x / (x^2 + 1)

Average Error: 14.5 → 0.0

Time: 5.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -988765961200.682373046875:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 596.0424036054944281204370781779289245605:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]

C

double f(double x) {
        double r3598533 = x;
        double r3598534 = r3598533 * r3598533;
        double r3598535 = 1.0;
        double r3598536 = r3598534 + r3598535;
        double r3598537 = r3598533 / r3598536;
        return r3598537;
}

↓

double f(double x) {
        double r3598538 = x;
        double r3598539 = -988765961200.6824;
        bool r3598540 = r3598538 <= r3598539;
        double r3598541 = 1.0;
        double r3598542 = r3598541 / r3598538;
        double r3598543 = 1.0;
        double r3598544 = 5.0;
        double r3598545 = pow(r3598538, r3598544);
        double r3598546 = r3598543 / r3598545;
        double r3598547 = r3598538 * r3598538;
        double r3598548 = r3598547 * r3598538;
        double r3598549 = r3598543 / r3598548;
        double r3598550 = r3598546 - r3598549;
        double r3598551 = r3598542 + r3598550;
        double r3598552 = 596.0424036054944;
        bool r3598553 = r3598538 <= r3598552;
        double r3598554 = r3598543 + r3598547;
        double r3598555 = r3598538 / r3598554;
        double r3598556 = r3598553 ? r3598555 : r3598551;
        double r3598557 = r3598540 ? r3598551 : r3598556;
        return r3598557;
}

Error

Bits error versus x

Target

Original	14.5
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -988765961200.6824 or 596.0424036054944 < x

   1. Initial program 30.4

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

   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

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

   if -988765961200.6824 < x < 596.0424036054944

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -988765961200.682373046875:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 596.0424036054944281204370781779289245605:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019192 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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