Average Error: 15.5 → 0.0
Time: 21.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 780.2110112890140953822992742061614990234\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 780.2110112890140953822992742061614990234\right):\\
\;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\

\end{array}
double f(double x) {
        double r97317 = x;
        double r97318 = r97317 * r97317;
        double r97319 = 1.0;
        double r97320 = r97318 + r97319;
        double r97321 = r97317 / r97320;
        return r97321;
}
↓
double f(double x) {
        double r97322 = x;
        double r97323 = -3416824020916.5903;
        bool r97324 = r97322 <= r97323;
        double r97325 = 780.2110112890141;
        bool r97326 = r97322 <= r97325;
        double r97327 = !r97326;
        bool r97328 = r97324 || r97327;
        double r97329 = 1.0;
        double r97330 = r97329 / r97322;
        double r97331 = 1.0;
        double r97332 = 5.0;
        double r97333 = pow(r97322, r97332);
        double r97334 = r97331 / r97333;
        double r97335 = 3.0;
        double r97336 = pow(r97322, r97335);
        double r97337 = r97331 / r97336;
        double r97338 = r97334 - r97337;
        double r97339 = r97330 + r97338;
        double r97340 = fma(r97322, r97322, r97331);
        double r97341 = r97329 / r97340;
        double r97342 = r97322 * r97341;
        double r97343 = r97328 ? r97339 : r97342;
        return r97343;
}

Error

Bits error versus x

Target

Original15.5
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3416824020916.5903 or 780.2110112890141 < x

    1. Initial program 31.5

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied div-inv31.5

      \[\leadsto \color{blue}{x \cdot \frac{1}{x \cdot x + 1}}\]

    4. Simplified31.5

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

    if -3416824020916.5903 < x < 780.2110112890141

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied div-inv0.0

      \[\leadsto \color{blue}{x \cdot \frac{1}{x \cdot x + 1}}\]

    4. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3416824020916.59033203125 \lor \neg \left(x \le 780.2110112890140953822992742061614990234\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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