Average Error: 14.6 → 0.0
Time: 12.8s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.683354923764348 \cdot 10^{+38}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 486.478699858977:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.683354923764348 \cdot 10^{+38}:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 486.478699858977:\\
\;\;\;\;\frac{x}{x \cdot x + 1}\\

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

\end{array}
double f(double x) {
        double r2681537 = x;
        double r2681538 = r2681537 * r2681537;
        double r2681539 = 1.0;
        double r2681540 = r2681538 + r2681539;
        double r2681541 = r2681537 / r2681540;
        return r2681541;
}
↓
double f(double x) {
        double r2681542 = x;
        double r2681543 = -7.683354923764348e+38;
        bool r2681544 = r2681542 <= r2681543;
        double r2681545 = 1.0;
        double r2681546 = r2681545 / r2681542;
        double r2681547 = r2681546 / r2681542;
        double r2681548 = r2681547 / r2681542;
        double r2681549 = r2681546 - r2681548;
        double r2681550 = 5.0;
        double r2681551 = pow(r2681542, r2681550);
        double r2681552 = r2681545 / r2681551;
        double r2681553 = r2681549 + r2681552;
        double r2681554 = 486.478699858977;
        bool r2681555 = r2681542 <= r2681554;
        double r2681556 = r2681542 * r2681542;
        double r2681557 = r2681556 + r2681545;
        double r2681558 = r2681542 / r2681557;
        double r2681559 = r2681555 ? r2681558 : r2681553;
        double r2681560 = r2681544 ? r2681553 : r2681559;
        return r2681560;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.6
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -7.683354923764348e+38 or 486.478699858977 < x

    1. Initial program 31.7

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

    3. Applied add-sqr-sqrt31.7

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

    4. Applied *-un-lft-identity31.7

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

    5. Applied times-frac31.6

      \[\leadsto \color{blue}{\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}}\]
    6. Using strategy rm

    7. Applied pow1/231.6

      \[\leadsto \frac{1}{\color{blue}{{\left(x \cdot x + 1\right)}^{\frac{1}{2}}}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\]

    8. Applied pow-flip31.7

      \[\leadsto \color{blue}{{\left(x \cdot x + 1\right)}^{\left(-\frac{1}{2}\right)}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\]

    9. Simplified31.7

      \[\leadsto {\left(x \cdot x + 1\right)}^{\color{blue}{\frac{-1}{2}}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\]

    10. Taylor expanded around inf 0.0

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

    11. Simplified0.0

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

    if -7.683354923764348e+38 < x < 486.478699858977

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.683354923764348 \cdot 10^{+38}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 486.478699858977:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{\frac{1}{x}}{x}}{x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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