Average Error: 15.1 → 0.0
Time: 7.8s
Precision: 64
\[\frac{x}{x \cdot x + 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2678435.1094305497:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 11907527.731338572:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]
\frac{x}{x \cdot x + 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2678435.1094305497:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\mathbf{elif}\;x \le 11907527.731338572:\\
\;\;\;\;\frac{x}{1 + x \cdot x}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\

\end{array}
double f(double x) {
        double r1088071 = x;
        double r1088072 = r1088071 * r1088071;
        double r1088073 = 1.0;
        double r1088074 = r1088072 + r1088073;
        double r1088075 = r1088071 / r1088074;
        return r1088075;
}
↓
double f(double x) {
        double r1088076 = x;
        double r1088077 = -2678435.1094305497;
        bool r1088078 = r1088076 <= r1088077;
        double r1088079 = 1.0;
        double r1088080 = r1088079 / r1088076;
        double r1088081 = r1088076 * r1088076;
        double r1088082 = r1088080 / r1088081;
        double r1088083 = r1088080 - r1088082;
        double r1088084 = 5.0;
        double r1088085 = pow(r1088076, r1088084);
        double r1088086 = r1088079 / r1088085;
        double r1088087 = r1088083 + r1088086;
        double r1088088 = 11907527.731338572;
        bool r1088089 = r1088076 <= r1088088;
        double r1088090 = r1088079 + r1088081;
        double r1088091 = r1088076 / r1088090;
        double r1088092 = r1088089 ? r1088091 : r1088087;
        double r1088093 = r1088078 ? r1088087 : r1088092;
        return r1088093;
}

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 < -2678435.1094305497 or 11907527.731338572 < x

    1. Initial program 30.4

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

    2. Taylor expanded around inf 0.0

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

    3. Simplified0.0

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

    if -2678435.1094305497 < x < 11907527.731338572

    1. Initial program 0.0

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2678435.1094305497:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 11907527.731338572:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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