Average Error: 15.2 → 0.0
Time: 13.2s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.8453524061497666 \cdot 10^{+21}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 13951219.631737411:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.8453524061497666 \cdot 10^{+21}:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 13951219.631737411:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r1885504 = x;
        double r1885505 = r1885504 * r1885504;
        double r1885506 = 1.0;
        double r1885507 = r1885505 + r1885506;
        double r1885508 = r1885504 / r1885507;
        return r1885508;
}
↓
double f(double x) {
        double r1885509 = x;
        double r1885510 = -1.8453524061497666e+21;
        bool r1885511 = r1885509 <= r1885510;
        double r1885512 = 1.0;
        double r1885513 = r1885512 / r1885509;
        double r1885514 = r1885509 * r1885509;
        double r1885515 = r1885513 / r1885514;
        double r1885516 = r1885513 - r1885515;
        double r1885517 = 5.0;
        double r1885518 = pow(r1885509, r1885517);
        double r1885519 = r1885512 / r1885518;
        double r1885520 = r1885516 + r1885519;
        double r1885521 = 13951219.631737411;
        bool r1885522 = r1885509 <= r1885521;
        double r1885523 = fma(r1885509, r1885509, r1885512);
        double r1885524 = r1885509 / r1885523;
        double r1885525 = r1885522 ? r1885524 : r1885520;
        double r1885526 = r1885511 ? r1885520 : r1885525;
        return r1885526;
}

Error

Bits error versus x

Target

Original15.2
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.8453524061497666e+21 or 13951219.631737411 < x

    1. Initial program 31.5

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

    2. Simplified31.5

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

    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

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

    if -1.8453524061497666e+21 < x < 13951219.631737411

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.8453524061497666 \cdot 10^{+21}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 13951219.631737411:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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