Average Error: 14.8 → 0.0
Time: 33.0s
Precision: 64
Internal Precision: 576
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x}{x \cdot x + 1} \le -1.3944464986082 \cdot 10^{-309}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1^2 + x^2}^*}}{\sqrt{x \cdot x + 1}}\\ \mathbf{if}\;\frac{x}{x \cdot x + 1} \le 0.0:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1^2 + x^2}^*}}{\sqrt{x \cdot x + 1}}\\ \end{array}\]

Error

Bits error versus x

Target

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ x (+ (* x x) 1)) < -1.3944464986082e-309 or 0.0 < (/ x (+ (* x x) 1))

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    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

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{1^2 + x^2}^*}}}{\sqrt{x \cdot x + 1}}\]

    if -1.3944464986082e-309 < (/ x (+ (* x x) 1)) < 0.0

    1. Initial program 59.5

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

    2. Taylor expanded around inf 0

      \[\leadsto \color{blue}{\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}}\]
  3. Recombined 2 regimes into one program.

Runtime

Time bar (total: 33.0s)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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