Average Error: 22.5 → 29.4
Time: 26.0s
Precision: 64
Internal precision: 1408
\[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le 3509096337649261.5:\\ \;\;\;\;\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 - {\left({\left(\sqrt[3]{{x}^{\left(\frac{1}{3}\right)}}\right)}^3\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{{x}^{4}}} \cdot \frac{1}{3} - \left(\left(\sqrt[3]{\frac{1}{x}} - {x}^{\frac{-1}{3}}\right) + \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right)\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes.

  2. if x < 3509096337649261.5

    1. Initial program 1.3

      \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    2. Using strategy rm

    3. Applied flip-- 1.3

      \[\leadsto \color{blue}{\frac{{\left({\left(x + 1\right)}^{\left(\frac{1}{3}\right)}\right)}^2 - {\left({x}^{\left(\frac{1}{3}\right)}\right)}^2}{{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} + {x}^{\left(\frac{1}{3}\right)}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt 1.4

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

    if 3509096337649261.5 < x

    1. Initial program 46.0

      \[{\left(x + 1\right)}^{\left(\frac{1}{3}\right)} - {x}^{\left(\frac{1}{3}\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt 53.8

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

    4. Applied taylor 47.3

      \[\leadsto \left({x}^{\frac{-1}{3}} + \frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)\]

    5. Taylor expanded around inf 47.3

      \[\leadsto \color{blue}{\left({x}^{\frac{-1}{3}} + \frac{1}{3} \cdot {\left(\frac{1}{{x}^{4}}\right)}^{\frac{1}{3}}\right) - \left(\frac{1}{9} \cdot {\left(\frac{1}{{x}^{7}}\right)}^{\frac{1}{3}} + {\left(\frac{1}{x}\right)}^{\frac{1}{3}}\right)}\]

    6. Applied simplify 60.4

      \[\leadsto \color{blue}{\sqrt[3]{\frac{1}{{x}^{4}}} \cdot \frac{1}{3} - \left(\left(\sqrt[3]{\frac{1}{x}} - {x}^{\frac{-1}{3}}\right) + \frac{1}{9} \cdot \sqrt[3]{\frac{1}{{x}^{7}}}\right)}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 26.0s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (x)
  :name "NMSE problem 3.3.4"
  :pre (>= x 0)
  (- (pow (+ x 1) (/ 1 3)) (pow x (/ 1 3))))