Average Error: 9.6 → 0.1
Time: 24.8s
Precision: 64
Internal Precision: 1088
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -136164714.82789978:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{x} \cdot \frac{1}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 5787.699880756418:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x \cdot \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x}}{x \cdot x}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 3 regimes

  2. if x < -136164714.82789978

    1. Initial program 19.6

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

    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

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

    5. Applied div-inv0.1

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

    if -136164714.82789978 < x < 5787.699880756418

    1. Initial program 0.3

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

    3. Applied frac-sub0.3

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

    4. Applied frac-add0.0

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

    if 5787.699880756418 < x

    1. Initial program 19.3

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

    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

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -136164714.82789978:\\ \;\;\;\;\frac{2}{{x}^{5}} + \left(\frac{2}{x} \cdot \frac{1}{x \cdot x} + \frac{2}{{x}^{7}}\right)\\ \mathbf{elif}\;x \le 5787.699880756418:\\ \;\;\;\;\frac{\left(x - 1\right) \cdot \left(x - \left(x + 1\right) \cdot 2\right) + x \cdot \left(x + 1\right)}{\left(x - 1\right) \cdot \left(x \cdot \left(x + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{2}{{x}^{7}} + \frac{\frac{2}{x}}{x \cdot x}\right) + \frac{2}{{x}^{5}}\\ \end{array}\]

Runtime

Time bar (total: 24.8s)Debug logProfile

herbie shell --seed 2018220 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

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

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