x / (x^2 + 1)

Average Error: 15.2 → 0.0

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -491659454900747.938 \lor \neg \left(x \le 1492.2784419761988\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

C

double f(double x) {
        double r55095 = x;
        double r55096 = r55095 * r55095;
        double r55097 = 1.0;
        double r55098 = r55096 + r55097;
        double r55099 = r55095 / r55098;
        return r55099;
}

↓

double f(double x) {
        double r55100 = x;
        double r55101 = -491659454900747.94;
        bool r55102 = r55100 <= r55101;
        double r55103 = 1492.2784419761988;
        bool r55104 = r55100 <= r55103;
        double r55105 = !r55104;
        bool r55106 = r55102 || r55105;
        double r55107 = 1.0;
        double r55108 = 1.0;
        double r55109 = 5.0;
        double r55110 = pow(r55100, r55109);
        double r55111 = r55108 / r55110;
        double r55112 = 3.0;
        double r55113 = pow(r55100, r55112);
        double r55114 = r55108 / r55113;
        double r55115 = r55111 - r55114;
        double r55116 = r55108 / r55100;
        double r55117 = fma(r55107, r55115, r55116);
        double r55118 = r55100 * r55100;
        double r55119 = r55118 + r55107;
        double r55120 = r55100 / r55119;
        double r55121 = r55106 ? r55117 : r55120;
        return r55121;
}

Error

Bits error versus x

Target

Original	15.2
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -491659454900747.94 or 1492.2784419761988 < x

   1. Initial program 30.7

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

   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

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

   if -491659454900747.94 < x < 1492.2784419761988

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -491659454900747.938 \lor \neg \left(x \le 1492.2784419761988\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020018 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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