Average Error: 15.4 → 0.0
Time: 11.9s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2101569.4261024096:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 471.29821069866506:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2101569.4261024096:\\
\;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 471.29821069866506:\\
\;\;\;\;\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 r2096838 = x;
        double r2096839 = r2096838 * r2096838;
        double r2096840 = 1.0;
        double r2096841 = r2096839 + r2096840;
        double r2096842 = r2096838 / r2096841;
        return r2096842;
}
↓
double f(double x) {
        double r2096843 = x;
        double r2096844 = -2101569.4261024096;
        bool r2096845 = r2096843 <= r2096844;
        double r2096846 = 1.0;
        double r2096847 = r2096846 / r2096843;
        double r2096848 = r2096847 / r2096843;
        double r2096849 = r2096847 * r2096848;
        double r2096850 = r2096847 - r2096849;
        double r2096851 = 5.0;
        double r2096852 = pow(r2096843, r2096851);
        double r2096853 = r2096846 / r2096852;
        double r2096854 = r2096850 + r2096853;
        double r2096855 = 471.29821069866506;
        bool r2096856 = r2096843 <= r2096855;
        double r2096857 = fma(r2096843, r2096843, r2096846);
        double r2096858 = r2096843 / r2096857;
        double r2096859 = r2096856 ? r2096858 : r2096854;
        double r2096860 = r2096845 ? r2096854 : r2096859;
        return r2096860;
}

Error

Bits error versus x

Target

Original15.4
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -2101569.4261024096 or 471.29821069866506 < x

    1. Initial program 30.9

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

    2. Simplified30.9

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

    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 -2101569.4261024096 < x < 471.29821069866506

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2101569.4261024096:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 471.29821069866506:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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