x / (x^2 + 1)

Average Error: 15.0 → 0.0

Time: 2.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -11396411610485.0371 \lor \neg \left(x \le 408.77011000092864\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

C

double f(double x) {
        double r66612 = x;
        double r66613 = r66612 * r66612;
        double r66614 = 1.0;
        double r66615 = r66613 + r66614;
        double r66616 = r66612 / r66615;
        return r66616;
}

↓

double f(double x) {
        double r66617 = x;
        double r66618 = -11396411610485.037;
        bool r66619 = r66617 <= r66618;
        double r66620 = 408.77011000092864;
        bool r66621 = r66617 <= r66620;
        double r66622 = !r66621;
        bool r66623 = r66619 || r66622;
        double r66624 = 1.0;
        double r66625 = 1.0;
        double r66626 = 5.0;
        double r66627 = pow(r66617, r66626);
        double r66628 = r66625 / r66627;
        double r66629 = 3.0;
        double r66630 = pow(r66617, r66629);
        double r66631 = r66625 / r66630;
        double r66632 = r66628 - r66631;
        double r66633 = r66625 / r66617;
        double r66634 = fma(r66624, r66632, r66633);
        double r66635 = r66617 * r66617;
        double r66636 = r66635 + r66624;
        double r66637 = r66617 / r66636;
        double r66638 = r66623 ? r66634 : r66637;
        return r66638;
}

Error

Bits error versus x

Target

Original	15.0
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -11396411610485.037 or 408.77011000092864 < x

   1. Initial program 30.2

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

   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 -11396411610485.037 < x < 408.77011000092864

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -11396411610485.0371 \lor \neg \left(x \le 408.77011000092864\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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