x / (x^2 + 1)

Average Error: 15.1 → 0.0

Time: 7.3s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 509.71560468432989\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]

C

double f(double x) {
        double r69431 = x;
        double r69432 = r69431 * r69431;
        double r69433 = 1.0;
        double r69434 = r69432 + r69433;
        double r69435 = r69431 / r69434;
        return r69435;
}

↓

double f(double x) {
        double r69436 = x;
        double r69437 = -62379821099783.21;
        bool r69438 = r69436 <= r69437;
        double r69439 = 509.7156046843299;
        bool r69440 = r69436 <= r69439;
        double r69441 = !r69440;
        bool r69442 = r69438 || r69441;
        double r69443 = 1.0;
        double r69444 = r69443 / r69436;
        double r69445 = 1.0;
        double r69446 = 5.0;
        double r69447 = pow(r69436, r69446);
        double r69448 = r69445 / r69447;
        double r69449 = 3.0;
        double r69450 = pow(r69436, r69449);
        double r69451 = r69445 / r69450;
        double r69452 = r69448 - r69451;
        double r69453 = r69444 + r69452;
        double r69454 = r69436 * r69436;
        double r69455 = r69454 + r69445;
        double r69456 = r69443 / r69455;
        double r69457 = r69436 * r69456;
        double r69458 = r69442 ? r69453 : r69457;
        return r69458;
}

Error

Bits error versus x

Target

Original	15.1
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -62379821099783.21 or 509.7156046843299 < x

   1. Initial program 30.9

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

   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 -62379821099783.21 < x < 509.7156046843299

   1. Initial program 0.0

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

   3. Applied div-inv 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -62379821099783.2109 \lor \neg \left(x \le 509.71560468432989\right):\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(FPCore (x)
  :name "x / (x^2 + 1)"
  :precision binary64

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

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