Average Error: 15.1 → 0.0
Time: 39.7s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2169385582642.869:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 429.5871414147981:\\ \;\;\;\;\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1} \cdot \left(x \cdot x - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2169385582642.869:\\
\;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\

\mathbf{elif}\;x \le 429.5871414147981:\\
\;\;\;\;\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1} \cdot \left(x \cdot x - 1\right)\\

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

\end{array}
double f(double x) {
        double r9630628 = x;
        double r9630629 = r9630628 * r9630628;
        double r9630630 = 1.0;
        double r9630631 = r9630629 + r9630630;
        double r9630632 = r9630628 / r9630631;
        return r9630632;
}
↓
double f(double x) {
        double r9630633 = x;
        double r9630634 = -2169385582642.869;
        bool r9630635 = r9630633 <= r9630634;
        double r9630636 = 1.0;
        double r9630637 = 5.0;
        double r9630638 = pow(r9630633, r9630637);
        double r9630639 = r9630636 / r9630638;
        double r9630640 = r9630636 / r9630633;
        double r9630641 = r9630639 + r9630640;
        double r9630642 = r9630633 * r9630633;
        double r9630643 = r9630642 * r9630633;
        double r9630644 = r9630636 / r9630643;
        double r9630645 = r9630641 - r9630644;
        double r9630646 = 429.5871414147981;
        bool r9630647 = r9630633 <= r9630646;
        double r9630648 = r9630642 * r9630642;
        double r9630649 = r9630648 - r9630636;
        double r9630650 = r9630633 / r9630649;
        double r9630651 = r9630642 - r9630636;
        double r9630652 = r9630650 * r9630651;
        double r9630653 = r9630647 ? r9630652 : r9630645;
        double r9630654 = r9630635 ? r9630645 : r9630653;
        return r9630654;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Split input into 2 regimes

  2. if x < -2169385582642.869 or 429.5871414147981 < x

    1. Initial program 30.5

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

    2. Taylor expanded around -inf 0.0

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

    3. Simplified0.0

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

    if -2169385582642.869 < x < 429.5871414147981

    1. Initial program 0.0

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

    3. Applied flip-+0.0

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

    4. Applied associate-/r/0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2169385582642.869:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \mathbf{elif}\;x \le 429.5871414147981:\\ \;\;\;\;\frac{x}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) - 1} \cdot \left(x \cdot x - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{{x}^{5}} + \frac{1}{x}\right) - \frac{1}{\left(x \cdot x\right) \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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