Average Error: 15.0 → 0.0
Time: 2.5s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1025.28762287299992 \lor \neg \left(x \le 764.667521381814254\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1025.28762287299992 \lor \neg \left(x \le 764.667521381814254\right):\\
\;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\

\end{array}
double f(double x) {
        double r53883 = x;
        double r53884 = r53883 * r53883;
        double r53885 = 1.0;
        double r53886 = r53884 + r53885;
        double r53887 = r53883 / r53886;
        return r53887;
}
↓
double f(double x) {
        double r53888 = x;
        double r53889 = -1025.287622873;
        bool r53890 = r53888 <= r53889;
        double r53891 = 764.6675213818143;
        bool r53892 = r53888 <= r53891;
        double r53893 = !r53892;
        bool r53894 = r53890 || r53893;
        double r53895 = 1.0;
        double r53896 = r53895 / r53888;
        double r53897 = 1.0;
        double r53898 = 3.0;
        double r53899 = pow(r53888, r53898);
        double r53900 = r53897 / r53899;
        double r53901 = 5.0;
        double r53902 = pow(r53888, r53901);
        double r53903 = r53895 / r53902;
        double r53904 = r53897 * r53903;
        double r53905 = r53900 - r53904;
        double r53906 = r53896 - r53905;
        double r53907 = r53888 * r53888;
        double r53908 = r53907 + r53897;
        double r53909 = sqrt(r53908);
        double r53910 = r53888 / r53909;
        double r53911 = r53910 / r53909;
        double r53912 = r53894 ? r53906 : r53911;
        return r53912;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -1025.287622873 or 764.6675213818143 < x

    1. Initial program 30.0

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -1025.287622873 < x < 764.6675213818143

    1. Initial program 0.0

      \[\frac{x}{x \cdot x + 1}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{x}{\color{blue}{\sqrt{x \cdot x + 1} \cdot \sqrt{x \cdot x + 1}}}\]

    4. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1025.28762287299992 \lor \neg \left(x \le 764.667521381814254\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - 1 \cdot \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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