Average Error: 14.5 → 0.0
Time: 11.0s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -988765961200.682373046875:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 596.0424036054944281204370781779289245605:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -988765961200.682373046875:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\

\mathbf{elif}\;x \le 596.0424036054944281204370781779289245605:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r2042906 = x;
        double r2042907 = r2042906 * r2042906;
        double r2042908 = 1.0;
        double r2042909 = r2042907 + r2042908;
        double r2042910 = r2042906 / r2042909;
        return r2042910;
}
↓
double f(double x) {
        double r2042911 = x;
        double r2042912 = -988765961200.6824;
        bool r2042913 = r2042911 <= r2042912;
        double r2042914 = 1.0;
        double r2042915 = 5.0;
        double r2042916 = pow(r2042911, r2042915);
        double r2042917 = r2042914 / r2042916;
        double r2042918 = r2042911 * r2042911;
        double r2042919 = r2042914 / r2042918;
        double r2042920 = r2042919 / r2042911;
        double r2042921 = r2042917 - r2042920;
        double r2042922 = 1.0;
        double r2042923 = r2042922 / r2042911;
        double r2042924 = r2042921 + r2042923;
        double r2042925 = 596.0424036054944;
        bool r2042926 = r2042911 <= r2042925;
        double r2042927 = fma(r2042911, r2042911, r2042914);
        double r2042928 = r2042911 / r2042927;
        double r2042929 = r2042926 ? r2042928 : r2042924;
        double r2042930 = r2042913 ? r2042924 : r2042929;
        return r2042930;
}

Error

Bits error versus x

Target

Original14.5
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -988765961200.6824 or 596.0424036054944 < x

    1. Initial program 30.4

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

    2. Simplified30.4

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -988765961200.6824 < x < 596.0424036054944

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -988765961200.682373046875:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 596.0424036054944281204370781779289245605:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))