Average Error: 15.1 → 0.0
Time: 9.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -957088378446.058:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 8137.871959944802:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -957088378446.058:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 8137.871959944802:\\
\;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\

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

\end{array}
double f(double x) {
        double r899252 = x;
        double r899253 = r899252 * r899252;
        double r899254 = 1.0;
        double r899255 = r899253 + r899254;
        double r899256 = r899252 / r899255;
        return r899256;
}
↓
double f(double x) {
        double r899257 = x;
        double r899258 = -957088378446.058;
        bool r899259 = r899257 <= r899258;
        double r899260 = 1.0;
        double r899261 = r899260 / r899257;
        double r899262 = r899257 * r899257;
        double r899263 = r899261 / r899262;
        double r899264 = r899261 - r899263;
        double r899265 = 5.0;
        double r899266 = pow(r899257, r899265);
        double r899267 = r899260 / r899266;
        double r899268 = r899264 + r899267;
        double r899269 = 8137.871959944802;
        bool r899270 = r899257 <= r899269;
        double r899271 = fma(r899257, r899257, r899260);
        double r899272 = r899257 / r899271;
        double r899273 = r899270 ? r899272 : r899268;
        double r899274 = r899259 ? r899268 : r899273;
        return r899274;
}

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 < -957088378446.058 or 8137.871959944802 < x

    1. Initial program 31.2

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

    2. Simplified31.2

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

    3. Taylor expanded around inf 0.0

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

    4. Simplified0.0

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

    if -957088378446.058 < x < 8137.871959944802

    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 add-sqr-sqrt0.0

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

    5. Applied associate-/r*0.0

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

    7. Applied div-inv0.0

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

    8. Applied associate-/l*0.0

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

    9. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -957088378446.058:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 8137.871959944802:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(x, x, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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