Average Error: 14.8 → 0.0
Time: 11.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -57137068010548903280640:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 8124.998191315608892182353883981704711914:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -57137068010548903280640:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\

\mathbf{elif}\;x \le 8124.998191315608892182353883981704711914:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r2142397 = x;
        double r2142398 = r2142397 * r2142397;
        double r2142399 = 1.0;
        double r2142400 = r2142398 + r2142399;
        double r2142401 = r2142397 / r2142400;
        return r2142401;
}
↓
double f(double x) {
        double r2142402 = x;
        double r2142403 = -5.71370680105489e+22;
        bool r2142404 = r2142402 <= r2142403;
        double r2142405 = 1.0;
        double r2142406 = 5.0;
        double r2142407 = pow(r2142402, r2142406);
        double r2142408 = r2142405 / r2142407;
        double r2142409 = r2142402 * r2142402;
        double r2142410 = r2142405 / r2142409;
        double r2142411 = r2142410 / r2142402;
        double r2142412 = r2142408 - r2142411;
        double r2142413 = 1.0;
        double r2142414 = r2142413 / r2142402;
        double r2142415 = r2142412 + r2142414;
        double r2142416 = 8124.998191315609;
        bool r2142417 = r2142402 <= r2142416;
        double r2142418 = fma(r2142402, r2142402, r2142405);
        double r2142419 = r2142402 / r2142418;
        double r2142420 = r2142417 ? r2142419 : r2142415;
        double r2142421 = r2142404 ? r2142415 : r2142420;
        return r2142421;
}

Error

Bits error versus x

Target

Original14.8
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.71370680105489e+22 or 8124.998191315609 < x

    1. Initial program 30.6

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

    2. Simplified30.6

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

    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

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

    if -5.71370680105489e+22 < x < 8124.998191315609

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -57137068010548903280640:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \mathbf{elif}\;x \le 8124.998191315608892182353883981704711914:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} - \frac{\frac{1}{x \cdot x}}{x}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))