Average Error: 29.8 → 8.6
Time: 10.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3699.679152965851:\\ \;\;\;\;\mathsf{fma}\left(0.333333333333333315, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111\\ \mathbf{elif}\;x \le 3.48590003758747111 \cdot 10^{-6}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3699.679152965851:\\
\;\;\;\;\mathsf{fma}\left(0.333333333333333315, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111\\

\mathbf{elif}\;x \le 3.48590003758747111 \cdot 10^{-6}:\\
\;\;\;\;e^{\log \left(\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{x}\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{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 r70624 = x;
        double r70625 = 1.0;
        double r70626 = r70624 + r70625;
        double r70627 = cbrt(r70626);
        double r70628 = cbrt(r70624);
        double r70629 = r70627 - r70628;
        return r70629;
}
↓
double f(double x) {
        double r70630 = x;
        double r70631 = -3699.6791529658512;
        bool r70632 = r70630 <= r70631;
        double r70633 = 0.3333333333333333;
        double r70634 = 1.0;
        double r70635 = 2.0;
        double r70636 = pow(r70630, r70635);
        double r70637 = r70634 / r70636;
        double r70638 = cbrt(r70637);
        double r70639 = 8.0;
        double r70640 = pow(r70630, r70639);
        double r70641 = r70634 / r70640;
        double r70642 = cbrt(r70641);
        double r70643 = 0.06172839506172839;
        double r70644 = r70642 * r70643;
        double r70645 = fma(r70633, r70638, r70644);
        double r70646 = 5.0;
        double r70647 = pow(r70630, r70646);
        double r70648 = r70634 / r70647;
        double r70649 = cbrt(r70648);
        double r70650 = 0.1111111111111111;
        double r70651 = r70649 * r70650;
        double r70652 = r70645 - r70651;
        double r70653 = 3.485900037587471e-06;
        bool r70654 = r70630 <= r70653;
        double r70655 = 1.0;
        double r70656 = r70630 + r70655;
        double r70657 = cbrt(r70656);
        double r70658 = r70657 * r70657;
        double r70659 = cbrt(r70658);
        double r70660 = cbrt(r70657);
        double r70661 = cbrt(r70630);
        double r70662 = -r70661;
        double r70663 = fma(r70659, r70660, r70662);
        double r70664 = log(r70663);
        double r70665 = exp(r70664);
        double r70666 = r70657 + r70661;
        double r70667 = r70657 * r70666;
        double r70668 = 0.6666666666666666;
        double r70669 = pow(r70630, r70668);
        double r70670 = r70667 + r70669;
        double r70671 = r70655 / r70670;
        double r70672 = r70654 ? r70665 : r70671;
        double r70673 = r70632 ? r70652 : r70672;
        return r70673;
}

Error

Bits error versus x

Derivation

  1. Split input into 3 regimes

  2. if x < -3699.6791529658512

    1. Initial program 60.1

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

    2. Taylor expanded around inf 45.0

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

    3. Simplified29.7

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

    if -3699.6791529658512 < x < 3.485900037587471e-06

    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-exp-log0.1

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

    if 3.485900037587471e-06 < x

    1. Initial program 58.4

      \[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
    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 simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3699.679152965851:\\ \;\;\;\;\mathsf{fma}\left(0.333333333333333315, \sqrt[3]{\frac{1}{{x}^{2}}}, \sqrt[3]{\frac{1}{{x}^{8}}} \cdot 0.061728395061728392\right) - \sqrt[3]{\frac{1}{{x}^{5}}} \cdot 0.1111111111111111\\ \mathbf{elif}\;x \le 3.48590003758747111 \cdot 10^{-6}:\\ \;\;\;\;e^{\log \left(\mathsf{fma}\left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}, \sqrt[3]{\sqrt[3]{x + 1}}, -\sqrt[3]{x}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt[3]{x + 1} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right) + {x}^{\frac{2}{3}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  :precision binary64
  (- (cbrt (+ x 1)) (cbrt x)))