Average Error: 10.0 → 0.2
Time: 56.8s
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.081462638229255:\\ \;\;\;\;\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{if}\;x \le 98.29124047196814:\\ \;\;\;\;\frac{1}{x - 1} + (\left(\sqrt{\frac{1}{1 + x}}\right) \cdot \left(\sqrt{\frac{1}{1 + x}}\right) + \left(\frac{-2}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original10.0
Target0.3
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -1.081462638229255

    1. Initial program 20.1

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

    2. Taylor expanded around inf 0.8

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

    3. Applied simplify0.3

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

    if -1.081462638229255 < x < 98.29124047196814

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied prod-diff0.0

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

    6. Applied associate-+l+0.0

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

    7. Applied simplify0.0

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

    if 98.29124047196814 < x

    1. Initial program 19.7

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

    2. Taylor expanded around inf 0.6

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

    3. Applied simplify0.1

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

    5. Applied div-inv0.1

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

    6. Applied associate-/l*0.6

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

    7. Applied simplify0.6

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

  4. Applied simplify0.2

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;x \le -1.081462638229255:\\ \;\;\;\;\left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right) + \frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{if}\;x \le 98.29124047196814:\\ \;\;\;\;\frac{1}{x - 1} + (\left(\sqrt{\frac{1}{1 + x}}\right) \cdot \left(\sqrt{\frac{1}{1 + x}}\right) + \left(\frac{-2}{x}\right))_*\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{3}} + \left(\frac{2}{{x}^{5}} + \frac{2}{{x}^{7}}\right)\\ \end{array}}\]

Runtime

Time bar (total: 56.8s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +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))))