Average Error: 10.1 → 0.2
Time: 33.9s
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -116.8302555943657:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 105.48450295986906:\\ \;\;\;\;(1 \cdot \left(\frac{1}{x - 1}\right) + \left(\frac{1}{x} \cdot \left(-2\right)\right))_* + \frac{1}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x \cdot x}}{x}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]

Error

Bits error versus x

Target

Original10.1
Target0.2
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -116.8302555943657

    1. Initial program 19.9

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

    2. Initial simplification19.9

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

    3. Taylor expanded around -inf 0.5

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

    4. Simplified0.1

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

    6. Applied associate-/r*0.1

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

    7. Taylor expanded around -inf 0.5

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

    if -116.8302555943657 < x < 105.48450295986906

    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 div-inv0.0

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

    5. Applied *-un-lft-identity0.0

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

    6. Applied prod-diff0.0

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

    7. Applied associate-+r+0.0

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

    8. Simplified0.0

      \[\leadsto \left(\frac{1}{x + 1} + (1 \cdot \left(\frac{1}{x - 1}\right) + \left(-\frac{1}{x} \cdot 2\right))_*\right) + \color{blue}{0}\]

    if 105.48450295986906 < x

    1. Initial program 20.2

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

    2. Initial simplification20.2

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

    3. Taylor expanded around -inf 0.4

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

    4. Simplified0.1

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

    6. Applied associate-/r*0.1

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

    8. Applied associate-/l/0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -116.8302555943657:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 105.48450295986906:\\ \;\;\;\;(1 \cdot \left(\frac{1}{x - 1}\right) + \left(\frac{1}{x} \cdot \left(-2\right)\right))_* + \frac{1}{1 + x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x \cdot x}}{x}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]

Runtime

Time bar (total: 33.9s)Debug logProfile

herbie shell --seed 2018230 +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))))