 1.0;
        double r67233 = 2.0;
        double r67234 = pow(r67228, r67233);
        double r67235 = r67232 / r67234;
        double r67236 = 0.3333333333333333;
        double r67237 = pow(r67235, r67236);
        double r67238 = r67231 * r67237;
        double r67239 = 0.06172839506172839;
        double r67240 = 8.0;
        double r67241 = pow(r67228, r67240);
        double r67242 = r67232 / r67241;
        double r67243 = pow(r67242, r67236);
        double r67244 = r67239 * r67243;
        double r67245 = r67238 + r67244;
        double r67246 = 0.1111111111111111;
        double r67247 = 5.0;
        double r67248 = pow(r67228, r67247);
        double r67249 = r67232 / r67248;
        double r67250 = pow(r67249, r67236);
        double r67251 = r67246 * r67250;
        double r67252 = r67245 - r67251;
        double r67253 = 2.228167258219939e-07;
        bool r67254 = r67228 <= r67253;
        double r67255 = 1.0;
        double r67256 = r67228 + r67255;
        double r67257 = cbrt(r67256);
        double r67258 = r67257 * r67257;
        double r67259 = cbrt(r67258);
        double r67260 = cbrt(r67257);
        double r67261 = r67259 * r67260;
        double r67262 = cbrt(r67228);
        double r67263 = r67261 - r67262;
        double r67264 = log(r67263);
        double r67265 = exp(r67264);
        double r67266 = 0.0;
        double r67267 = r67266 + r67255;
        double r67268 = r67257 + r67262;
        double r67269 = r67257 * r67268;
        double r67270 = 0.6666666666666666;
        double r67271 = pow(r67228, r67270);
        double r67272 = r67269 + r67271;
        double r67273 = r67267 / r67272;
        double r67274 = r67254 ? r67265 : r67273;
        double r67275 = r67230 ? r67252 : r67274;
        return r67275;
}

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 < -4.44245620170887e+61

    1. Initial program 61.2

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

    2. Taylor expanded around inf 38.9

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

    if -4.44245620170887e+61 < x < 2.228167258219939e-07

    1. Initial program 5.0

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

    3. Applied add-cube-cbrt5.0

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

    4. Applied cbrt-prod5.0

      \[\leadsto \color{blue}{\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}}} - \sqrt[3]{x}\]
    5. Using strategy rm

    6. Applied add-exp-log5.0

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

    if 2.228167258219939e-07 < x

    1. Initial program 58.3

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

    3. Applied flip3--58.2

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

    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. Simplified4.4

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.4424562017088699 \cdot 10^{61}:\\ \;\;\;\;\left(0.333333333333333315 \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + 0.061728395061728392 \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - 0.1111111111111111 \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}\\ \mathbf{elif}\;x \le 2.22816725821993897 \cdot 10^{-7}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 + 1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))