Average Error: 14.9 → 0.0
Time: 9.3s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -473716199.82409686:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 570.517048492422:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -473716199.82409686:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 570.517048492422:\\
\;\;\;\;\frac{x}{1 + x \cdot x}\\

\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 r1377044 = x;
        double r1377045 = r1377044 * r1377044;
        double r1377046 = 1.0;
        double r1377047 = r1377045 + r1377046;
        double r1377048 = r1377044 / r1377047;
        return r1377048;
}
↓
double f(double x) {
        double r1377049 = x;
        double r1377050 = -473716199.82409686;
        bool r1377051 = r1377049 <= r1377050;
        double r1377052 = 1.0;
        double r1377053 = r1377052 / r1377049;
        double r1377054 = r1377049 * r1377049;
        double r1377055 = r1377053 / r1377054;
        double r1377056 = r1377053 - r1377055;
        double r1377057 = 5.0;
        double r1377058 = pow(r1377049, r1377057);
        double r1377059 = r1377052 / r1377058;
        double r1377060 = r1377056 + r1377059;
        double r1377061 = 570.517048492422;
        bool r1377062 = r1377049 <= r1377061;
        double r1377063 = r1377052 + r1377054;
        double r1377064 = r1377049 / r1377063;
        double r1377065 = r1377062 ? r1377064 : r1377060;
        double r1377066 = r1377051 ? r1377060 : r1377065;
        return r1377066;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -473716199.82409686 or 570.517048492422 < x

    1. Initial program 30.8

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

    3. Applied add-sqr-sqrt30.8

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

    4. Applied associate-/r*30.7

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

    if -473716199.82409686 < x < 570.517048492422

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -473716199.82409686:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 570.517048492422:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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