Average Error: 14.9 → 0.0
Time: 24.4s
Precision: 64
Internal Precision: 384
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5.64710349421599 \cdot 10^{+21}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{if}\;x \le 524.0272384802016:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \end{array}\]

Error

Bits error versus x

Target

Original14.9
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.64710349421599e+21 or 524.0272384802016 < 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 -5.64710349421599e+21 < x < 524.0272384802016

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

Runtime

Time bar (total: 24.4s)Debug logProfile

herbie shell --seed '#(1063282112 2455465480 4141627379 3773598652 1647277307 776739644)' 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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