Average Error: 9.8 → 0.2
Time: 36.7s
Precision: 64
Internal Precision: 128
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -110.6000428433042:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 119.2725197811658:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\sqrt{\frac{\frac{2}{x}}{x}}}{\frac{x}{\sqrt{\frac{\frac{2}{x}}{x}}}}\right)\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -110.6000428433042

    1. Initial program 19.9

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

    2. Taylor expanded around -inf 0.4

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

    3. Simplified0.1

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

    4. Taylor expanded around 0 0.4

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

    if -110.6000428433042 < x < 119.2725197811658

    1. Initial program 0.0

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

    if 119.2725197811658 < x

    1. Initial program 20.4

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

    2. Taylor expanded around -inf 0.6

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

    3. Simplified0.1

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

    5. Applied associate-/r*0.1

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

    7. Applied add-sqr-sqrt0.2

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

    8. Applied associate-/l*0.2

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -110.6000428433042:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{3}} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 119.2725197811658:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{{x}^{7}} + \frac{\sqrt{\frac{\frac{2}{x}}{x}}}{\frac{x}{\sqrt{\frac{\frac{2}{x}}{x}}}}\right)\\ \end{array}\]

Runtime

Time bar (total: 36.7s)Debug logProfile

herbie shell --seed 2018348 +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))))