Average Error: 29.2 → 8.9
Time: 14.7s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4221.179793202217297221068292856216430664:\\ \;\;\;\;\mathsf{fma}\left(0.06172839506172839163511412152729462832212, \sqrt[3]{\frac{1}{{x}^{8}}}, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \left(-0.1111111111111111049432054187491303309798\right)\right)\right)\\ \mathbf{elif}\;x \le 2.627646657846368341371692551461380915612 \cdot 10^{-7}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{1 + x}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4221.179793202217297221068292856216430664:\\
\;\;\;\;\mathsf{fma}\left(0.06172839506172839163511412152729462832212, \sqrt[3]{\frac{1}{{x}^{8}}}, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \left(-0.1111111111111111049432054187491303309798\right)\right)\right)\\

\mathbf{elif}\;x \le 2.627646657846368341371692551461380915612 \cdot 10^{-7}:\\
\;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\

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

\end{array}
double f(double x) {
        double r84415 = x;
        double r84416 = 1.0;
        double r84417 = r84415 + r84416;
        double r84418 = cbrt(r84417);
        double r84419 = cbrt(r84415);
        double r84420 = r84418 - r84419;
        return r84420;
}
↓
double f(double x) {
        double r84421 = x;
        double r84422 = -4221.179793202217;
        bool r84423 = r84421 <= r84422;
        double r84424 = 0.06172839506172839;
        double r84425 = 1.0;
        double r84426 = 8.0;
        double r84427 = pow(r84421, r84426);
        double r84428 = r84425 / r84427;
        double r84429 = cbrt(r84428);
        double r84430 = r84421 * r84421;
        double r84431 = r84425 / r84430;
        double r84432 = cbrt(r84431);
        double r84433 = 0.3333333333333333;
        double r84434 = 5.0;
        double r84435 = pow(r84421, r84434);
        double r84436 = r84425 / r84435;
        double r84437 = cbrt(r84436);
        double r84438 = 0.1111111111111111;
        double r84439 = -r84438;
        double r84440 = r84437 * r84439;
        double r84441 = fma(r84432, r84433, r84440);
        double r84442 = fma(r84424, r84429, r84441);
        double r84443 = 2.6276466578463683e-07;
        bool r84444 = r84421 <= r84443;
        double r84445 = 1.0;
        double r84446 = r84445 + r84421;
        double r84447 = cbrt(r84446);
        double r84448 = cbrt(r84421);
        double r84449 = r84447 - r84448;
        double r84450 = 0.6666666666666666;
        double r84451 = pow(r84421, r84450);
        double r84452 = r84448 + r84447;
        double r84453 = r84452 * r84447;
        double r84454 = r84451 + r84453;
        double r84455 = r84445 / r84454;
        double r84456 = r84444 ? r84449 : r84455;
        double r84457 = r84423 ? r84442 : r84456;
        return r84457;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -4221.179793202217

    1. Initial program 60.1

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

    3. Applied add-cube-cbrt60.3

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

    4. Applied cbrt-prod60.6

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

    5. Simplified60.6

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

    6. Simplified60.6

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

    7. Taylor expanded around inf 45.5

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

    8. Simplified31.7

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

    if -4221.179793202217 < x < 2.6276466578463683e-07

    1. Initial program 0.1

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

    if 2.6276466578463683e-07 < x

    1. Initial program 58.4

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

    3. Applied flip3--58.3

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

    4. Simplified1.0

      \[\leadsto \frac{\color{blue}{1 + 0}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \left(\sqrt[3]{x} \cdot \sqrt[3]{x} + \sqrt[3]{x + 1} \cdot \sqrt[3]{x}\right)}\]

    5. Simplified4.4

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

  4. Final simplification8.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4221.179793202217297221068292856216430664:\\ \;\;\;\;\mathsf{fma}\left(0.06172839506172839163511412152729462832212, \sqrt[3]{\frac{1}{{x}^{8}}}, \mathsf{fma}\left(\sqrt[3]{\frac{1}{x \cdot x}}, 0.3333333333333333148296162562473909929395, \sqrt[3]{\frac{1}{{x}^{5}}} \cdot \left(-0.1111111111111111049432054187491303309798\right)\right)\right)\\ \mathbf{elif}\;x \le 2.627646657846368341371692551461380915612 \cdot 10^{-7}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{\frac{2}{3}} + \left(\sqrt[3]{x} + \sqrt[3]{1 + x}\right) \cdot \sqrt[3]{1 + x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1.0)) (cbrt x)))