Average Error: 29.6 → 8.8
Time: 19.5s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5010.644987029124422406312078237533569336:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\ \mathbf{elif}\;x \le 3.197944084307661311537829273987654232769 \cdot 10^{-4}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, {x}^{\frac{2}{3}}\right)}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5010.644987029124422406312078237533569336:\\
\;\;\;\;\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 3.197944084307661311537829273987654232769 \cdot 10^{-4}:\\
\;\;\;\;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{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 r41405 = x;
        double r41406 = 1.0;
        double r41407 = r41405 + r41406;
        double r41408 = cbrt(r41407);
        double r41409 = cbrt(r41405);
        double r41410 = r41408 - r41409;
        return r41410;
}
↓
double f(double x) {
        double r41411 = x;
        double r41412 = -5010.644987029124;
        bool r41413 = r41411 <= r41412;
        double r41414 = 0.3333333333333333;
        double r41415 = 1.0;
        double r41416 = 2.0;
        double r41417 = pow(r41411, r41416);
        double r41418 = r41415 / r41417;
        double r41419 = cbrt(r41418);
        double r41420 = 8.0;
        double r41421 = pow(r41411, r41420);
        double r41422 = r41415 / r41421;
        double r41423 = cbrt(r41422);
        double r41424 = 0.06172839506172839;
        double r41425 = r41423 * r41424;
        double r41426 = fma(r41414, r41419, r41425);
        double r41427 = 5.0;
        double r41428 = pow(r41411, r41427);
        double r41429 = r41415 / r41428;
        double r41430 = cbrt(r41429);
        double r41431 = 0.1111111111111111;
        double r41432 = r41430 * r41431;
        double r41433 = r41426 - r41432;
        double r41434 = 0.00031979440843076613;
        bool r41435 = r41411 <= r41434;
        double r41436 = 1.0;
        double r41437 = r41411 + r41436;
        double r41438 = cbrt(r41437);
        double r41439 = r41438 * r41438;
        double r41440 = cbrt(r41439);
        double r41441 = cbrt(r41438);
        double r41442 = r41440 * r41441;
        double r41443 = cbrt(r41411);
        double r41444 = r41442 - r41443;
        double r41445 = log(r41444);
        double r41446 = exp(r41445);
        double r41447 = r41436 + r41411;
        double r41448 = cbrt(r41447);
        double r41449 = r41448 + r41443;
        double r41450 = 0.6666666666666666;
        double r41451 = pow(r41411, r41450);
        double r41452 = fma(r41449, r41448, r41451);
        double r41453 = r41436 / r41452;
        double r41454 = r41435 ? r41446 : r41453;
        double r41455 = r41413 ? r41433 : r41454;
        return r41455;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -5010.644987029124

    1. Initial program 60.0

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

    2. Taylor expanded around inf 45.2

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

    3. 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 -5010.644987029124 < x < 0.00031979440843076613

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied cbrt-prod0.1

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

    5. Applied fma-neg0.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{x}\right)}\]
    6. Using strategy rm

    7. Applied add-cube-cbrt0.1

      \[\leadsto \mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\color{blue}{\left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right) \cdot \sqrt[3]{\sqrt[3]{x}}}\right)\]
    8. Using strategy rm

    9. Applied add-exp-log0.1

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

    10. Simplified0.1

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

    if 0.00031979440843076613 < x

    1. Initial program 58.8

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

    3. Applied flip3--58.7

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

    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

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

  4. Final simplification8.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5010.644987029124422406312078237533569336:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.06172839506172839163511412152729462832212\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111049432054187491303309798\\ \mathbf{elif}\;x \le 3.197944084307661311537829273987654232769 \cdot 10^{-4}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \sqrt[3]{\sqrt[3]{x + 1}} - \sqrt[3]{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\sqrt[3]{1 + x} + \sqrt[3]{x}, \sqrt[3]{1 + x}, {x}^{\frac{2}{3}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))