Average Error: 30.2 → 16.1
Time: 53.9s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{x} \le -559.9900762991593 \lor \neg \left(\sqrt[3]{x} \le 10.627650893067266\right):\\ \;\;\;\;(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if (cbrt x) < -559.9900762991593 or 10.627650893067266 < (cbrt x)

    1. Initial program 60.5

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

    3. Applied add-cbrt-cube60.6

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

    5. Applied cbrt-prod60.8

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

    6. Taylor expanded around inf 38.9

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

    7. Simplified31.5

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

    if -559.9900762991593 < (cbrt x) < 10.627650893067266

    1. Initial program 0.2

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

    3. Applied add-cbrt-cube0.3

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

    5. Applied add-sqr-sqrt0.9

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

    6. Applied fma-neg0.9

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

  4. Final simplification16.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt[3]{x} \le -559.9900762991593 \lor \neg \left(\sqrt[3]{x} \le 10.627650893067266\right):\\ \;\;\;\;(\left(\sqrt[3]{\frac{1}{{x}^{5}}}\right) \cdot \frac{-1}{9} + \left((\frac{1}{3} \cdot \left(\sqrt[3]{\frac{1}{x \cdot x}}\right) + \left(\frac{5}{81} \cdot \sqrt[3]{\frac{1}{{x}^{8}}}\right))_*\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt{\sqrt[3]{x + 1}}\right) \cdot \left(\sqrt{\sqrt[3]{x + 1}}\right) + \left(-\sqrt[3]{\sqrt[3]{x} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 53.9s)Debug logProfile

herbie shell --seed 2018277 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))