Average Error: 9.6 → 0.1
Time: 46.5s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -127.92652278327832:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 105.50043142207437:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \end{array}\]

Error

Bits error versus x

Target

Original9.6
Target0.3
Herbie0.1
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -127.92652278327832 or 105.50043142207437 < x

    1. Initial program 19.7

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

    3. Applied flip--53.0

      \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} \cdot \frac{1}{x + 1} - \frac{2}{x} \cdot \frac{2}{x}}{\frac{1}{x + 1} + \frac{2}{x}}} + \frac{1}{x - 1}\]

    4. Applied frac-add54.2

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

    5. Simplified25.4

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

    6. Taylor expanded around inf 0.5

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

    7. Simplified0.1

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

    if -127.92652278327832 < x < 105.50043142207437

    1. Initial program 0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -127.92652278327832:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 105.50043142207437:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019093 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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