x / (x^2 + 1)

Average Error: 14.6 → 0.0

Time: 20.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.195002563386160441668121103642730572028 \cdot 10^{62} \lor \neg \left(x \le 507.0100903930748472703271545469760894775\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

C

double f(double x) {
        double r63367 = x;
        double r63368 = r63367 * r63367;
        double r63369 = 1.0;
        double r63370 = r63368 + r63369;
        double r63371 = r63367 / r63370;
        return r63371;
}

↓

double f(double x) {
        double r63372 = x;
        double r63373 = -3.1950025633861604e+62;
        bool r63374 = r63372 <= r63373;
        double r63375 = 507.01009039307485;
        bool r63376 = r63372 <= r63375;
        double r63377 = !r63376;
        bool r63378 = r63374 || r63377;
        double r63379 = 1.0;
        double r63380 = r63379 / r63372;
        double r63381 = 1.0;
        double r63382 = 3.0;
        double r63383 = pow(r63372, r63382);
        double r63384 = r63381 / r63383;
        double r63385 = 5.0;
        double r63386 = pow(r63372, r63385);
        double r63387 = r63381 / r63386;
        double r63388 = r63384 - r63387;
        double r63389 = r63380 - r63388;
        double r63390 = r63372 * r63372;
        double r63391 = r63390 + r63381;
        double r63392 = r63372 / r63391;
        double r63393 = r63378 ? r63389 : r63392;
        return r63393;
}

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 < -3.1950025633861604e+62 or 507.01009039307485 < x

   1. Initial program 33.3

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

   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

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

   if -3.1950025633861604e+62 < x < 507.01009039307485

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.195002563386160441668121103642730572028 \cdot 10^{62} \lor \neg \left(x \le 507.0100903930748472703271545469760894775\right):\\ \;\;\;\;\frac{1}{x} - \left(\frac{1}{{x}^{3}} - \frac{1}{{x}^{5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x \cdot x + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (x)
  :name "x / (x^2 + 1)"

  :herbie-target
  (/ 1.0 (+ x (/ 1.0 x)))

  (/ x (+ (* x x) 1.0)))