Average Error: 14.3 → 0.0
Time: 19.5s
Precision: 64
Internal Precision: 320
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5728221888143.946 \lor \neg \left(x \le 131309240.38858211\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \end{array}\]

Error

Bits error versus x

Target

Original14.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5728221888143.946 or 131309240.38858211 < 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 -5728221888143.946 < x < 131309240.38858211

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5728221888143.946 \lor \neg \left(x \le 131309240.38858211\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{(x \cdot x + 1)_*}\\ \end{array}\]

Runtime

Time bar (total: 19.5s)Debug logProfile

herbie shell --seed 2018252 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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