x / (x^2 + 1)

Average Error: 15.2 → 0.0

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.8453524061497666 \cdot 10^{+21}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 13951219.631737411:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

C

double f(double x) {
        double r1083643 = x;
        double r1083644 = r1083643 * r1083643;
        double r1083645 = 1.0;
        double r1083646 = r1083644 + r1083645;
        double r1083647 = r1083643 / r1083646;
        return r1083647;
}

↓

double f(double x) {
        double r1083648 = x;
        double r1083649 = -1.8453524061497666e+21;
        bool r1083650 = r1083648 <= r1083649;
        double r1083651 = 1.0;
        double r1083652 = r1083651 / r1083648;
        double r1083653 = r1083648 * r1083648;
        double r1083654 = r1083652 / r1083653;
        double r1083655 = r1083652 - r1083654;
        double r1083656 = 5.0;
        double r1083657 = pow(r1083648, r1083656);
        double r1083658 = r1083651 / r1083657;
        double r1083659 = r1083655 + r1083658;
        double r1083660 = 13951219.631737411;
        bool r1083661 = r1083648 <= r1083660;
        double r1083662 = r1083651 + r1083653;
        double r1083663 = r1083648 / r1083662;
        double r1083664 = r1083661 ? r1083663 : r1083659;
        double r1083665 = r1083650 ? r1083659 : r1083664;
        return r1083665;
}

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 < -1.8453524061497666e+21 or 13951219.631737411 < x

   1. Initial program 31.5

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

   2. Taylor expanded around -inf 0.0

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

   3. Simplified 0.0

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

   if -1.8453524061497666e+21 < x < 13951219.631737411

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.8453524061497666 \cdot 10^{+21}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \mathbf{elif}\;x \le 13951219.631737411:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} - \frac{\frac{1}{x}}{x \cdot x}\right) + \frac{1}{{x}^{5}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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