Average Error: 15.2 → 0.0
Time: 11.8s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -955471611474.0786:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 129599.43046274735:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -955471611474.0786:\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\

\mathbf{elif}\;x \le 129599.43046274735:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r1683616 = x;
        double r1683617 = r1683616 * r1683616;
        double r1683618 = 1.0;
        double r1683619 = r1683617 + r1683618;
        double r1683620 = r1683616 / r1683619;
        return r1683620;
}
↓
double f(double x) {
        double r1683621 = x;
        double r1683622 = -955471611474.0786;
        bool r1683623 = r1683621 <= r1683622;
        double r1683624 = 1.0;
        double r1683625 = 5.0;
        double r1683626 = pow(r1683621, r1683625);
        double r1683627 = r1683624 / r1683626;
        double r1683628 = r1683624 / r1683621;
        double r1683629 = r1683621 * r1683621;
        double r1683630 = r1683621 * r1683629;
        double r1683631 = r1683624 / r1683630;
        double r1683632 = r1683628 - r1683631;
        double r1683633 = r1683627 + r1683632;
        double r1683634 = 129599.43046274735;
        bool r1683635 = r1683621 <= r1683634;
        double r1683636 = fma(r1683621, r1683621, r1683624);
        double r1683637 = r1683621 / r1683636;
        double r1683638 = r1683635 ? r1683637 : r1683633;
        double r1683639 = r1683623 ? r1683633 : r1683638;
        return r1683639;
}

Error

Bits error versus x

Target

Original15.2
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -955471611474.0786 or 129599.43046274735 < x

    1. Initial program 30.7

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

    2. Simplified30.7

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -955471611474.0786 < x < 129599.43046274735

    1. Initial program 0.0

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

    2. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -955471611474.0786:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 129599.43046274735:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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