Average Error: 29.9 → 0.6
Time: 12.4s
Precision: 64
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1} + \sqrt[3]{x} \cdot \left(\sqrt[3]{x + 1} + \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right)}\]
\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]{-1} \cdot \sqrt[3]{-x}\right)}
double f(double x) {
        double r1299429 = x;
        double r1299430 = 1.0;
        double r1299431 = r1299429 + r1299430;
        double r1299432 = cbrt(r1299431);
        double r1299433 = cbrt(r1299429);
        double r1299434 = r1299432 - r1299433;
        return r1299434;
}
↓
double f(double x) {
        double r1299435 = 1.0;
        double r1299436 = x;
        double r1299437 = r1299436 + r1299435;
        double r1299438 = cbrt(r1299437);
        double r1299439 = r1299438 * r1299438;
        double r1299440 = cbrt(r1299436);
        double r1299441 = -1.0;
        double r1299442 = cbrt(r1299441);
        double r1299443 = -r1299436;
        double r1299444 = cbrt(r1299443);
        double r1299445 = r1299442 * r1299444;
        double r1299446 = r1299438 + r1299445;
        double r1299447 = r1299440 * r1299446;
        double r1299448 = r1299439 + r1299447;
        double r1299449 = r1299435 / r1299448;
        return r1299449;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 29.9

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

  3. Applied flip3--29.8

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

  4. Simplified0.6

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

  5. Simplified0.6

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

  6. Taylor expanded around -inf 32.2

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

  7. Simplified0.6

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

  8. Final simplification0.6

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

Reproduce

herbie shell --seed 2019156 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))