Average Error: 14.6 → 0.0
Time: 10.4s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -503.944307784687680396018549799919128418 \lor \neg \left(x \le 446.6796391904089205127093009650707244873\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\ \end{array}\]
\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{\frac{x}{\sqrt{x \cdot x + 1}}}{\sqrt{x \cdot x + 1}}\\

\end{array}
double f(double x) {
        double r54249 = x;
        double r54250 = r54249 * r54249;
        double r54251 = 1.0;
        double r54252 = r54250 + r54251;
        double r54253 = r54249 / r54252;
        return r54253;
}
↓
double f(double x) {
        double r54254 = x;
        double r54255 = -503.9443077846877;
        bool r54256 = r54254 <= r54255;
        double r54257 = 446.6796391904089;
        bool r54258 = r54254 <= r54257;
        double r54259 = !r54258;
        bool r54260 = r54256 || r54259;
        double r54261 = 1.0;
        double r54262 = r54261 / r54254;
        double r54263 = 1.0;
        double r54264 = 5.0;
        double r54265 = pow(r54254, r54264);
        double r54266 = r54263 / r54265;
        double r54267 = 3.0;
        double r54268 = pow(r54254, r54267);
        double r54269 = r54263 / r54268;
        double r54270 = r54266 - r54269;
        double r54271 = r54262 + r54270;
        double r54272 = r54254 * r54254;
        double r54273 = r54272 + r54263;
        double r54274 = sqrt(r54273);
        double r54275 = r54254 / r54274;
        double r54276 = r54275 / r54274;
        double r54277 = r54260 ? r54271 : r54276;
        return r54277;
}

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 associate-/r*0.0

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

    5. Simplified0.0

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

  4. Final simplification0.0

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

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)))