Average Error: 14.4 → 0.0
Time: 9.7s
Precision: 64
Internal Precision: 384
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -76951863.5324409:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{if}\;x \le 35688.954472079415:\\ \;\;\;\;\frac{\frac{x}{\sqrt{(x \cdot x + 1)_*}}}{\sqrt{(x \cdot x + 1)_*}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \end{array}\]

Error

Bits error versus x

Target

Original14.4
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -76951863.5324409 or 35688.954472079415 < x

    1. Initial program 30.0

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

    2. Applied simplify30.0

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

    3. Taylor expanded around inf 0.0

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

    if -76951863.5324409 < x < 35688.954472079415

    1. Initial program 0.0

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

    2. Applied simplify0.0

      \[\leadsto \color{blue}{\frac{x}{(x \cdot x + 1)_*}}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.0

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

    5. Applied associate-/r*0.0

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

Runtime

Time bar (total: 9.7s)Debug logProfile

herbie shell --seed '#(1070227846 1561819246 480764335 4016816270 2602869839 2117310382)' +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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