Average Error: 10.1 → 1.1
Time: 59.9s
Precision: 64
Internal precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
⬇
\[\begin{array}{l} \mathbf{if}\;x \le -0.17532047635385054:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^3}\\ \mathbf{if}\;x \le 5.980014352468749 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(\left(x - 2\right) - \left(x + x\right)\right) + \left(x \cdot x + x\right)}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{2}{{x}^{5}}\right) + \frac{2}{{x}^3}\\ \end{array}\]

Error

Bits error versus x

Target

Original10.1
Comparison0.4
Herbie1.1
\[ \frac{2}{x \cdot \left({x}^2 - 1\right)} \]

Derivation

  1. Split input into 2 regimes.

  2. if x < -0.17532047635385054 or 5.980014352468749e+22 < x

    1. Initial program 18.9

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

    2. Applied taylor 0.6

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

    3. Taylor expanded around inf 0.6

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

    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. Applied simplify 0.6

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

    if -0.17532047635385054 < x < 5.980014352468749e+22

    1. Initial program 1.8

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

    3. Applied frac-sub 1.9

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

    4. Applied frac-add 1.5

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

    5. Applied simplify 1.6

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

Runtime

Time bar (total: 59.9s) Debug logProfile

Please include this information when filing a bug report:

herbie shell --seed '#(1064651971 495577305 2200811460 13024471 864198081 231948279)'
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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