Average Error: 15.6 → 0.2
Time: 12.6s
Precision: 64
Internal Precision: 128
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.9966163161558536 \lor \neg \left(x \le 1.0115407637064349\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} + x\right) - {x}^{3}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.6
Target0.1
Herbie0.2
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.9966163161558536 or 1.0115407637064349 < x

    1. Initial program 30.6

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

    2. Simplified30.6

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]

    3. Taylor expanded around -inf 0.3

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

    if -0.9966163161558536 < x < 1.0115407637064349

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]

    3. Taylor expanded around 0 0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.9966163161558536 \lor \neg \left(x \le 1.0115407637064349\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left({x}^{5} + x\right) - {x}^{3}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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