Average Error: 14.9 → 0.0
Time: 6.8s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -31939.41578853163:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 450.35164288801093:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1 + x \cdot x}}}{\sqrt{1 + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -31939.41578853163:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 450.35164288801093:\\
\;\;\;\;\frac{\frac{x}{\sqrt{1 + x \cdot x}}}{\sqrt{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 r935524 = x;
        double r935525 = r935524 * r935524;
        double r935526 = 1.0;
        double r935527 = r935525 + r935526;
        double r935528 = r935524 / r935527;
        return r935528;
}
↓
double f(double x) {
        double r935529 = x;
        double r935530 = -31939.41578853163;
        bool r935531 = r935529 <= r935530;
        double r935532 = 1.0;
        double r935533 = r935532 / r935529;
        double r935534 = r935529 * r935529;
        double r935535 = r935533 / r935534;
        double r935536 = r935533 - r935535;
        double r935537 = 5.0;
        double r935538 = pow(r935529, r935537);
        double r935539 = r935532 / r935538;
        double r935540 = r935536 + r935539;
        double r935541 = 450.35164288801093;
        bool r935542 = r935529 <= r935541;
        double r935543 = r935532 + r935534;
        double r935544 = sqrt(r935543);
        double r935545 = r935529 / r935544;
        double r935546 = r935545 / r935544;
        double r935547 = r935542 ? r935546 : r935540;
        double r935548 = r935531 ? r935540 : r935547;
        return r935548;
}

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 < -31939.41578853163 or 450.35164288801093 < x

    1. Initial program 30.4

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

    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

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

    if -31939.41578853163 < x < 450.35164288801093

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -31939.41578853163:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 450.35164288801093:\\ \;\;\;\;\frac{\frac{x}{\sqrt{1 + x \cdot x}}}{\sqrt{1 + x \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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