Average Error: 14.5 → 0.0
Time: 10.2s
Precision: 64
Internal Precision: 128
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.971673281926759 \cdot 10^{+19} \lor \neg \left(x \le 568.7808384339279\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.5
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.971673281926759e+19 or 568.7808384339279 < x

    1. Initial program 31.0

      \[\frac{x}{x \cdot x + 1}\]

    2. Taylor expanded around -inf 0.0

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

    if -5.971673281926759e+19 < x < 568.7808384339279

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

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

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

    4. Applied associate-/r*0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.971673281926759 \cdot 10^{+19} \lor \neg \left(x \le 568.7808384339279\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Runtime

Time bar (total: 10.2s)Debug logProfile

herbie shell --seed 2018348 
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1 (+ x (/ 1 x)))

  (/ x (+ (* x x) 1)))