Average Error: 14.6 → 0.0
Time: 35.9s
Precision: 64
Internal Precision: 384
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1870122.656240604:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{if}\;x \le 60103344.098054335:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1^2 + x^2}^*}}{\sqrt{x \cdot x + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \end{array}\]

Error

Bits error versus x

Target

Original14.6
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1870122.656240604 or 60103344.098054335 < x

    1. Initial program 29.9

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

    2. Taylor expanded around inf 0.0

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

    if -1870122.656240604 < x < 60103344.098054335

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]

    4. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]

    5. Applied simplify0.0

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

Runtime

Time bar (total: 35.9s)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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