Average Error: 15.2 → 0.0
Time: 13.8s
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 r875205 = x;
        double r875206 = r875205 * r875205;
        double r875207 = 1.0;
        double r875208 = r875206 + r875207;
        double r875209 = r875205 / r875208;
        return r875209;
}
↓
double f(double x) {
        double r875210 = x;
        double r875211 = -1.8453524061497666e+21;
        bool r875212 = r875210 <= r875211;
        double r875213 = 1.0;
        double r875214 = r875213 / r875210;
        double r875215 = r875210 * r875210;
        double r875216 = r875214 / r875215;
        double r875217 = r875214 - r875216;
        double r875218 = 5.0;
        double r875219 = pow(r875210, r875218);
        double r875220 = r875213 / r875219;
        double r875221 = r875217 + r875220;
        double r875222 = 13951219.631737411;
        bool r875223 = r875210 <= r875222;
        double r875224 = fma(r875210, r875210, r875213);
        double r875225 = r875210 / r875224;
        double r875226 = r875223 ? r875225 : r875221;
        double r875227 = r875212 ? r875221 : r875226;
        return r875227;
}

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

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

    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)))