Average Error: 15.2 → 0.0
Time: 7.4s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -215710684808493064519680 \lor \neg \left(x \le 467.5510035009052671739482320845127105713\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -215710684808493064519680 \lor \neg \left(x \le 467.5510035009052671739482320845127105713\right):\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\

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

\end{array}
double f(double x) {
        double r39212 = x;
        double r39213 = r39212 * r39212;
        double r39214 = 1.0;
        double r39215 = r39213 + r39214;
        double r39216 = r39212 / r39215;
        return r39216;
}
↓
double f(double x) {
        double r39217 = x;
        double r39218 = -2.1571068480849306e+23;
        bool r39219 = r39217 <= r39218;
        double r39220 = 467.55100350090527;
        bool r39221 = r39217 <= r39220;
        double r39222 = !r39221;
        bool r39223 = r39219 || r39222;
        double r39224 = 1.0;
        double r39225 = 5.0;
        double r39226 = pow(r39217, r39225);
        double r39227 = r39224 / r39226;
        double r39228 = 1.0;
        double r39229 = r39228 / r39217;
        double r39230 = r39227 + r39229;
        double r39231 = 3.0;
        double r39232 = pow(r39217, r39231);
        double r39233 = r39224 / r39232;
        double r39234 = r39230 - r39233;
        double r39235 = fma(r39217, r39217, r39224);
        double r39236 = r39217 / r39235;
        double r39237 = r39223 ? r39234 : r39236;
        return r39237;
}

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 < -2.1571068480849306e+23 or 467.55100350090527 < x

    1. Initial program 31.7

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

    2. Simplified31.7

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

    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

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

    if -2.1571068480849306e+23 < x < 467.55100350090527

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -215710684808493064519680 \lor \neg \left(x \le 467.5510035009052671739482320845127105713\right):\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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