Average Error: 9.7 → 0.1
Time: 42.3s
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -110.34035026761937 \lor \neg \left(x \le 107.7956188699589\right):\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{1}{x} \cdot \frac{\frac{2}{x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \left(\frac{1}{x - 1} - \frac{2}{x}\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.7
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -110.34035026761937 or 107.7956188699589 < x

    1. Initial program 19.1

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

    2. Initial simplification19.1

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

    4. Applied clear-num19.1

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

    5. Taylor expanded around -inf 0.5

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

    6. Simplified0.1

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

    8. Applied associate-/r*0.1

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

    10. Applied div-inv0.2

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

    if -110.34035026761937 < x < 107.7956188699589

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

    4. Applied clear-num0.0

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -110.34035026761937 \lor \neg \left(x \le 107.7956188699589\right):\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{1}{x} \cdot \frac{\frac{2}{x}}{x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + 1} + \left(\frac{1}{x - 1} - \frac{2}{x}\right)\\ \end{array}\]

Runtime

Time bar (total: 42.3s)Debug logProfile

herbie shell --seed 2018227 +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))))