Average Error: 9.8 → 0.1
Time: 41.2s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1487.4772934733048 \lor \neg \left(x \le 2101.3708044277932\right):\\ \;\;\;\;\frac{\frac{2}{{x}^{4}} - \left(\frac{\frac{2}{x}}{x \cdot x} - \frac{\frac{2}{x}}{x}\right)}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + x} + \frac{2}{x}}{\frac{\left(x - 1\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}{1 + \left(\frac{-2}{x} \cdot \left(-1 + x\right) + \frac{-1 + x}{1 + x}\right)}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -1487.4772934733048 or 2101.3708044277932 < x

    1. Initial program 19.3

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

    3. Applied flip--53.9

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

    4. Applied frac-add54.7

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

    5. Simplified25.0

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

    6. Taylor expanded around -inf 0.2

      \[\leadsto \frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \color{blue}{\left(\left(2 \cdot \frac{1}{{x}^{4}} + 2 \cdot \frac{1}{{x}^{2}}\right) - 2 \cdot \frac{1}{{x}^{3}}\right)}}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\]

    7. Simplified0.2

      \[\leadsto \frac{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \color{blue}{\left(\frac{2}{{x}^{4}} - \left(\frac{\frac{2}{x}}{x \cdot x} - \frac{\frac{2}{x}}{x}\right)\right)}}{\left(\frac{1}{x + 1} + \frac{2}{x}\right) \cdot \left(x - 1\right)}\]
    8. Using strategy rm

    9. Applied times-frac0.2

      \[\leadsto \color{blue}{\frac{\frac{1}{x + 1} + \frac{2}{x}}{\frac{1}{x + 1} + \frac{2}{x}} \cdot \frac{\frac{2}{{x}^{4}} - \left(\frac{\frac{2}{x}}{x \cdot x} - \frac{\frac{2}{x}}{x}\right)}{x - 1}}\]

    10. Simplified0.2

      \[\leadsto \color{blue}{1} \cdot \frac{\frac{2}{{x}^{4}} - \left(\frac{\frac{2}{x}}{x \cdot x} - \frac{\frac{2}{x}}{x}\right)}{x - 1}\]

    if -1487.4772934733048 < x < 2101.3708044277932

    1. Initial program 0.1

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

    3. Applied flip--30.4

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

    4. Applied frac-add30.4

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

    5. Simplified30.4

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

    7. Applied associate-/l*0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1487.4772934733048 \lor \neg \left(x \le 2101.3708044277932\right):\\ \;\;\;\;\frac{\frac{2}{{x}^{4}} - \left(\frac{\frac{2}{x}}{x \cdot x} - \frac{\frac{2}{x}}{x}\right)}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{1 + x} + \frac{2}{x}}{\frac{\left(x - 1\right) \cdot \left(\frac{1}{1 + x} + \frac{2}{x}\right)}{1 + \left(\frac{-2}{x} \cdot \left(-1 + x\right) + \frac{-1 + x}{1 + x}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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