Average Error: 10.0 → 0.5
Time: 17.0s
Precision: 64
Internal precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -6.667433711053936 \cdot 10^{-08}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{1}{x} \cdot \frac{2}{{x}^2}\\ \mathbf{if}\;x \le 1183236732512159.2:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{1}{x} \cdot \frac{2}{{x}^2}\\ \end{array}\]

Error

Bits error versus x

Derivation

  1. Split input into 2 regimes.

  2. if x < -6.667433711053936e-08 or 1183236732512159.2 < x

    1. Initial program 19.3

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

    2. Applied taylor 0.5

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

    3. Taylor expanded around inf 0.5

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

    4. Applied simplify 0.1

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

    6. Applied *-un-lft-identity 0.1

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

    7. Applied times-frac 0.2

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

    8. Applied simplify 0.1

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

    if -6.667433711053936e-08 < x < 1183236732512159.2

    1. Initial program 0.8

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  3. Recombined 2 regimes into one program.
  4. Removed slow pow expressions

Runtime

Time bar (total: 17.0s) Debug log

Please include this information when filing a bug report:

herbie shell --seed '#(3052192724 3812927732 3686175817 630908657 2373248591 511094450)'
(FPCore (x)
  :name "NMSE problem 3.3.3"
  :pre (!= x 0 1 -1)
  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))