Average Error: 10.1 → 0.1
Time: 4.2m
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -97.01646070172622:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 110.98110507549657:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \left((\left(-\sqrt[3]{\frac{2}{x}}\right) \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) + \left(\left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) \cdot \sqrt[3]{\frac{2}{x}}\right))_* + \frac{1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original10.1
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -97.01646070172622 or 110.98110507549657 < x

    1. Initial program 20.1

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

    3. Applied add-cube-cbrt52.0

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

    4. Applied add-cube-cbrt54.1

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

    5. Applied prod-diff53.9

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

    6. Applied associate-+l+53.9

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

    7. Simplified39.1

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

    8. Taylor expanded around -inf 0.6

      \[\leadsto \color{blue}{2 \cdot \frac{1}{{x}^{7}} + \left(2 \cdot \frac{1}{{x}^{3}} + 2 \cdot \frac{1}{{x}^{5}}\right)}\]

    9. Simplified0.1

      \[\leadsto \color{blue}{\frac{2}{{x}^{7}} + \left(\frac{\frac{2}{x}}{x \cdot x} + \frac{2}{{x}^{5}}\right)}\]

    if -97.01646070172622 < x < 110.98110507549657

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt1.3

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

    4. Applied add-cube-cbrt1.3

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

    5. Applied prod-diff1.3

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

    6. Applied associate-+l+1.3

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

    7. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -97.01646070172622:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 110.98110507549657:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \left((\left(-\sqrt[3]{\frac{2}{x}}\right) \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) + \left(\left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) \cdot \sqrt[3]{\frac{2}{x}}\right))_* + \frac{1}{x - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \end{array}\]

Runtime

Time bar (total: 4.2m)Debug logProfile

herbie shell --seed 2018234 +o rules:numerics
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))