Average Error: 15.2 → 0.0
Time: 18.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4887615356376792.0:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 454.0640131308542:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4887615356376792.0:\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\

\mathbf{elif}\;x \le 454.0640131308542:\\
\;\;\;\;\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 r1723135 = x;
        double r1723136 = r1723135 * r1723135;
        double r1723137 = 1.0;
        double r1723138 = r1723136 + r1723137;
        double r1723139 = r1723135 / r1723138;
        return r1723139;
}
↓
double f(double x) {
        double r1723140 = x;
        double r1723141 = -4887615356376792.0;
        bool r1723142 = r1723140 <= r1723141;
        double r1723143 = 1.0;
        double r1723144 = 5.0;
        double r1723145 = pow(r1723140, r1723144);
        double r1723146 = r1723143 / r1723145;
        double r1723147 = r1723143 / r1723140;
        double r1723148 = r1723140 * r1723140;
        double r1723149 = r1723140 * r1723148;
        double r1723150 = r1723143 / r1723149;
        double r1723151 = r1723147 - r1723150;
        double r1723152 = r1723146 + r1723151;
        double r1723153 = 454.0640131308542;
        bool r1723154 = r1723140 <= r1723153;
        double r1723155 = fma(r1723140, r1723140, r1723143);
        double r1723156 = r1723140 / r1723155;
        double r1723157 = r1723154 ? r1723156 : r1723152;
        double r1723158 = r1723142 ? r1723152 : r1723157;
        return r1723158;
}

Error

Bits error versus x

Target

Original15.2
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4887615356376792.0 or 454.0640131308542 < x

    1. Initial program 30.7

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

    2. Simplified30.7

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

    4. Applied add-sqr-sqrt30.7

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

    5. Applied associate-/r*30.6

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -4887615356376792.0 < x < 454.0640131308542

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4887615356376792.0:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \mathbf{elif}\;x \le 454.0640131308542:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{x \cdot \left(x \cdot x\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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