Average Error: 10.2 → 0.2
Time: 1.2m
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -141.9708462546792:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^{3}}\\ \mathbf{if}\;x \le 117.44589041658848:\\ \;\;\;\;\frac{\left(x - 2\right) - x \cdot 2}{\left(x + 1\right) \cdot x} + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{\frac{\frac{2}{x}}{x}}{x}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -141.9708462546792

    1. Initial program 20.5

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

    2. Taylor expanded around inf 0.4

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

    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*0.5

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

    7. Applied simplify0.4

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

    if -141.9708462546792 < x < 117.44589041658848

    1. Initial program 0.1

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

    3. Applied frac-sub0.1

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

    4. Applied simplify0.1

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

    if 117.44589041658848 < x

    1. Initial program 20.5

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

    2. Taylor expanded around inf 0.4

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

    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 associate-/r*0.1

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018198 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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