Average Error: 9.8 → 0.2
Time: 2.7m
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -2.5526076135737913 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(x + 2\right) - x \cdot 2}{x \cdot \left(x - 1\right)} + \frac{1}{x + 1}\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 3.609331750466219 \cdot 10^{-22}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right) + \frac{2}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 2\right) - x \cdot 2}{x \cdot \left(x - 1\right)} + \frac{1}{x + 1}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.8
Target0.3
Herbie0.2
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < -2.5526076135737913e-13 or 3.609331750466219e-22 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1)))

    1. Initial program 0.3

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

    2. Initial simplification0.3

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

    4. Applied frac-sub0.3

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

    5. Simplified0.3

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

    if -2.5526076135737913e-13 < (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))) < 3.609331750466219e-22

    1. Initial program 19.6

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

    2. Initial simplification19.6

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

    4. Applied flip-+19.6

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

    5. Taylor expanded around inf 0.5

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

    6. Simplified0.1

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

    8. Applied associate-/r*0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le -2.5526076135737913 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(x + 2\right) - x \cdot 2}{x \cdot \left(x - 1\right)} + \frac{1}{x + 1}\\ \mathbf{elif}\;\frac{1}{x - 1} + \left(\frac{1}{x + 1} - \frac{2}{x}\right) \le 3.609331750466219 \cdot 10^{-22}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{\frac{2}{x}}{x}}{x}\right) + \frac{2}{{x}^{5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 2\right) - x \cdot 2}{x \cdot \left(x - 1\right)} + \frac{1}{x + 1}\\ \end{array}\]

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018225 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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