Average Error: 15.0 → 0.0
Time: 17.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -247355.42006732605:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 402.2628134253315:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -247355.42006732605:\\
\;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\

\mathbf{elif}\;x \le 402.2628134253315:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r1283391 = x;
        double r1283392 = r1283391 * r1283391;
        double r1283393 = 1.0;
        double r1283394 = r1283392 + r1283393;
        double r1283395 = r1283391 / r1283394;
        return r1283395;
}
↓
double f(double x) {
        double r1283396 = x;
        double r1283397 = -247355.42006732605;
        bool r1283398 = r1283396 <= r1283397;
        double r1283399 = 1.0;
        double r1283400 = r1283399 / r1283396;
        double r1283401 = 5.0;
        double r1283402 = pow(r1283396, r1283401);
        double r1283403 = r1283399 / r1283402;
        double r1283404 = r1283396 * r1283396;
        double r1283405 = r1283400 / r1283404;
        double r1283406 = r1283403 - r1283405;
        double r1283407 = r1283400 + r1283406;
        double r1283408 = 402.2628134253315;
        bool r1283409 = r1283396 <= r1283408;
        double r1283410 = fma(r1283396, r1283396, r1283399);
        double r1283411 = r1283396 / r1283410;
        double r1283412 = r1283409 ? r1283411 : r1283407;
        double r1283413 = r1283398 ? r1283407 : r1283412;
        return r1283413;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -247355.42006732605 or 402.2628134253315 < x

    1. Initial program 30.3

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

    2. Simplified30.3

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

    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

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

    if -247355.42006732605 < x < 402.2628134253315

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -247355.42006732605:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 402.2628134253315:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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