Average Error: 29.8 → 0.5
Time: 10.8s
Precision: 64
Internal Precision: 128
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 3.931414823910018 \cdot 10^{-06}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (- (cbrt (+ x 1)) (cbrt x)) < 3.931414823910018e-06

    1. Initial program 60.8

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

    2. Taylor expanded around 0 61.2

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

    3. Simplified60.8

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

    4. Taylor expanded around -inf 62.4

      \[\leadsto \color{blue}{\left(e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)} + \frac{1}{3} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{x}\right) - \left({\left(x \cdot -1\right)}^{\frac{1}{3}} \cdot \sqrt[3]{-1} + \frac{1}{9} \cdot \frac{e^{\frac{1}{3} \cdot \left(\log -1 - \log \left(\frac{-1}{x}\right)\right)}}{{x}^{2}}\right)}\]

    5. Simplified0.6

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

    if 3.931414823910018e-06 < (- (cbrt (+ x 1)) (cbrt x))

    1. Initial program 0.4

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

    2. Taylor expanded around 0 31.1

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

    3. Simplified0.4

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{1 + x} - \sqrt[3]{x} \le 3.931414823910018 \cdot 10^{-06}:\\ \;\;\;\;\left(\sqrt[3]{x} - \sqrt[3]{-1} \cdot \sqrt[3]{-x}\right) + \left(\frac{1}{3} - \frac{\frac{1}{9}}{x}\right) \cdot \frac{\sqrt[3]{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{1 + x} - \sqrt[3]{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))