Average Error: 15.1 → 0.0
Time: 16.5s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.33425693437648922400944129830191617921 \cdot 10^{154}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 217598.0712392236164305359125137329101562:\\ \;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.33425693437648922400944129830191617921 \cdot 10^{154}:\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\

\mathbf{elif}\;x \le 217598.0712392236164305359125137329101562:\\
\;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\

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

\end{array}
double f(double x) {
        double r3007771 = x;
        double r3007772 = r3007771 * r3007771;
        double r3007773 = 1.0;
        double r3007774 = r3007772 + r3007773;
        double r3007775 = r3007771 / r3007774;
        return r3007775;
}
↓
double f(double x) {
        double r3007776 = x;
        double r3007777 = -1.3342569343764892e+154;
        bool r3007778 = r3007776 <= r3007777;
        double r3007779 = 1.0;
        double r3007780 = 5.0;
        double r3007781 = pow(r3007776, r3007780);
        double r3007782 = r3007779 / r3007781;
        double r3007783 = 1.0;
        double r3007784 = r3007783 / r3007776;
        double r3007785 = r3007779 / r3007776;
        double r3007786 = r3007776 * r3007776;
        double r3007787 = r3007785 / r3007786;
        double r3007788 = r3007784 - r3007787;
        double r3007789 = r3007782 + r3007788;
        double r3007790 = 217598.07123922362;
        bool r3007791 = r3007776 <= r3007790;
        double r3007792 = fma(r3007776, r3007776, r3007779);
        double r3007793 = sqrt(r3007792);
        double r3007794 = r3007776 / r3007793;
        double r3007795 = r3007794 / r3007793;
        double r3007796 = r3007791 ? r3007795 : r3007789;
        double r3007797 = r3007778 ? r3007789 : r3007796;
        return r3007797;
}

Error

Bits error versus x

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -1.3342569343764892e+154 or 217598.07123922362 < x

    1. Initial program 40.5

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

    2. Simplified40.5

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

    4. Applied add-sqr-sqrt40.5

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

    5. Applied associate-/r*40.5

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -1.3342569343764892e+154 < x < 217598.07123922362

    1. Initial program 0.1

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

    2. Simplified0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied associate-/r*0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.33425693437648922400944129830191617921 \cdot 10^{154}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \mathbf{elif}\;x \le 217598.0712392236164305359125137329101562:\\ \;\;\;\;\frac{\frac{x}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}}{\sqrt{\mathsf{fma}\left(x, x, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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