Average Error: 14.8 → 0.0
Time: 7.9s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -57137068010548903280640:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 8745.591563834146654698997735977172851562:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -57137068010548903280640:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\

\mathbf{elif}\;x \le 8745.591563834146654698997735977172851562:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r3483722 = x;
        double r3483723 = r3483722 * r3483722;
        double r3483724 = 1.0;
        double r3483725 = r3483723 + r3483724;
        double r3483726 = r3483722 / r3483725;
        return r3483726;
}
↓
double f(double x) {
        double r3483727 = x;
        double r3483728 = -5.71370680105489e+22;
        bool r3483729 = r3483727 <= r3483728;
        double r3483730 = 1.0;
        double r3483731 = 5.0;
        double r3483732 = pow(r3483727, r3483731);
        double r3483733 = r3483730 / r3483732;
        double r3483734 = 1.0;
        double r3483735 = r3483734 / r3483727;
        double r3483736 = r3483733 + r3483735;
        double r3483737 = r3483727 * r3483727;
        double r3483738 = r3483737 * r3483727;
        double r3483739 = r3483730 / r3483738;
        double r3483740 = r3483736 - r3483739;
        double r3483741 = 8745.591563834147;
        bool r3483742 = r3483727 <= r3483741;
        double r3483743 = fma(r3483727, r3483727, r3483730);
        double r3483744 = r3483727 / r3483743;
        double r3483745 = r3483742 ? r3483744 : r3483740;
        double r3483746 = r3483729 ? r3483740 : r3483745;
        return r3483746;
}

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 8745.591563834147 < 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

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

    if -5.71370680105489e+22 < x < 8745.591563834147

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -57137068010548903280640:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 8745.591563834146654698997735977172851562:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

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)))