Average Error: 29.9 → 14.8
Time: 1.1m
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3168.4030408053313:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{5}{81}\right))_*\right))_*\\ \mathbf{elif}\;x \le 3949.2773822279364:\\ \;\;\;\;\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]
double f(double x) {
        double r3996523 = x;
        double r3996524 = 1.0;
        double r3996525 = r3996523 + r3996524;
        double r3996526 = cbrt(r3996525);
        double r3996527 = cbrt(r3996523);
        double r3996528 = r3996526 - r3996527;
        return r3996528;
}
↓
double f(double x) {
        double r3996529 = x;
        double r3996530 = -3168.4030408053313;
        bool r3996531 = r3996529 <= r3996530;
        double r3996532 = -0.1111111111111111;
        double r3996533 = 1.0;
        double r3996534 = 5.0;
        double r3996535 = pow(r3996529, r3996534);
        double r3996536 = r3996533 / r3996535;
        double r3996537 = cbrt(r3996536);
        double r3996538 = 0.3333333333333333;
        double r3996539 = r3996529 * r3996529;
        double r3996540 = r3996533 / r3996539;
        double r3996541 = cbrt(r3996540);
        double r3996542 = r3996539 * r3996539;
        double r3996543 = r3996542 * r3996542;
        double r3996544 = r3996533 / r3996543;
        double r3996545 = cbrt(r3996544);
        double r3996546 = 0.06172839506172839;
        double r3996547 = r3996545 * r3996546;
        double r3996548 = fma(r3996538, r3996541, r3996547);
        double r3996549 = fma(r3996532, r3996537, r3996548);
        double r3996550 = 3949.2773822279364;
        bool r3996551 = r3996529 <= r3996550;
        double r3996552 = r3996529 + r3996533;
        double r3996553 = cbrt(r3996552);
        double r3996554 = cbrt(r3996529);
        double r3996555 = r3996554 * r3996554;
        double r3996556 = r3996554 * r3996555;
        double r3996557 = cbrt(r3996556);
        double r3996558 = cbrt(r3996557);
        double r3996559 = r3996558 * r3996558;
        double r3996560 = r3996559 * r3996558;
        double r3996561 = r3996553 - r3996560;
        double r3996562 = r3996551 ? r3996561 : r3996549;
        double r3996563 = r3996531 ? r3996549 : r3996562;
        return r3996563;
}


\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3168.4030408053313:\\
\;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{5}{81}\right))_*\right))_*\\

\mathbf{elif}\;x \le 3949.2773822279364:\\
\;\;\;\;\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{5}{81}\right))_*\right))_*\\

\end{array}

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -3168.4030408053313 or 3949.2773822279364 < x

    1. Initial program 60.3

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

    2. Taylor expanded around inf 37.7

      \[\leadsto \color{blue}{\left(\frac{1}{3} \cdot {\left(\frac{1}{{x}^{2}}\right)}^{\frac{1}{3}} + \frac{5}{81} \cdot {\left(\frac{1}{{x}^{8}}\right)}^{\frac{1}{3}}\right) - \frac{1}{9} \cdot {\left(\frac{1}{{x}^{5}}\right)}^{\frac{1}{3}}}\]

    3. Simplified29.8

      \[\leadsto \color{blue}{(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{5}{81}\right))_*\right))_*}\]

    if -3168.4030408053313 < x < 3949.2773822279364

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    5. Applied add-cube-cbrt0.1

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

  4. Final simplification14.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3168.4030408053313:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{5}{81}\right))_*\right))_*\\ \mathbf{elif}\;x \le 3949.2773822279364:\\ \;\;\;\;\sqrt[3]{x + 1} - \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;(\frac{-1}{9} \cdot \left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\sqrt[3]{\frac{1}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)}} \cdot \frac{5}{81}\right))_*\right))_*\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))