x / (x^2 + 1)

Average Error: 15.2 → 0.0

Time: 2.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -76798468.31292221 \lor \neg \left(x \le 8077604.2444988824\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

C

double f(double x) {
        double r68620 = x;
        double r68621 = r68620 * r68620;
        double r68622 = 1.0;
        double r68623 = r68621 + r68622;
        double r68624 = r68620 / r68623;
        return r68624;
}

↓

double f(double x) {
        double r68625 = x;
        double r68626 = -76798468.31292221;
        bool r68627 = r68625 <= r68626;
        double r68628 = 8077604.244498882;
        bool r68629 = r68625 <= r68628;
        double r68630 = !r68629;
        bool r68631 = r68627 || r68630;
        double r68632 = 1.0;
        double r68633 = 1.0;
        double r68634 = 5.0;
        double r68635 = pow(r68625, r68634);
        double r68636 = r68633 / r68635;
        double r68637 = 3.0;
        double r68638 = pow(r68625, r68637);
        double r68639 = r68633 / r68638;
        double r68640 = r68636 - r68639;
        double r68641 = r68633 / r68625;
        double r68642 = fma(r68632, r68640, r68641);
        double r68643 = r68625 * r68625;
        double r68644 = r68643 + r68632;
        double r68645 = r68625 / r68644;
        double r68646 = r68631 ? r68642 : r68645;
        return r68646;
}

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 < -76798468.31292221 or 8077604.244498882 < x

   1. Initial program 31.3

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

   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

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

   if -76798468.31292221 < x < 8077604.244498882

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -76798468.31292221 \lor \neg \left(x \le 8077604.2444988824\right):\\ \;\;\;\;\mathsf{fma}\left(1, \frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}, \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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