Average Error: 14.8 → 0.0
Time: 31.9s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8735329382739425.0:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 604.0333324436991:\\ \;\;\;\;x \cdot \frac{1}{(x \cdot x + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8735329382739425.0:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\

\mathbf{elif}\;x \le 604.0333324436991:\\
\;\;\;\;x \cdot \frac{1}{(x \cdot x + 1)_*}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\

\end{array}
double f(double x) {
        double r17922864 = x;
        double r17922865 = r17922864 * r17922864;
        double r17922866 = 1.0;
        double r17922867 = r17922865 + r17922866;
        double r17922868 = r17922864 / r17922867;
        return r17922868;
}
↓
double f(double x) {
        double r17922869 = x;
        double r17922870 = -8735329382739425.0;
        bool r17922871 = r17922869 <= r17922870;
        double r17922872 = 1.0;
        double r17922873 = 5.0;
        double r17922874 = pow(r17922869, r17922873);
        double r17922875 = r17922872 / r17922874;
        double r17922876 = r17922872 / r17922869;
        double r17922877 = r17922875 + r17922876;
        double r17922878 = r17922869 * r17922869;
        double r17922879 = r17922878 * r17922869;
        double r17922880 = r17922872 / r17922879;
        double r17922881 = r17922877 - r17922880;
        double r17922882 = 604.0333324436991;
        bool r17922883 = r17922869 <= r17922882;
        double r17922884 = fma(r17922869, r17922869, r17922872);
        double r17922885 = r17922872 / r17922884;
        double r17922886 = r17922869 * r17922885;
        double r17922887 = r17922883 ? r17922886 : r17922881;
        double r17922888 = r17922871 ? r17922881 : r17922887;
        return r17922888;
}

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 < -8735329382739425.0 or 604.0333324436991 < x

    1. Initial program 30.7

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

    2. Simplified30.7

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

    4. Applied add-sqr-sqrt30.7

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

    5. Applied associate-/r*30.6

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

    6. Taylor expanded around -inf 0.0

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

    7. Simplified0.0

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

    if -8735329382739425.0 < x < 604.0333324436991

    1. Initial program 0.0

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

    2. Simplified0.0

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

    4. Applied div-inv0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8735329382739425.0:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 604.0333324436991:\\ \;\;\;\;x \cdot \frac{1}{(x \cdot x + 1)_*}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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