Average Error: 14.7 → 0.0
Time: 8.6s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -5442929637181724477620224000 \lor \neg \left(x \le 474.5306634629978930206561926752328872681\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{{x}^{3} \cdot x - 1 \cdot 1}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -5442929637181724477620224000 \lor \neg \left(x \le 474.5306634629978930206561926752328872681\right):\\
\;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{{x}^{3} \cdot x - 1 \cdot 1}\\

\end{array}
double f(double x) {
        double r70674 = x;
        double r70675 = r70674 * r70674;
        double r70676 = 1.0;
        double r70677 = r70675 + r70676;
        double r70678 = r70674 / r70677;
        return r70678;
}
↓
double f(double x) {
        double r70679 = x;
        double r70680 = -5.442929637181724e+27;
        bool r70681 = r70679 <= r70680;
        double r70682 = 474.5306634629979;
        bool r70683 = r70679 <= r70682;
        double r70684 = !r70683;
        bool r70685 = r70681 || r70684;
        double r70686 = 1.0;
        double r70687 = 5.0;
        double r70688 = pow(r70679, r70687);
        double r70689 = r70686 / r70688;
        double r70690 = 1.0;
        double r70691 = r70690 / r70679;
        double r70692 = 3.0;
        double r70693 = pow(r70679, r70692);
        double r70694 = r70686 / r70693;
        double r70695 = r70691 - r70694;
        double r70696 = r70689 + r70695;
        double r70697 = r70679 * r70679;
        double r70698 = r70697 - r70686;
        double r70699 = r70693 * r70679;
        double r70700 = r70686 * r70686;
        double r70701 = r70699 - r70700;
        double r70702 = r70679 / r70701;
        double r70703 = r70698 * r70702;
        double r70704 = r70685 ? r70696 : r70703;
        return r70704;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.7
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -5.442929637181724e+27 or 474.5306634629979 < x

    1. Initial program 30.9

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

    3. Applied flip-+49.3

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

    4. Applied associate-/r/49.3

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

    5. Simplified49.3

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

    6. Taylor expanded around inf 0.0

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

    7. Simplified0.0

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

    if -5.442929637181724e+27 < x < 474.5306634629979

    1. Initial program 0.0

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

    3. Applied flip-+0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{x}{{x}^{3} \cdot x - 1 \cdot 1}} \cdot \left(x \cdot x - 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5442929637181724477620224000 \lor \neg \left(x \le 474.5306634629978930206561926752328872681\right):\\ \;\;\;\;\frac{1}{{x}^{5}} + \left(\frac{1}{x} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x - 1\right) \cdot \frac{x}{{x}^{3} \cdot x - 1 \cdot 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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