x / (x^2 + 1)

Average Error: 14.6 → 0.0

Time: 6.2s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.333300421862763794492150706903049815887 \cdot 10^{154} \lor \neg \left(x \le 982.8796450984181092280778102576732635498\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\ \end{array}\]

C

double f(double x) {
        double r72884 = x;
        double r72885 = r72884 * r72884;
        double r72886 = 1.0;
        double r72887 = r72885 + r72886;
        double r72888 = r72884 / r72887;
        return r72888;
}

↓

double f(double x) {
        double r72889 = x;
        double r72890 = -1.3333004218627638e+154;
        bool r72891 = r72889 <= r72890;
        double r72892 = 982.8796450984181;
        bool r72893 = r72889 <= r72892;
        double r72894 = !r72893;
        bool r72895 = r72891 || r72894;
        double r72896 = 1.0;
        double r72897 = r72896 / r72889;
        double r72898 = 1.0;
        double r72899 = 5.0;
        double r72900 = pow(r72889, r72899);
        double r72901 = r72898 / r72900;
        double r72902 = 3.0;
        double r72903 = pow(r72889, r72902);
        double r72904 = r72898 / r72903;
        double r72905 = r72901 - r72904;
        double r72906 = r72897 + r72905;
        double r72907 = r72889 * r72889;
        double r72908 = r72907 + r72898;
        double r72909 = sqrt(r72908);
        double r72910 = r72896 / r72909;
        double r72911 = r72889 / r72909;
        double r72912 = r72910 * r72911;
        double r72913 = r72895 ? r72906 : r72912;
        return r72913;
}

Error

Bits error versus x

Target

Original	14.6
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -1.3333004218627638e+154 or 982.8796450984181 < x

   1. Initial program 39.3

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

   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. Simplified 0.0

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

   if -1.3333004218627638e+154 < x < 982.8796450984181

   1. Initial program 0.1

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

   3. Applied add-sqr-sqrt 0.1

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

   4. Applied *-un-lft-identity 0.1

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

   5. Applied times-frac 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.333300421862763794492150706903049815887 \cdot 10^{154} \lor \neg \left(x \le 982.8796450984181092280778102576732635498\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{x \cdot x + 1}} \cdot \frac{x}{\sqrt{x \cdot x + 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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