Average Error: 14.3 → 0.0
Time: 7.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.450745238348297 \cdot 10^{+24}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 665.7378871081933:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.450745238348297 \cdot 10^{+24}:\\
\;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 665.7378871081933:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r1897030 = x;
        double r1897031 = r1897030 * r1897030;
        double r1897032 = 1.0;
        double r1897033 = r1897031 + r1897032;
        double r1897034 = r1897030 / r1897033;
        return r1897034;
}
↓
double f(double x) {
        double r1897035 = x;
        double r1897036 = -7.450745238348297e+24;
        bool r1897037 = r1897035 <= r1897036;
        double r1897038 = 1.0;
        double r1897039 = r1897038 / r1897035;
        double r1897040 = r1897039 / r1897035;
        double r1897041 = r1897039 * r1897040;
        double r1897042 = r1897039 - r1897041;
        double r1897043 = 5.0;
        double r1897044 = pow(r1897035, r1897043);
        double r1897045 = r1897038 / r1897044;
        double r1897046 = r1897042 + r1897045;
        double r1897047 = 665.7378871081933;
        bool r1897048 = r1897035 <= r1897047;
        double r1897049 = fma(r1897035, r1897035, r1897038);
        double r1897050 = r1897035 / r1897049;
        double r1897051 = r1897048 ? r1897050 : r1897046;
        double r1897052 = r1897037 ? r1897046 : r1897051;
        return r1897052;
}

Error

Bits error versus x

Target

Original14.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.450745238348297e+24 or 665.7378871081933 < 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

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

    if -7.450745238348297e+24 < x < 665.7378871081933

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.450745238348297 \cdot 10^{+24}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 665.7378871081933:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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