Average Error: 14.6 → 0.0
Time: 11.2s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -503.944307784687680396018549799919128418 \lor \neg \left(x \le 446.6796391904089205127093009650707244873\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -503.944307784687680396018549799919128418 \lor \neg \left(x \le 446.6796391904089205127093009650707244873\right):\\
\;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\

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

\end{array}
double f(double x) {
        double r60396 = x;
        double r60397 = r60396 * r60396;
        double r60398 = 1.0;
        double r60399 = r60397 + r60398;
        double r60400 = r60396 / r60399;
        return r60400;
}
↓
double f(double x) {
        double r60401 = x;
        double r60402 = -503.9443077846877;
        bool r60403 = r60401 <= r60402;
        double r60404 = 446.6796391904089;
        bool r60405 = r60401 <= r60404;
        double r60406 = !r60405;
        bool r60407 = r60403 || r60406;
        double r60408 = 1.0;
        double r60409 = r60408 / r60401;
        double r60410 = 1.0;
        double r60411 = 5.0;
        double r60412 = pow(r60401, r60411);
        double r60413 = r60410 / r60412;
        double r60414 = 3.0;
        double r60415 = pow(r60401, r60414);
        double r60416 = r60410 / r60415;
        double r60417 = r60413 - r60416;
        double r60418 = r60409 + r60417;
        double r60419 = r60401 * r60401;
        double r60420 = r60419 + r60410;
        double r60421 = sqrt(r60420);
        double r60422 = r60408 / r60421;
        double r60423 = r60401 / r60421;
        double r60424 = r60422 * r60423;
        double r60425 = r60407 ? r60418 : r60424;
        return r60425;
}

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 < -503.9443077846877 or 446.6796391904089 < x

    1. Initial program 29.5

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -503.9443077846877 < x < 446.6796391904089

    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 *-un-lft-identity0.0

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

    5. Applied times-frac0.0

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

    6. Simplified0.0

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

    7. Simplified0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -503.944307784687680396018549799919128418 \lor \neg \left(x \le 446.6796391904089205127093009650707244873\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))