Average Error: 9.6 → 0.1
Time: 41.0s
Precision: 64
Internal Precision: 1152
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -115.09004239056316:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{if}\;x \le 103.10500246290717:\\ \;\;\;\;\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original9.6
Target0.2
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -115.09004239056316

    1. Initial program 19.7

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

    2. Taylor expanded around inf 0.5

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

    3. Applied simplify0.1

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

    if -115.09004239056316 < x < 103.10500246290717

    1. Initial program 0.1

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

    if 103.10500246290717 < x

    1. Initial program 19.8

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

    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

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

    5. Applied associate-/r*0.1

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

Runtime

Time bar (total: 41.0s)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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