x / (x^2 + 1)

Average Error: 14.3 → 0.0

Time: 8.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -267060974.6176845133304595947265625:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 508.8749887332332946243695914745330810547:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]

C

double f(double x) {
        double r2856590 = x;
        double r2856591 = r2856590 * r2856590;
        double r2856592 = 1.0;
        double r2856593 = r2856591 + r2856592;
        double r2856594 = r2856590 / r2856593;
        return r2856594;
}

↓

double f(double x) {
        double r2856595 = x;
        double r2856596 = -267060974.6176845;
        bool r2856597 = r2856595 <= r2856596;
        double r2856598 = 1.0;
        double r2856599 = r2856598 / r2856595;
        double r2856600 = 1.0;
        double r2856601 = 5.0;
        double r2856602 = pow(r2856595, r2856601);
        double r2856603 = r2856600 / r2856602;
        double r2856604 = r2856595 * r2856595;
        double r2856605 = r2856604 * r2856595;
        double r2856606 = r2856600 / r2856605;
        double r2856607 = r2856603 - r2856606;
        double r2856608 = r2856599 + r2856607;
        double r2856609 = 508.8749887332333;
        bool r2856610 = r2856595 <= r2856609;
        double r2856611 = r2856600 + r2856604;
        double r2856612 = r2856595 / r2856611;
        double r2856613 = r2856610 ? r2856612 : r2856608;
        double r2856614 = r2856597 ? r2856608 : r2856613;
        return r2856614;
}

Error

Bits error versus x

Target

Original	14.3
Target	0.1
Herbie	0.0

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

Derivation

1. Split input into 2 regimes

2. if x < -267060974.6176845 or 508.8749887332333 < x

   1. Initial program 29.6

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

   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 -267060974.6176845 < x < 508.8749887332333

   1. Initial program 0.0

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

4. Final simplification 0.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -267060974.6176845133304595947265625:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \mathbf{elif}\;x \le 508.8749887332332946243695914745330810547:\\ \;\;\;\;\frac{x}{1 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} + \left(\frac{1}{{x}^{5}} - \frac{1}{\left(x \cdot x\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "x / (x^2 + 1)"

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

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