Average Error: 29.5 → 0.4
Time: 22.7s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4030.9229650993684 \lor \neg \left(x \le 4165.8468806330775\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{\frac{x \cdot x}{\frac{1}{3}}}\right) \cdot \left(\frac{-1}{3} + \frac{\frac{5}{27}}{x}\right) + \left(\frac{\frac{1}{3}}{x} \cdot \sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes

  2. if x < -4030.9229650993684 or 4165.8468806330775 < x

    1. Initial program 60.2

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

    2. Initial simplification60.2

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

    4. Applied add-exp-log60.2

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

    5. Taylor expanded around inf 33.9

      \[\leadsto \color{blue}{\left(\frac{5}{27} \cdot \frac{e^{\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \log \left(\frac{1}{x}\right)}}{{x}^{2}} + e^{\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \log \left(\frac{1}{x}\right)}\right) - \frac{1}{3} \cdot \frac{e^{\log \left(\frac{1}{3} \cdot {x}^{\frac{1}{3}}\right) + \log \left(\frac{1}{x}\right)}}{x}}\]

    6. Simplified0.6

      \[\leadsto \color{blue}{(\left(\frac{\sqrt[3]{x}}{\frac{x \cdot x}{\frac{1}{3}}}\right) \cdot \left(\frac{\frac{5}{27}}{x} + \frac{-1}{3}\right) + \left(\sqrt[3]{x} \cdot \frac{\frac{1}{3}}{x}\right))_*}\]

    if -4030.9229650993684 < x < 4165.8468806330775

    1. Initial program 0.1

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

    2. Initial simplification0.1

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

    4. Applied add-cube-cbrt0.1

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

    5. Applied cbrt-prod0.1

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

    6. Applied fma-neg0.1

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4030.9229650993684 \lor \neg \left(x \le 4165.8468806330775\right):\\ \;\;\;\;(\left(\frac{\sqrt[3]{x}}{\frac{x \cdot x}{\frac{1}{3}}}\right) \cdot \left(\frac{-1}{3} + \frac{\frac{5}{27}}{x}\right) + \left(\frac{\frac{1}{3}}{x} \cdot \sqrt[3]{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;(\left(\sqrt[3]{\sqrt[3]{1 + x} \cdot \sqrt[3]{1 + x}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{1 + x}}\right) + \left(-\sqrt[3]{x}\right))_*\\ \end{array}\]

Runtime

Time bar (total: 22.7s)Debug logProfile

herbie shell --seed 2018255 +o rules:numerics
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))