Average Error: 15.1 → 0.0
Time: 7.1s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 509.71560468432989\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 509.71560468432989\right):\\
\;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\

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

\end{array}
double f(double x) {
        double r60283 = x;
        double r60284 = r60283 * r60283;
        double r60285 = 1.0;
        double r60286 = r60284 + r60285;
        double r60287 = r60283 / r60286;
        return r60287;
}
↓
double f(double x) {
        double r60288 = x;
        double r60289 = -62379821099783.21;
        bool r60290 = r60288 <= r60289;
        double r60291 = 509.7156046843299;
        bool r60292 = r60288 <= r60291;
        double r60293 = !r60292;
        bool r60294 = r60290 || r60293;
        double r60295 = 1.0;
        double r60296 = r60295 / r60288;
        double r60297 = 1.0;
        double r60298 = 5.0;
        double r60299 = pow(r60288, r60298);
        double r60300 = r60297 / r60299;
        double r60301 = 3.0;
        double r60302 = pow(r60288, r60301);
        double r60303 = r60297 / r60302;
        double r60304 = r60300 - r60303;
        double r60305 = r60296 + r60304;
        double r60306 = r60288 * r60288;
        double r60307 = r60306 + r60297;
        double r60308 = r60295 / r60307;
        double r60309 = r60288 * r60308;
        double r60310 = r60294 ? r60305 : r60309;
        return r60310;
}

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 < -62379821099783.21 or 509.7156046843299 < x

    1. Initial program 30.9

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

    3. Applied div-inv31.0

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

    4. Taylor expanded around inf 0.0

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

    5. Simplified0.0

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

    if -62379821099783.21 < x < 509.7156046843299

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 509.71560468432989\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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