Average Error: 15.2 → 0.0
Time: 8.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -480639372.445875644683837890625 \lor \neg \left(x \le 418.3054548803003740431449841707944869995\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -480639372.445875644683837890625 \lor \neg \left(x \le 418.3054548803003740431449841707944869995\right):\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\

\end{array}
double f(double x) {
        double r47302 = x;
        double r47303 = r47302 * r47302;
        double r47304 = 1.0;
        double r47305 = r47303 + r47304;
        double r47306 = r47302 / r47305;
        return r47306;
}
↓
double f(double x) {
        double r47307 = x;
        double r47308 = -480639372.44587564;
        bool r47309 = r47307 <= r47308;
        double r47310 = 418.3054548803004;
        bool r47311 = r47307 <= r47310;
        double r47312 = !r47311;
        bool r47313 = r47309 || r47312;
        double r47314 = 1.0;
        double r47315 = r47314 / r47307;
        double r47316 = 1.0;
        double r47317 = 5.0;
        double r47318 = pow(r47307, r47317);
        double r47319 = r47316 / r47318;
        double r47320 = r47315 + r47319;
        double r47321 = 3.0;
        double r47322 = pow(r47307, r47321);
        double r47323 = r47316 / r47322;
        double r47324 = r47320 - r47323;
        double r47325 = fma(r47307, r47307, r47316);
        double r47326 = r47314 / r47325;
        double r47327 = log1p(r47326);
        double r47328 = expm1(r47327);
        double r47329 = r47307 * r47328;
        double r47330 = r47313 ? r47324 : r47329;
        return r47330;
}

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 < -480639372.44587564 or 418.3054548803004 < x

    1. Initial program 30.8

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

    2. Simplified30.8

      \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x, 1\right)}}\]
    3. Using strategy rm

    4. Applied div-inv30.8

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

    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

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

    if -480639372.44587564 < x < 418.3054548803004

    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. Using strategy rm

    4. Applied div-inv0.0

      \[\leadsto \color{blue}{x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}}\]
    5. Using strategy rm

    6. Applied expm1-log1p-u0.0

      \[\leadsto x \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -480639372.445875644683837890625 \lor \neg \left(x \le 418.3054548803003740431449841707944869995\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{fma}\left(x, x, 1\right)}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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