Average Error: 14.7 → 0.0
Time: 8.8s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -6092285729639.72168 \lor \neg \left(x \le 897.371029930506097\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -6092285729639.72168 \lor \neg \left(x \le 897.371029930506097\right):\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\

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

\end{array}
double f(double x) {
        double r95687 = x;
        double r95688 = r95687 * r95687;
        double r95689 = 1.0;
        double r95690 = r95688 + r95689;
        double r95691 = r95687 / r95690;
        return r95691;
}
↓
double f(double x) {
        double r95692 = x;
        double r95693 = -6092285729639.722;
        bool r95694 = r95692 <= r95693;
        double r95695 = 897.3710299305061;
        bool r95696 = r95692 <= r95695;
        double r95697 = !r95696;
        bool r95698 = r95694 || r95697;
        double r95699 = 1.0;
        double r95700 = 5.0;
        double r95701 = pow(r95692, r95700);
        double r95702 = r95699 / r95701;
        double r95703 = 1.0;
        double r95704 = r95703 / r95692;
        double r95705 = 3.0;
        double r95706 = pow(r95692, r95705);
        double r95707 = r95699 / r95706;
        double r95708 = r95704 - r95707;
        double r95709 = r95702 + r95708;
        double r95710 = fma(r95692, r95692, r95699);
        double r95711 = r95703 / r95710;
        double r95712 = r95692 * r95711;
        double r95713 = r95698 ? r95709 : r95712;
        return r95713;
}

Error

Bits error versus x

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -6092285729639.722 or 897.3710299305061 < x

    1. Initial program 30.2

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

    2. Simplified30.2

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

    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

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

    if -6092285729639.722 < x < 897.3710299305061

    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

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6092285729639.72168 \lor \neg \left(x \le 897.371029930506097\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\mathsf{fma}\left(x, x, 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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