Average Error: 30.0 → 15.7
Time: 12.2s
Precision: 64
Internal Precision: 1344
\[\sqrt[3]{x + 1} - \sqrt[3]{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4126.085170941377 \lor \neg \left(x \le 3787.871668043524\right):\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81} + \left(\frac{1}{3} \cdot \sqrt[3]{\frac{1}{x \cdot x}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \left(e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)} - \sqrt[3]{x}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -4126.085170941377 or 3787.871668043524 < 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-cube-cbrt60.5

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

    5. Taylor expanded around inf 39.1

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

    6. Simplified31.5

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

    if -4126.085170941377 < x < 3787.871668043524

    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-cbrt-cube0.1

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

    6. Applied add-cube-cbrt0.1

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

    7. Applied cbrt-prod0.1

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

    9. Applied add-exp-log0.1

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

  4. Final simplification15.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4126.085170941377 \lor \neg \left(x \le 3787.871668043524\right):\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{8}}} \cdot \frac{5}{81} + \left(\frac{1}{3} \cdot \sqrt[3]{\frac{1}{x \cdot x}} - \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \left(e^{\log \left(\sqrt[3]{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}\right)} \cdot \sqrt[3]{\sqrt[3]{x + 1}}\right)} - \sqrt[3]{x}\\ \end{array}\]

Runtime

Time bar (total: 12.2s)Debug logProfile

BaselineHerbieOracleSpan%
Regimes30.115.715.714.499.7%
herbie shell --seed 2018297 
(FPCore (x)
  :name "2cbrt (problem 3.3.4)"
  (- (cbrt (+ x 1)) (cbrt x)))