Average Error: 30.2 → 9.0
Time: 17.3s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4661.327623294632758188527077436447143555:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\ \mathbf{elif}\;x \le 1.421809655538258285959601900927395945473 \cdot 10^{-4}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, {x}^{\frac{2}{3}}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4661.327623294632758188527077436447143555:\\
\;\;\;\;\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 1.421809655538258285959601900927395945473 \cdot 10^{-4}:\\
\;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\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 r48266 = x;
        double r48267 = 1.0;
        double r48268 = r48266 + r48267;
        double r48269 = cbrt(r48268);
        double r48270 = cbrt(r48266);
        double r48271 = r48269 - r48270;
        return r48271;
}
↓
double f(double x) {
        double r48272 = x;
        double r48273 = -4661.327623294633;
        bool r48274 = r48272 <= r48273;
        double r48275 = 0.3333333333333333;
        double r48276 = 1.0;
        double r48277 = 2.0;
        double r48278 = pow(r48272, r48277);
        double r48279 = r48276 / r48278;
        double r48280 = cbrt(r48279);
        double r48281 = 8.0;
        double r48282 = pow(r48272, r48281);
        double r48283 = r48276 / r48282;
        double r48284 = cbrt(r48283);
        double r48285 = 0.06172839506172839;
        double r48286 = r48284 * r48285;
        double r48287 = fma(r48275, r48280, r48286);
        double r48288 = 5.0;
        double r48289 = pow(r48272, r48288);
        double r48290 = r48276 / r48289;
        double r48291 = cbrt(r48290);
        double r48292 = 0.1111111111111111;
        double r48293 = r48291 * r48292;
        double r48294 = r48287 - r48293;
        double r48295 = 0.00014218096555382583;
        bool r48296 = r48272 <= r48295;
        double r48297 = 1.0;
        double r48298 = r48272 + r48297;
        double r48299 = cbrt(r48298);
        double r48300 = cbrt(r48272);
        double r48301 = cbrt(r48300);
        double r48302 = r48301 * r48301;
        double r48303 = r48302 * r48301;
        double r48304 = r48299 - r48303;
        double r48305 = exp(r48304);
        double r48306 = log(r48305);
        double r48307 = r48297 + r48272;
        double r48308 = cbrt(r48307);
        double r48309 = r48308 + r48300;
        double r48310 = 0.6666666666666666;
        double r48311 = pow(r48272, r48310);
        double r48312 = fma(r48309, r48308, r48311);
        double r48313 = r48297 / r48312;
        double r48314 = r48296 ? r48306 : r48313;
        double r48315 = r48274 ? r48294 : r48314;
        return r48315;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -4661.327623294633

    1. Initial program 60.3

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

    3. Applied add-cube-cbrt60.9

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

    4. Taylor expanded around inf 44.5

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

    5. Simplified31.1

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

    if -4661.327623294633 < x < 0.00014218096555382583

    1. Initial program 0.1

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

    3. Applied add-log-exp0.1

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

    4. Applied add-log-exp0.1

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

    5. Applied diff-log0.1

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

    6. Simplified0.1

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

    8. Applied add-cube-cbrt0.1

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

    if 0.00014218096555382583 < x

    1. Initial program 59.1

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

    3. Applied flip3--59.0

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

    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 simplification9.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4661.327623294632758188527077436447143555:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\ \mathbf{elif}\;x \le 1.421809655538258285959601900927395945473 \cdot 10^{-4}:\\ \;\;\;\;\log \left(e^{\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, {x}^{\frac{2}{3}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))