Average Error: 15.3 → 0.0
Time: 16.0s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -3.0022313884212326 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 129781.97799511286:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -3.0022313884212326 \cdot 10^{+17}:\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\

\mathbf{elif}\;x \le 129781.97799511286:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r2052329 = x;
        double r2052330 = r2052329 * r2052329;
        double r2052331 = 1.0;
        double r2052332 = r2052330 + r2052331;
        double r2052333 = r2052329 / r2052332;
        return r2052333;
}
↓
double f(double x) {
        double r2052334 = x;
        double r2052335 = -3.0022313884212326e+17;
        bool r2052336 = r2052334 <= r2052335;
        double r2052337 = 1.0;
        double r2052338 = 5.0;
        double r2052339 = pow(r2052334, r2052338);
        double r2052340 = r2052337 / r2052339;
        double r2052341 = r2052337 / r2052334;
        double r2052342 = r2052334 * r2052334;
        double r2052343 = r2052334 * r2052342;
        double r2052344 = r2052337 / r2052343;
        double r2052345 = r2052341 - r2052344;
        double r2052346 = r2052340 + r2052345;
        double r2052347 = 129781.97799511286;
        bool r2052348 = r2052334 <= r2052347;
        double r2052349 = fma(r2052334, r2052334, r2052337);
        double r2052350 = r2052334 / r2052349;
        double r2052351 = r2052348 ? r2052350 : r2052346;
        double r2052352 = r2052336 ? r2052346 : r2052351;
        return r2052352;
}

Error

Bits error versus x

Target

Original15.3
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -3.0022313884212326e+17 or 129781.97799511286 < x

    1. Initial program 31.4

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

    2. Simplified31.4

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

    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

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

    if -3.0022313884212326e+17 < x < 129781.97799511286

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.0022313884212326 \cdot 10^{+17}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 129781.97799511286:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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