Average Error: 15.3 → 0.0
Time: 8.9s
Precision: 64
Internal Precision: 320
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -31176.103077362146 \lor \neg \left(x \le 389.2449688832499\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original15.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -31176.103077362146 or 389.2449688832499 < x

    1. Initial program 30.4

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

    2. Initial simplification30.4

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

    3. Taylor expanded around -inf 0.0

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

    if -31176.103077362146 < x < 389.2449688832499

    1. Initial program 0.0

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

    2. Initial simplification0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -31176.103077362146 \lor \neg \left(x \le 389.2449688832499\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Runtime

Time bar (total: 8.9s)Debug logProfile

herbie shell --seed 2018257 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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