Average Error: 29.8 → 0.5
Time: 7.2s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\]
\sqrt[3]{x + 1} - \sqrt[3]{x}
↓
\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}
double f(double x) {
        double r76598 = x;
        double r76599 = 1.0;
        double r76600 = r76598 + r76599;
        double r76601 = cbrt(r76600);
        double r76602 = cbrt(r76598);
        double r76603 = r76601 - r76602;
        return r76603;
}
↓
double f(double x) {
        double r76604 = 1.0;
        double r76605 = x;
        double r76606 = r76605 + r76604;
        double r76607 = cbrt(r76606);
        double r76608 = r76607 * r76607;
        double r76609 = cbrt(r76605);
        double r76610 = r76607 + r76609;
        double r76611 = r76609 * r76610;
        double r76612 = r76608 + r76611;
        double r76613 = r76604 / r76612;
        return r76613;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.8

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

  3. Applied add-cbrt-cube29.8

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

  4. Simplified29.8

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

  6. Applied flip3--29.7

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

  7. Simplified29.0

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

  8. Simplified29.0

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

  10. Applied *-un-lft-identity29.0

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

  11. Applied *-un-lft-identity29.0

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

  12. Applied distribute-lft-out--29.0

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

  13. Simplified14.6

    \[\leadsto \sqrt[3]{{\left(\frac{1 \cdot \color{blue}{\left(0 + 1\right)}}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{x}\right)}\right)}^{3}}\]
  14. Using strategy rm

  15. Applied rem-cbrt-cube0.5

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

  16. Final simplification0.5

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

Reproduce

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