Average Error: 14.9 → 0.3
Time: 11.3s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.0026376323582022:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 1.020834055947729:\\ \;\;\;\;{x}^{5} + \left(x - x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.0026376323582022:\\
\;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 1.020834055947729:\\
\;\;\;\;{x}^{5} + \left(x - x \cdot \left(x \cdot x\right)\right)\\

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

\end{array}
double f(double x) {
        double r2160782 = x;
        double r2160783 = r2160782 * r2160782;
        double r2160784 = 1.0;
        double r2160785 = r2160783 + r2160784;
        double r2160786 = r2160782 / r2160785;
        return r2160786;
}
↓
double f(double x) {
        double r2160787 = x;
        double r2160788 = -1.0026376323582022;
        bool r2160789 = r2160787 <= r2160788;
        double r2160790 = 1.0;
        double r2160791 = r2160790 / r2160787;
        double r2160792 = r2160791 / r2160787;
        double r2160793 = r2160791 * r2160792;
        double r2160794 = r2160791 - r2160793;
        double r2160795 = 5.0;
        double r2160796 = pow(r2160787, r2160795);
        double r2160797 = r2160790 / r2160796;
        double r2160798 = r2160794 + r2160797;
        double r2160799 = 1.020834055947729;
        bool r2160800 = r2160787 <= r2160799;
        double r2160801 = r2160787 * r2160787;
        double r2160802 = r2160787 * r2160801;
        double r2160803 = r2160787 - r2160802;
        double r2160804 = r2160796 + r2160803;
        double r2160805 = r2160800 ? r2160804 : r2160798;
        double r2160806 = r2160789 ? r2160798 : r2160805;
        return r2160806;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.1
Herbie0.3
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -1.0026376323582022 or 1.020834055947729 < x

    1. Initial program 29.6

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

    2. Simplified29.6

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

    3. Taylor expanded around inf 0.2

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

    4. Simplified0.2

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

    if -1.0026376323582022 < x < 1.020834055947729

    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. Taylor expanded around 0 0.3

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

    4. Simplified0.3

      \[\leadsto \color{blue}{\left(x - x \cdot \left(x \cdot x\right)\right) + {x}^{5}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.0026376323582022:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 1.020834055947729:\\ \;\;\;\;{x}^{5} + \left(x - x \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{1}{x} \cdot \frac{\frac{1}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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