Average Error: 9.6 → 0.2
Time: 1.7m
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -97.97978611925785:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 115.87679664980179:\\ \;\;\;\;\left(\frac{1}{x - 1} + (\left(-\sqrt[3]{\frac{2}{x}}\right) \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) + \left(\left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) \cdot \sqrt[3]{\frac{2}{x}}\right))_*\right) + \left(\frac{1}{1 + x} - \frac{2}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\sqrt{\frac{2}{x \cdot x}}}{\frac{x}{\sqrt{\frac{2}{x \cdot x}}}}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original9.6
Target0.3
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -97.97978611925785

    1. Initial program 20.2

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    2. Taylor expanded around inf 0.6

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

    3. Simplified0.1

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

    5. Applied associate-/l/0.6

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

    if -97.97978611925785 < x < 115.87679664980179

    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

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

    if 115.87679664980179 < x

    1. Initial program 19.2

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

    2. Taylor expanded around inf 0.5

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

    3. Simplified0.1

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

    5. Applied add-sqr-sqrt0.2

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

    6. Applied associate-/l*0.2

      \[\leadsto \frac{2}{{x}^{7}} + \left(\color{blue}{\frac{\sqrt{\frac{2}{x \cdot x}}}{\frac{x}{\sqrt{\frac{2}{x \cdot x}}}}} + \frac{2}{{x}^{5}}\right)\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -97.97978611925785:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{2}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 115.87679664980179:\\ \;\;\;\;\left(\frac{1}{x - 1} + (\left(-\sqrt[3]{\frac{2}{x}}\right) \cdot \left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) + \left(\left(\sqrt[3]{\frac{2}{x}} \cdot \sqrt[3]{\frac{2}{x}}\right) \cdot \sqrt[3]{\frac{2}{x}}\right))_*\right) + \left(\frac{1}{1 + x} - \frac{2}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{7}} + \left(\frac{2}{{x}^{5}} + \frac{\sqrt{\frac{2}{x \cdot x}}}{\frac{x}{\sqrt{\frac{2}{x \cdot x}}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 +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))))