Average Error: 14.9 → 0.0
Time: 13.1s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4377789621347.087890625 \lor \neg \left(x \le 434.3260566585274204953748267143964767456\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{4} - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4377789621347.087890625 \lor \neg \left(x \le 434.3260566585274204953748267143964767456\right):\\
\;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x}{{x}^{4} - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)\\

\end{array}
double f(double x) {
        double r45970 = x;
        double r45971 = r45970 * r45970;
        double r45972 = 1.0;
        double r45973 = r45971 + r45972;
        double r45974 = r45970 / r45973;
        return r45974;
}
↓
double f(double x) {
        double r45975 = x;
        double r45976 = -4377789621347.088;
        bool r45977 = r45975 <= r45976;
        double r45978 = 434.3260566585274;
        bool r45979 = r45975 <= r45978;
        double r45980 = !r45979;
        bool r45981 = r45977 || r45980;
        double r45982 = 1.0;
        double r45983 = r45982 / r45975;
        double r45984 = 1.0;
        double r45985 = 5.0;
        double r45986 = pow(r45975, r45985);
        double r45987 = r45984 / r45986;
        double r45988 = r45983 + r45987;
        double r45989 = 3.0;
        double r45990 = pow(r45975, r45989);
        double r45991 = r45984 / r45990;
        double r45992 = r45988 - r45991;
        double r45993 = 4.0;
        double r45994 = pow(r45975, r45993);
        double r45995 = r45984 * r45984;
        double r45996 = r45994 - r45995;
        double r45997 = r45975 / r45996;
        double r45998 = r45975 * r45975;
        double r45999 = r45998 - r45984;
        double r46000 = r45997 * r45999;
        double r46001 = r45981 ? r45992 : r46000;
        return r46001;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original14.9
Target0.1
Herbie0.0
\[\frac{1}{x + \frac{1}{x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -4377789621347.088 or 434.3260566585274 < x

    1. Initial program 30.7

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

    3. Applied flip-+48.9

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

    4. Applied associate-/r/48.9

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

    5. Simplified48.9

      \[\leadsto \color{blue}{\frac{x}{{x}^{4} - 1 \cdot 1}} \cdot \left(x \cdot x - 1\right)\]

    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

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

    if -4377789621347.088 < x < 434.3260566585274

    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

      \[\leadsto \color{blue}{\frac{x}{{x}^{4} - 1 \cdot 1}} \cdot \left(x \cdot x - 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4377789621347.087890625 \lor \neg \left(x \le 434.3260566585274204953748267143964767456\right):\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{{x}^{5}}\right) - \frac{1}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{{x}^{4} - 1 \cdot 1} \cdot \left(x \cdot x - 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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